Stony Brook University Seminario de Geometr a ICMAT ...De nition.Complete, non-compact n-manifold...

Mass, Scalar Curvature, &

Kahler Geometry, II

Claude LeBrunStony Brook University

Seminario de GeometrıaICMAT, November 5, 2018

1

Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R

n −Dn)/Γi, where Γi ⊂ O(n), such that

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4

Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that

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................................................................................................................................................................................................................................... .......

....

gjk = δjk + O(|x|1−n2−ε)

gjk,` = O(|x|−n2−ε), s ∈ L1

9

Why consider ALE spaces?

10

Key examples:

11

Key examples:

Term ALE coined by Gibbons & Hawking, 1979.

12

Key examples:


They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.

13

vv

vvv

Data: ` points in R3. =⇒ V with ∆V = 0

14

vv

vvv


V =∑j=1

1

2%j

15

vv

vvv

.......

.......

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.......

.......

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.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................


V =∑j=1

1

2%j

F = ?dV curvature θ on P → R3 − {pts}.

16

vv

vvv

.......

.......

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.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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.......

.......

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.......

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g = V h + V −1θ2


17

vv

vvv

.......

.......

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.......

.......

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........................

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.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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.......

.......

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g = V (dx2 + dy2 + dz2) + V −1θ2


18

vv

vvv

.......

.......

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.......

.......

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.......

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........................

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.......

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.......

.......

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g = V h + V −1θ2


19

vv

vvv

.......

.......

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.......

.......

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.......

.......

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.......

.......

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........................


g = V h + V −1θ2

on P . Then take M4 = Riemannian completion.

20

vv

vvv

.....................................................................................................................................................................................

......................................................................................................................................................................................................................

..........................................................................................................................................................................................................

.......................

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........................

........................

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..........


g = V h + V −1θ2


21

vv

vvv

.....................................................................................................................................................................................

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g = V h + V −1θ2


22

vv

vvv

.....................................................................................................................................................................................

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...................

............. ............. ............. ............. ............. ............. ............. ............. .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............


g = V h + V −1θ2


23

vv

vvv

.....................................................................................................................................................................................

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g = V h + V −1θ2


24

Deform retracts to k = `− 1 copies of S2,

25

Deform retracts to k = `− 1 copies of S2,each with self-intersection −2,

26

Deform retracts to k = `− 1 copies of S2,each with self-intersection −2,meeting transversely, & forming connected set:

27


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

28


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Configuration dual to Dynkin diagram Ak:

29


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


• • • •.............................................................................................................................................................................................................................................................................................................

30


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


• • • •.............................................................................................................................................................................................................................................................................................................

Diffeotype:

31


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


• • • •.............................................................................................................................................................................................................................................................................................................

Diffeotype:

Plumb together k copies of T ∗S2

according to diagram.

32

Key examples:



33

Key examples:



Their examples have just one end, with

Γ ∼= Z` ⊂ SU(2) ⊂ O(4).

34

Key examples:




Γ ∼= Z` ⊂ SU(2) ⊂ O(4).

The G-H metrics are hyper-Kahler, and were soonindependently rediscovered by Hitchin.

35

Kahler metrics:

(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)

36

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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37

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

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...........................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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38

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

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39

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

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40

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

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41

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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42

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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43

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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44

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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45

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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46

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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47

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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48

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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49

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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50

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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51

Kahler metrics:

(Mn, g): Kahler ⇐⇒ holonomy ⊂ O(n)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

52

Kahler metrics:

(M2m, g): Kahler ⇐⇒ holonomy ⊂ U(m)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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53

Kahler metrics:

(M2m, g) Kahler ⇐⇒ holonomy ⊂ U(m)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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54

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

U(m) := O(2m) ∩GL(m,C)

55

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Makes tangent space a complex vector space!

56

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................


J : TM → TM, J2 = −identity

“almost-complex structure”

57

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Invariant under parallel transport!

58

Kahler metrics:


⇐⇒ ∃ almost complex-structure J with ∇J = 0and g(J ·, J ·) = g.

