Einstein Metrics, Harmonic Forms, & Symplectic Four ...parton/NTDG2014/lucidi/lebrun.pdfFour...

Einstein Metrics,

Harmonic Forms, &

Symplectic Four-Manifolds

Claude LeBrunStony Brook University

Sardegna, 2014/9/20

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Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.

2


r = λh

for some constant λ ∈ R.

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r = λh


“. . . the greatest blunder of my life!”

— A. Einstein, to G. Gamow

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r = λh


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r = λh


As punishment . . .

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r = λh


λ called Einstein constant.

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r = λh



Has same sign as the scalar curvature

s = rjj = Rijij.

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r = λh



Has same sign as the scalar curvature

s = rjj = Rijij.

volg(Bε(p))

cnεn= 1− s ε2

6(n+2)+ O(ε4)

r......................................................................................... ε

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Geometrization Problem:

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Given a smooth compact manifold Mn, can one

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Given a smooth compact manifold Mn, can onedecompose M in Einstein and collapsed pieces?

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When n = 3, answer is “Yes.”

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Proof by Ricci flow. Perelman et al.

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Proof by Ricci flow. Perelman et al.

Perhaps reasonable in other dimensions?

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Recognition Problem:

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Suppose Mn admits Einstein metric h.

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What, if anything, does h then tell us about M?

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Can we recognize M by looking at h?

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When n = 3, h has constant sectional curvature!

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So M has universal cover S3, R3, H3. . .

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But when n ≥ 5, situation seems hopeless.

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{Einstein metrics on Sn}/∼ is highly disconnected.

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{Einstein metrics on Sn}/∼ is highly disconnected.

When n ≥ 4, situation is more encouraging. . .

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Moduli Spaces of Einstein metrics

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E (M) = {Einstein h}/(Diffeos× R+)

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Known to be connected for certain 4-manifolds:

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M = T 4, K3, H4/Γ, CH2/Γ.

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M = T 4, K3, H4/Γ, CH2/Γ.

Berger, Hitchin, Besson-Courtois-Gallot, L.

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M = T 4, K3, H4/Γ, CH2/Γ.


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M = T 4, K3, H4/Γ, CH2/Γ.


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M = T 4, K3, H4/Γ, CH2/Γ.


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Four Dimensions is Exceptional

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When n = 4, Einstein metrics are genuinely non-trivial: not typically spaces of constant curvature.

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There are beautiful and subtle global obstructionsto the existence of Einstein metrics on 4-manifolds.

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But might allow for geometrization of 4-manifoldsby decomposition into Einstein and collapsed pieces.

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One key question:

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One key question:

Does enough rigidity really hold in dimension fourto make this a genuine geometrization?

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Symplectic 4-manifolds:

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A laboratory for exploring Einstein metrics.

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Kahler geometry is a rich source of examples.

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If M admits a Kahler metric, it of course admits asymplectic form ω.

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On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.

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Some Suggestive Questions. If (M4, ω) is asymplectic 4-manifold, when does M4 admit anEinstein metric h (unrelated to ω)? What if wealso require λ ≥ 0?

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Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2

K3

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

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Conventions:

CP2 = reverse oriented CP2.

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Conventions:


Connected sum #:

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Conventions:


Connected sum #:

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Conventions:


Connected sum #:

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Conventions:


Connected sum #:

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Conventions:


Connected sum #:

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Conventions:


Connected sum #:

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Conventions:


Connected sum #:

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

72


Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3,

K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Del Pezzo surfaces,K3 surface, Enriques surface,Abelian surface, Hyper-elliptic surfaces.

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But we understand some cases better than others!

M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

74

Definitive list . . .

M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

75


M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

76


M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

77


M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

78


M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:

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M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:

Every Einstein metric is Ricci-flat Kahler.

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M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:


Moduli space E (M)= {Einstein metrics}/Diffeomorphisms

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M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:


Moduli space E (M) = {Einstein h}/(Diffeos×R+)

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M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:


Moduli space E (M) completely understood.

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M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:


Moduli space E (M) connected!

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M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:



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Above the line:

M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:



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Above the line:

Know an Einstein metric on each manifold.

M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:



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Above the line:

Moduli space E (M) 6= ∅. Is it connected?

M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:



88

Above the line:

Moduli space E (M) 6= ∅. But is it connected?

M ===

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).

Below the line:



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Objective of this lecture:

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Describe modest recent progress on this issue.

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For M a Del Pezzo surface, will define explicit open

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U ⊂ {Riemannian metrics on M}

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such that

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such that

• Every known Einstein metric belongs to U ;

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such that


• These form a connected family, mod diffeos; and

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such that


• These form a connected family, mod diffeos; and

•No other Einstein metrics belong to U !