59

Kahler metrics:



⇐⇒ (M2m, g) is a complex manifold & ∃ J -invariantclosed 2-form ω such that g = ω(·, J ·).

60

Kahler metrics:




dω = 0

61

Kahler metrics:




[ω] ∈ H2(M)

“Kahler class”

62

Kahler metrics:




63

Kahler metrics:




⇐⇒ In local complex coordinates (z1, . . . , zm), ∃f (z)

g = −m∑

j,k=1

∂2f

∂zj∂zk

[dzj ⊗ dzk + dzk ⊗ dzj

]

64

Kahler metrics:





g = −m∑

j,k=1

∂2f

∂zj∂zk


]

65

Kahler metrics:





ω = im∑

j,k=1

∂2f

∂zj∂zkdzj ∧ dzk

66

Kahler metrics:





g = −m∑

j,k=1

∂2f

∂zj∂zk


]

67

Kahler metrics:




Kahler magic:

r = −m∑

j,k=1

∂2

∂zj∂zklog det[gpq]

[dzj⊗dzk+dzk⊗dzj

]

68

Kahler metrics:




Kahler magic:

If we define the Ricci form by

ρ = r(J ·, ·)then iρ is curvature of canonical line bundle Λm,0.

69

Kahler metrics:

(M2m, g): Ricci-flat Kahler⇐= holonomy⊂ SU(m)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

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...........................................................

...................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

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70

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

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71

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

........................................

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.....................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

SU(m) ⊂ U(m) : {A | detA = 1}

72

Kahler metrics:


s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

........................................

........................................

.....................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

73

Kahler metrics:

(M2m, g): Ricci-flat Kahler⇐⇒ holonomy⊂ SU(m)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

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........................................................

........................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

if M is simply connected.

74

Hyper-Kahler metrics:

(M4`, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(`)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

........................................

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75



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

........................................

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76



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

........................................

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.....................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(`) := O(4`) ∩GL(`,H)

77



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................

...................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

........................................

........................................

.....................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(`) ⊂ SU(2`)

78



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(`) ⊂ SU(2`)

in many ways! (For example, permute i, j, k. . . )

79



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(`) ⊂ SU(2`)

in many ways! (For example, permute i, j, k. . . )

80



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................

...................................................

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........................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(`) ⊂ SU(2`)

Ricci-flat and Kahler,

for many different complex structures!

81



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(`) ⊂ SU(2`)

82


(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)

s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................

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........................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(1) = SU(2)

83



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................

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........................................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(1) = SU(2)

When (M4, g) simply connected:

hyper-Kahler ⇐⇒ Ricci-flat Kahler.

84



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(1) = SU(2)

85



s

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................

Sp(1) = SU(2)



86

All these complex structures can be repackaged as

Penrose Twistor Space (Z6, J),

which is a complex 3-manifold

.....................................................................................................................................................................................................................................................................................................................................................................................................................................

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r

r Z

M 4

↓

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87



which is a complex 3-manifold

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r

r Z

M 4

↓

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88



which is a complex 3-manifold.

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89




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r Z

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Complex structure faithfully encodes the metric.

90




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Complex structure encodes metric mod homothety.

91




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92




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Constructing twistor space suffices for existence.

93

Hitchin’s Twistor Spaces:

94


H0(CP1,O(2)) = C3⊃ R3.

95


H0(CP1,O(2)) = C3 ⊃ R3.

96


H0(CP1,O(2)) = C3 ⊃ R3.

vv

vvv

So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).

97


H0(CP1,O(2)) = C3 ⊃ R3.

vv

vvv


Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)

98


H0(CP1,O(2)) = C3 ⊃ R3.

vv

vvv



xy = (z − P1) · · · (z − P`)

99


H0(CP1,O(2)) = C3 ⊃ R3.

vv

vvv



xy = (z − P1) · · · (z − P`)

100


H0(CP1,O(2)) = C3 ⊃ R3.

vv

vvv



xy = (z − P1) · · · (z − P`)is the twistor space of a Gibbons-Hawking metric.

101

Key examples:




Γ ∼= Z` ⊂ SU(2) ⊂ O(4).


102

Key examples:




Γ ∼= Z` ⊂ SU(2) ⊂ O(4).


Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).

103

Key examples:




Γ ∼= Z` ⊂ SU(2) ⊂ O(4).



This conjecture was proved by Kronheimer, 1986.

104

Given Γ ⊂ SU(2) finite subgroup,

105


106


the orbifold C2/Γ can be viewed as

107



singular complex surface ⊂ mathbbC3 by

108



singular complex surface ⊂ C3 by

choosing 3 generators of Γ-invariant polynomials.

109





110





Example.

[e2πi/m

e−2πi/m

]∈ SU(2)

111





Example.

[e2πi/m

e−2πi/m

]∈ SU(2)

generates Γ ∼= Zm.

112





Example.

[e2πi/m

e−2πi/m

]∈ SU(2)

generates Γ ∼= Zm. Setting

u = zm1 , v = zm2 , y = z1z2,

113





Example.

[e2πi/m

e−2πi/m

]∈ SU(2)


u = zm1 , v = zm2 , y = z1z2,

then identifies C2/Γ with

uv = ym.

114





Example.

[e2πi/m

e−2πi/m

]∈ SU(2)


w =1

2(zm1 −z

m2 ), x =

i

2(zm1 +zm2 ), y = z1z2,

then identifies C2/Γ with

w2 + x2 + ym = 0.

115

Felix Klein’s Table of Singularities (1884)

116


Zm ←→ w2 + x2 + ym = 0

117


Zm ←→ w2 + x2 + ym = 0

Dih∗m ←→ w2 + y(x2 + ym) = 0

118


Zm ←→ w2 + x2 + ym = 0

Dih∗m ←→ w2 + y(x2 + ym) = 0

T ∗ ←→ w2 + x3 + y4 = 0

119


Zm ←→ w2 + x2 + ym = 0

Dih∗m ←→ w2 + y(x2 + ym) = 0

T ∗ ←→ w2 + x3 + y4 = 0

O∗ ←→ w2 + x3 + xy3 = 0

120


Zm ←→ w2 + x2 + ym = 0

Dih∗m ←→ w2 + y(x2 + ym) = 0

T ∗ ←→ w2 + x3 + y4 = 0

O∗ ←→ w2 + x3 + xy3 = 0

I∗ ←→ w2 + x3 + y5 = 0

121

Prototypical Klein singularity:

• Smooth it, by deformation:

w2 + x2 + y2 = 0

122

Two ways to get rid of a singularity:


w2 + x2 + y2 = 0

123



w2 + x2 + y2 = 0

124



w2 + x2 + y2 = ε

125



w2 + x2 + y2 = ε

132



w2 + x2 + y2 = ε

• Resolve it, by blowing up, iteratively:

w2 + x2 + y2 = 0

133



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2

134



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2

135



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2

136



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2

143



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2

Usually these are topologically different.

144



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2


But for Klein singularities, they are diffeomorphic!

145



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2



Gorenstein singularities. Crepant Resolutions.

146



w2 + x2 + y2 = ε


w2 + x2 + y2 = 0

O(−2) → O(−1)↓ ↓

CP1 ↪→ CP2



Gorenstein singularities. Crepant Resolutions.

147

Resolutions of Klein Singularities:

148


∀ Klein singularity V ⊂ C3, ∃! resolution

149



V → V

150



V → V

with c1(T 1,0V ) = 0.

151



V → V

with c1(T 1,0V ) = 0.

Replaces origin with a union of CP1’s,each with self-intersection −2,meeting transversely, & forming connected set:

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152



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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153



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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154



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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155



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Intersection pattern dual to Dynkin diagram!

• • • • •••

........................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................

156



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................................................................................................................................................................................................................................................


• • • • •••

........................................................................................................................................................................................................................................................................................................................................................................................................

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157



V → V

with c1(T 1,0V ) = 0.