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Formulation will depend on. . .

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Special character of dimension 4:

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The Lie group SO(4) is not simple:

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so(4) ∼= so(3)⊕ so(3).

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so(4) ∼= so(3)⊕ so(3).

On oriented (M4, g), =⇒

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so(4) ∼= so(3)⊕ so(3).

On oriented (M4, g), =⇒Λ2 = Λ+ ⊕ Λ−

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so(4) ∼= so(3)⊕ so(3).


where Λ± are (±1)-eigenspaces of

? : Λ2→ Λ2,

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so(4) ∼= so(3)⊕ so(3).


where Λ± are (±1)-eigenspaces of

? : Λ2→ Λ2,

?2 = 1.

Λ+ self-dual 2-forms.Λ− anti-self-dual 2-forms.

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Riemann curvature of g

R : Λ2→ Λ2

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R : Λ2→ Λ2

splits into 4 irreducible pieces:

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R : Λ2→ Λ2


Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

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R : Λ2→ Λ2


Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

where

s = scalar curvature

r = trace-free Ricci curvature

W+ = self-dual Weyl curvature (conformally invariant)

W− = anti-self-dual Weyl curvature

109


R : Λ2→ Λ2


Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

where

s = scalar curvature

r = trace-free Ricci curvature

W+ = self-dual Weyl curvature (conformally invariant)

W− = anti-self-dual Weyl curvature ′′

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Smooth compactM4 has invariants b±(M), definedin terms of intersection pairing

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H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

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H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

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H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

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H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

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Hodge theory:

H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ? ϕ = 0}.

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Hodge theory:

H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ? ϕ = 0}.Since ? is involution of RHS, =⇒

H2(M,R) = H+g ⊕H−g ,

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Hodge theory:


H2(M,R) = H+g ⊕H−g ,

where

H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.

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Hodge theory:


H2(M,R) = H+g ⊕H−g ,

where

H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms. Then

b±(M) = dimH±g .

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H−g

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{a | a · a = 0} ⊂H2(M,R)

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H+g

H−g

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{a | a · a = 0} ⊂ H2(M,R)

121

H+g′

H−g′

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{a | a · a = 0} ⊂ H2(M,R)

122

H+g

H−g

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b±(M) = dimH±g .

123

H+g′

H−g′

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b±(M) = dimH±g .

124

H+g

H−g

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b±(M) = dimH±g .

125

H+g

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b±(M) = dimH±g .

126

More Technical Question. When does a com-pact complex surface (M4, J) admit an Einsteinmetric h which is Hermitian, in the sense that

127


h(·, ·) = h(J ·, J ·)?

128


h(·, ·) = h(J ·, J ·)?

Kahler if the 2-form

ω = h(J ·, ·)is closed:

dω = 0.

But we do not assume this!

129


h(·, ·) = h(J ·, J ·)?

130

Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”

131


More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that

c1(M4, J) = λ[ω].

132



c1(M4, J) = λ[ω].

Moreover, this metric is unique, up to isometry,if λ 6= 0.

133



c1(M4, J) = λ[ω].


Aubin, Yau, Siu, Tian . . . Kahler case.

134



c1(M4, J) = λ[ω].



Chen-L-Weber (2008), L (2012): non-Kahler case.

135



c1(M4, J) = λ[ω].



Chen-L-Weber (2008), L (2012): non-Kahler case.

Only two metrics arise in non-Kahler case!

136



c1(M4, J) = λ[ω].


Key Point. Metrics are actually conformally Kahler.

137



c1(M4, J) = λ[ω].



138



c1(M4, J) = λ[ω].



h = s−2g

139



c1(M4, J) = λ[ω].



Strictly 4-dimensional phenomenon!

140



c1(M4, J) = λ[ω].



Strictly 4-dimensional phenomenon!

“Riemannian Goldberg-Sachs Theorem.”

141

So, which complex surfaces admit

142

So, which complex surfaces admit

Einstein Hermitian metrics with λ > 0?

143

Del Pezzo surfaces:

144

Del Pezzo surfaces:

(M4, J) for which c1 is a Kahler class [ω].

145

Del Pezzo surfaces:

(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”

146

Del Pezzo surfaces:


Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.

147

Del Pezzo surfaces:



148

Del Pezzo surfaces:



149

Del Pezzo surfaces:



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CP2

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150

Blowing up:

151

Blowing up:

If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up

rN

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152

Blowing up:


rN

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153

Blowing up:


rN

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154

Blowing up:


M ≈ N#CP2

in which added CP1 has normal bundle O(−1).

rN

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155

Blowing up:


M ≈ N#CP2


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157

Blowing up:


M ≈ N#CP2


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159

Blowing up:


M ≈ N#CP2


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161

Del Pezzo surfaces:



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162

Del Pezzo surfaces:



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No 3 on a line, no 5 on conic, no 8 on nodal cubic.