•.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................................................................................................................................................................................................................................................


• • • • •••

........................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................

158



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................................................................................................................................................................................................................................................


• • • • •••

........................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................

159



V → V

with c1(T 1,0V ) = 0.


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................................................................................................................................................................................................................................................


• • • • •••

........................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................

160

Zk+1←→Ak • • • •.......................................................................................................................................................

161

Zk+1←→Ak • • • •.......................................................................................................................................................

Dih∗k−2←→Dk • • • ••.................................................................................................................

................................................

...........................................................

162

Zk+1←→Ak • • • •.......................................................................................................................................................

Dih∗k−2←→Dk • • • ••.................................................................................................................

................................................

...........................................................

T ∗←→E6 • • • •••................................................................................................................................................................................................................

.......

.......

.......

.......

.......

....

163

Zk+1←→Ak • • • •.......................................................................................................................................................

Dih∗k−2←→Dk • • • ••.................................................................................................................

................................................

...........................................................

T ∗←→E6 • • • •••................................................................................................................................................................................................................

.......

.......

.......

.......

.......

....

O∗←→E7 • • • ••• •..................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

....

164

Zk+1←→Ak • • • •.......................................................................................................................................................

Dih∗k−2←→Dk • • • ••.................................................................................................................

................................................

...........................................................

T ∗←→E6 • • • •••................................................................................................................................................................................................................

.......

.......

.......

.......

.......

....

O∗←→E7 • • • ••• •..................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

....

I∗←→E8 • • • ••• • •....................................................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

....

165

Key examples:




Γ ∼= Z` ⊂ SU(2) ⊂ O(4).



Proved by Kronheimer, who also showed (1989) thisgives complete classification of ALE hyper-Kahlers.

166



s

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Sp(1) = SU(2)



167




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But similar for scalar-flat Kahler surfaces (M4, g, J)!

169

Any scalar-flat Kahler surface (M4, g, J) has a


which is once again a complex 3-manifold.

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170




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Integrability condition for twistor space: W+ ≡ 0.

171




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For Kahler surfaces, |W+|2 = s2/24.

172




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For Kahler surfaces, integrable ⇐⇒ scalar-flat!

173




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Leads to constructions of explicit examples.

174




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Many simple examples are AE or ALE.

175

Some AE Scalar-Flat Kahler Surfaces:

176


(L ’91)

177


f

vv

vvv

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.............

178


f

vv

vvv

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.............

Data: k points inH3 and one point at infinity. =⇒V with ∆V = 0

179


vv

vvv

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Data: k points in H3 = upper half-space model.

180


vv

vvv

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Data: k points in H3. =⇒ V with ∆V = 0

V = 1 +

k∑j=1

Gj

181


vv

vvv

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V = 1 +

k∑j=1

1

e2%j − 1

182


vv

vvv

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V = 1 +

k∑j=1

Gj

183


vv

vvv

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F = ?dV curvature θ on P → H3 − {pts}.

184


vv

vvv

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g = z2(V h + V −1θ2

)

185


vv

vvv

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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


g = z2

(Vdx2 + dy2 + dz2

z2+ V −1θ2

)

186


vv

vvv

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


g = V (dx2 + dy2 + dz2) + z2V −1θ2

187


vv

vvv

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


g = z2(V h + V −1θ2

)

188


vv

vvv

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.......

.......

........................................................

..............................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................

.................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Riemannian completion is AE scalar-flat Kahler.

g = z2(V h + V −1θ2

)

189


vv

vvv

.................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

......

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............


g = z2(V h + V −1θ2

)

190


vv

vvv

C×× ×××

.................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

......

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............


g = z2(V h + V −1θ2

)

191


C2

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.............................

......................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. C×× ×××


g = z2(V h + V −1θ2

)

192


C2

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.............................

......................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. C×× ×××


g = z2(V h + V −1θ2

)

194


C2

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.............................

......................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............

−1−1 −1−1−1

........................................................................................................

........................................................................................................

........................................................................................................

........................................................................................................

........................................................................................................


g = z2(V h + V −1θ2

)

195


vv

vvv

C×× ×××

.................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

......

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............

.............


g = z2(V h + V −1θ2

)

196


C2

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.............................

......................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............

−1−1 −1−1−1

........................................................................................................

........................................................................................................

........................................................................................................

........................................................................................................

........................................................................................................


g = z2(V h + V −1θ2

)

197

Some ALE Scalar-Flat Kahler Surfaces:

198


(L ’91)

199


200


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

...........................

...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............

Data: k + 1 points in H3. =⇒ V with ∆V = 0

201


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

...........................

...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............


V = 1 + `G0 +

k∑j=1

Gj

202


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

...........................

...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............


V = 1 +`

e2%0 − 1+

k∑j=1

1

e2%j − 1

203


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

...........................

...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............


V = 1 + `G0 +

k∑j=1

Gj

204


f

vv

vvv

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.......

........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

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....................................

..............................................................................................................................................................................................................................................................................

....

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........................................

..............................................................................................................................................................................................................................................................................................................................................................................

..........

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

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...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............


F = ?dV curvature θ on P → H3 − {pts}.

205


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

...........................

...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............


g =1

4 sinh2 %0

(V h + V −1θ2

)

206


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

...........................

...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............

Riemannian completion is ALE scalar-flat Kahler.

g =1

4 sinh2 %0

(V h + V −1θ2

)

207


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

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.........................

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...............................

......................................

....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............

Riemannian completion is ALE with Γ = Z`.

V = 1 +`

e2%0 − 1+

k∑j=1

1

e2%j − 1

208


f

vv

vvv

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................

...................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................

....................

......................

.......................

.........................

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...............................

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....................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................

...............................................................

..............................................

....................................

....................................................................................

...................

............. .............

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

.............

.................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ......

..........................

..........................

............................................

.................................................................

×

×

×

×

×


V = 1 +`

e2%0 − 1+

k∑j=1

1

e2%j − 1

209


Blow up of Chern-class −` line bundle over CP1 atk points on zero section Σ.

......................................................................................................................................................................................................................................................................................

..................................

................

............................................................................................................................................................................................................................................................................................................................................................

............................................

........................

.............

............................................................................................................................................................................................................................................................................................................................................................

............................................

........................

.............

Σ−`

...................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................


V = 1 +`

e2%0 − 1+

k∑j=1

1

e2%j − 1

210



......................................................................................................................................................................................................................................................................................

..................................

................

............................................................................................................................................................................................................................................................................................................................................................

............................................

........................

.............

............................................................................................................................................................................................................................................................................................................................................................

............................................

........................

.............

×× × Σ−`

...................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................


V = 1 +`

e2%0 − 1+

k∑j=1

1

e2%j − 1

211



......................................................................................................................................................................................................................................................................................

..................................

................

............................................................................................................................................................................................................................................................................................................................................................

............................................

........................

.............

............................................................................................................................................................................................................................................................................................................................................................

............................................

........................

.............

E1

−1−1−1

E2Ek

Σ−`− k

...................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

......


V = 1 +`

e2%0 − 1+

k∑j=1

1

e2%j − 1

212


Penrose Twistor Space (Z, J),


.....................................................................................................................................................................................................................................................................................................................................................................................................................................

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213

Twistor Spaces for These Metrics:

H0(CP1 × CP1,O(1, 1)) = C4⊃ R1,3 ⊃ H3

214


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3⊃ H3

215


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

216


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

f

vv

vvv

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.............

So k + 1 points in H3 give rise to

217


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

f

vv

vvv

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.............

So k + 1 points in H3 give rise to

P0, P1, . . . , Pk ∈ H0(CP1 × CP1,O(1, 1)).

218


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

219


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,

220


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,

let Z be the hypersurface

xy = P `0 P1 · · · Pk.

221


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,


xy = P `0 P1 · · · Pk.

Then twistor space Z obtained from Z by

222


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,


xy = P `0 P1 · · · Pk.