163

Del Pezzo surfaces:



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164

Del Pezzo surfaces:



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165

Del Pezzo surfaces:



166

Del Pezzo surfaces:



For each topological type:

167

Del Pezzo surfaces:




Moduli space of such (M4, J) is connected.

168

Del Pezzo surfaces:




Moduli space of such (M4, J) is connected.

Just a point if b2(M) ≤ 5.

169

Every del Pezzo surface has b+ = 1. ⇐⇒

170


Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

171



dω = 0, ?ω = ω.

172



dω = 0, ?ω = ω.

173



dω = 0, ?ω = ω.

Such a form defines a symplectic structure, exceptat points where ω = 0.

174



dω = 0, ?ω = ω.


175



dω = 0, ?ω = ω.


Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.

176



dω = 0, ?ω = ω.



177



dω = 0, ?ω = ω.



178



dω = 0, ?ω = ω.



179



dω = 0, ?ω = ω.



180



dω = 0, ?ω = ω.



• open condition;

181



dω = 0, ?ω = ω.



• open condition;

• conformally invariant; and

182



dω = 0, ?ω = ω.



• open condition;


• holds in Kahler case.

183

Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy

184


185


186


187


W+(ω, ω) > 0

188


W+(ω, ω) > 0

at every point of M .

189


W+(ω, ω) > 0


Notice that

190


W+(ω, ω) > 0


Notice that

• =⇒ ω is symplectic form.

191


W+(ω, ω) > 0


Notice that

• =⇒ ω is symplectic form.

• =⇒ W+ is everywhere non-zero.

192


W+(ω, ω) > 0


193


W+(ω, ω) > 0


However,

194


W+(ω, ω) > 0


However,

• recall that W+ is trace-free; so

195


W+(ω, ω) > 0


However,

• recall that W+ is trace-free; so

• wherever W+ non-zero, has positive directions.

196


W+(ω, ω) > 0


197


W+(ω, ω) > 0


Thus, we are really just demanding that

198


W+(ω, ω) > 0



• ω 6= 0 everywhere;

199


W+(ω, ω) > 0




•W+ 6= everywhere; and

200


W+(ω, ω) > 0




•W+ 6= everywhere; and

•W+ and ω are everywhere roughly aligned.

201


W+(ω, ω) > 0


202


W+(ω, ω) > 0


• open condition;

203


W+(ω, ω) > 0


• open condition;


204


W+(ω, ω) > 0


• open condition;


• holds for (almost) Kahler metrics with s > 0.

205


W+(ω, ω) > 0


• open condition;



Kahler =⇒ W+ =

− s12− s

12s6

206


W+(ω, ω) > 0


• open condition;



207


W+(ω, ω) > 0


• open condition;



208

Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).

209


210


211


212


213


In other words, h is known, and is either

214



•Kahler-Einstein, with λ > 0; or

215




• Page metric on CP2#CP2; or

216





• CLW metric on CP2#2CP2.

217





• CLW metric on CP2#2CP2.

Conversely, all these are of positive symplectic type.

218

For M4 a Del Pezzo surface, set

219


considered as smooth compact oriented 4-manifold,

220


221


222


E (M) = {Einstein h on M}/(Diffeos× R+)

223



E +ω (M) = {Einstein h with W+(ω, ω) > 0}/∼

224




Corollary. E +ω (M) is connected. Moreover, if

b2(M) ≤ 5, then E +ω (M) = {point}.

225






226






227






Corollary.E +ω (M) is exactly one connected com-

ponent of E (M).

228

Method of Proof.

229

Key Observation:

230

Key Observation:

By second Bianchi identity,

231

Key Observation:


h Einstein =⇒ δW+ = (δW )+ = 0.

232

Key Observation:



(δW )bcd := −∇aW abcd = −∇[crd]b +

1

6hb[c∇d]s

233

Key Observation:




1

6hb[c∇d]s

Our strategy:

234

Key Observation:




1

6hb[c∇d]s

Our strategy:

study weaker equation

235

Key Observation:




1

6hb[c∇d]s

Our strategy:


δW+ = 0

236

Key Observation:




1

6hb[c∇d]s

Our strategy:


δW+ = 0

as proxy for Einstein equation.

237

Now suppose that ω 6= 0 everywhere.

238


Rescale h to obtain g with |ω| ≡√

2:

239



2:

g =1√2|ω|h.