• removing curve in zero section cut out by P0,

223


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,


xy = P `0 P1 · · · Pk.



• adding two rational curves at infinity, and

224


H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3

InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,


xy = P `0 P1 · · · Pk.



• adding two rational curves at infinity, and

•making small resolutions of isolated singularities.

225




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226




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Lots more ALE scalar-flat Kahler surfaces now known:

227




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Joyce, Calderbank-Singer, Lock-Viaclovsky. . .

228




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Joyce, Calderbank-Singer, Lock-Viaclovsky. . .

But full classification remains an open problem.

229


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gjk,` = O(|x|−n2−ε), s ∈ L1

230

Definition. The mass (at a given end) of anALE n-manifold is defined to be

m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

231


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

232


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

233


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

234


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

235


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where

236


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where

• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;

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.....................

...............................

237


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where


...............................................................................................................................................................................................................................................................................................................................

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238


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where


• ν is the outpointing Euclidean unit normal;and

239


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where



• αE is the volume (n− 1)-from induced by theEuclidean metric.

240


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.

241


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





242


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





gjk,` = O(|x|−n2−ε), s ∈ L1

243


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





244


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





245


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




Chrusciel-type fall-off:

gjk − δjk ∈ C1−τ , τ >

n− 2

2

246


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where





247


m(M, g) := lim%→∞

Γ(n2)

4(n− 1)πn/2

∫Σ(%)

[gij,i − gii,j

]νjαE

where




We’ll see a new proof of this in the Kahler case.

248

In fact, we’ll eventually prove:

249


Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

250



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where

• s = scalar curvature;

• dµ = metric volume form;

• c1 = c1(M,J) ∈ H2(M) is first Chern class;

• [ω] ∈ H2(M) is Kahler class of (g, J); and

• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).

251



m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

where

• s = scalar curvature;

• dµ = metric volume form;

• c1 = c1(M,J) ∈ H2(M) is first Chern class;

• [ω] ∈ H2(M) is Kahler class of (g, J); and

• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).

• ♣ : H2(M)∼=−→ H2

c (M) inverse of natural map.

252

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

253

Scalar-flat Kahler case:

m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

254

Scalar-flat Kahler surface:

m(M, g) = − 1

3π〈♣(c1), [ω]〉

255


m(M, g) = − 1

3π〈♣(c1), [ω]〉

Today: What does this mean in practice?

256


m(M, g) = − 1

3π〈♣(c1), [ω]〉

257


m(M, g) = − 1

3π〈♣(c1), [ω]〉

258


m(M, g) = − 1

3π〈♣(c1), [ω]〉

Gravitational instantons?

259


m(M, g) = − 1

3π〈♣(c1), [ω]〉


Ricci flat! =⇒ c1 = 0.

260


m(M, g) = − 1

3π〈♣(c1), [ω]〉



261


m(M, g) = − 1

3π〈♣(c1), [ω]〉



Mass automatically vanishes!

262


m(M, g) = − 1

3π〈♣(c1), [ω]〉




Bartnik: Ricci-flat =⇒ faster fall-off of metric!

263


m(M, g) = − 1

3π〈♣(c1), [ω]〉




Bartnik: Ricci-flat =⇒ mass vanishes!

264


m(M, g) = − 1

3π〈♣(c1), [ω]〉

Today: What does this mean in practice?

265


m(M, g) = − 1

3π〈♣(c1), [ω]〉

Today: Exploit Poincare duality. . .

266

Proposition. Let (M, g, J) be an ALE scalar-flat Kahler surface. Let E1, . . . E` be a basis forH2(M,R), and let Q = [Qjk] = [Ej · Ek] be thecorresponding intersection matrix. If we definea1, . . . , a` by

267


268


269


270

Proposition. Let (M, g, J) be an ALE scalar-flat Kahler surface. Let E1, . . . E` be a basis forH2(M,R), and let Q = [Qjk] = [Ej · Ek] be thecorresponding intersection matrix. If we definea1, . . . , a` by a1

...a`

= Q−1∫E1c1

...∫E`c1

271


...a`

= Q−1∫E1c1

...∫E`c1

then the mass of (M, g) is given by

272


...a`

= Q−1∫E1c1

...∫E`c1


m(M, g) = − 1

3π

∑j=1

aj

∫Ej

[ω]

273


...a`

= Q−1∫E1c1

...∫E`c1


m(M, g) = − 1

3π

∑j=1

aj

∫Ej

[ω]

where [ω] denotes the Kahler class of (M, g, J).

274


m(M, g) = − 1

3π〈♣(c1), [ω]〉

275


m(M, g) = − 1

3π〈♣(c1), [ω]〉

Example. Blow up C2 at k points.

276


m(M, g) = − 1

3π〈♣(c1), [ω]〉


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.............×× ×××

277


m(M, g) = − 1

3π〈♣(c1), [ω]〉


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×××××

278


m(M, g) = − 1

3π〈♣(c1), [ω]〉


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.............×× ×××

279


m(M, g) = − 1

3π〈♣(c1), [ω]〉


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−1−1 −1−1−1

E4E3 E5E2E1 ........................................................................................................

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280


...a`

= Q−1∫E1c1

...∫E`c1


m(M, g) = − 1

3π

∑j=1

aj

∫Ej

[ω]


281


m(M, g) = − 1

3π〈♣(c1), [ω]〉


m(M, g) =1

3π

∑j=1

∫Ej

[ω]

282


m(M, g) = − 1

3π〈♣(c1), [ω]〉


m(M, g) =1

3π

∑j=1

∫Ej

[ω]

Always positive! (AE): Positive mass theorem.

283


m(M, g) = − 1

3π〈♣(c1), [ω]〉


m(M, g) =1

3π

∑j=1

∫Ej

[ω]

Always positive! (AE): Positive mass theorem.

284


m(M, g) = − 1

3π〈♣(c1), [ω]〉

285


m(M, g) = − 1

3π〈♣(c1), [ω]〉

Example. Blow up Chern-class −` line bundleover CP1 at k points on zero section Σ.

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Σ−`

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m(M, g) = − 1

3π〈♣(c1), [ω]〉


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×× × Σ−`

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m(M, g) = − 1

3π〈♣(c1), [ω]〉


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E1

−1−1−1

E2Ek

Σ−`− k

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...a`

= Q−1∫E1c1

...∫E`c1


m(M, g) = − 1

3π

∑j=1

aj

∫Ej

[ω]


289


m(M, g) = − 1

3π〈♣(c1), [ω]〉


m(M, g) =

290


m(M, g) = − 1

3π〈♣(c1), [ω]〉


m(M, g) =1

3π`

(2− `)∫

Σω+2

k∑j=1

∫Ej

ω

.

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m(M, g) = − 1

3π〈♣(c1), [ω]〉


m(M, g) =1

3π`

(2− `)∫

Σω + 2

k∑j=1

∫Ej

ω

.

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...a`

= Q−1∫E1c1

...∫E`c1


m(M, g) = − 1

3π

∑j=1

aj

∫Ej

[ω]


293

Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then zapm(M, g) ≤ 0, with = iff g is Ricci-flat.

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Examples: Hirzebruch-Jung resolution of C2/Z`.

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(z1, z2) 7→ (e2πi/`z1, e2πik/`z2)

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Calderbank-Singer metrics generalize for k 6= ±1.

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Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.

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301


...a`

= Q−1∫E1c1

...∫E`c1


m(M, g) = − 1

3π

∑j=1

aj

∫Ej

[ω]


302


V. Alexeev: Q−1 term-by-term ≤ 0 for these.

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V. Alexeev: Q−1 term-by-term ≤ 0 for these.

Brought to our attention by C. Spotti.

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m(M, g) = − 1

3π〈♣(c1), [ω]〉

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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1

+(m− 1)!

4(2m− 1)πm

∫Msgdµg

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End, Part II

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Stony Brook University Seminario de Geometr a ICMAT ...De nition.Complete, non-compact n-manifold...

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