240



2:

g =1√2|ω|h.

This g is almost-Kahler:

241



2:

g =1√2|ω|h.


related to symplectic form ω by

242



2:

g =1√2|ω|h.



g = ω(·, J ·)

243



2:

g =1√2|ω|h.



g = ω(·, J ·)

for some g-preserving almost-complex structure J .

244

Weitzenbock for harmonic self-dual 2-form ω:

245


0 = ∇∗∇ω − 2W+(ω, ·) +s

3ω

246


0 = ∇∗∇ω − 2W+(ω, ·) +s

3ω

Taking inner product with ω:

247


0 = ∇∗∇ω − 2W+(ω, ·) +s

3ω


0 =1

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

s

3|ω|2

248


0 = ∇∗∇ω − 2W+(ω, ·) +s

3ω


0 =1

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

s

3|ω|2

In almost-Kahler case, |ω|2 ≡ 2, so

249


0 = ∇∗∇ω − 2W+(ω, ·) +s

3ω


0 =1

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

s

3|ω|2


1

2|∇ω|2 = W+(ω, ω)− s

3

250


0 = ∇∗∇ω − 2W+(ω, ·) +s

3ω


0 =1

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

s

3|ω|2


1

2|∇ω|2 = W+(ω, ω)− s

3

W+(ω, ω) ≥ s

3

251

Equation δW+ = 0? not conformally invariant!

252

Equation δW+ = 0 not conformally invariant!

253


If h = f2g satisfies

254



δW+ = 0

255



δW+ = 0

then g instead satisfies

256



δW+ = 0


δ(fW+) = 0

257



δW+ = 0


δ(fW+) = 0

which in turn implies the Weitzenbock formula

258



δW+ = 0


δ(fW+) = 0


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

259



δW+ = 0


δ(fW+) = 0


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

for fW+ ∈ End(Λ+).

260

Now take inner product of Weitzenbock formula

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

261


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

with 2ω⊗ω and integrate by parts, using identity

262


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

with 2ω⊗ω and integrate by parts, using identity

263


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

with 2ω ⊗ ω and integrate by parts, using identity

264


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

265


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

266


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

0 =

∫M

(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2

)f dµ,

267


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

0 =

∫M

(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2

)f dµ,

where W+(ω)⊥ = projection of W+(ω, ·) to ω⊥.

268


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

0 =

∫M

(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2

)f dµ

269


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

0 ≥∫M

(−sW+(ω, ω) + 3

[W+(ω, ω)

]2 )f dµ

270


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

0 ≥ 3

∫MW+(ω, ω)

(W+(ω, ω)− s

3

)f dµ

271


0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I


〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .

This yields

0 ≥ 3

∫MW+(ω, ω)

(1

2|∇ω|2

)f dµ

272

Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then

273


274


275


276


0 ≥∫MW+(ω, ω)|∇ω|2f dµ

277


0 ≥∫MW+(ω, ω)|∇ω|2f dµ

Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If

278


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


279


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


280


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0

281


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0

for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.

282


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0


283


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0


284


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0


285


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0


Kahler =⇒ W+ =

− s12− s

12s6

286


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0


287


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0


Conversely, any Kahler (M4, g, ω) with s > 0 sat-isfies W+(ω, ω) > 0, and f = c/s solves

288


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0



289


0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0



∇(fW+) = 0 =⇒ δ(fW+) = 0.

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0 ≥∫MW+(ω, ω)|∇ω|2f dµ


δ(fW+) = 0



∇(fW+) = 0 =⇒ δ(fW+) = 0.

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Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If

W+(ω, ω) > 0

for some self-dual harmonic 2-form ω, then

h = s−2g

for a unique Kahler metric g of scalar curvatures > 0.

Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.

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W+(ω, ω) > 0


h = s−2g



293


W+(ω, ω) > 0


h = s−2g



294


W+(ω, ω) > 0


h = s−2g



295


W+(ω, ω) > 0


h = s−2g



296


W+(ω, ω) > 0


h = s−2g



297


W+(ω, ω) > 0


h = s−2g



298


W+(ω, ω) > 0


h = s−2g



299


W+(ω, ω) > 0


h = s−2g



300


W+(ω, ω) > 0


h = s−2g



301


W+(ω, ω) > 0


h = s−2g



302


W+(ω, ω) > 0


h = s−2g



303


W+(ω, ω) > 0


h = s−2g



304


W+(ω, ω) > 0


h = s−2g



305


W+(ω, ω) > 0


h = s−2g



306


W+(ω, ω) > 0


h = s−2g



307


W+(ω, ω) > 0


h = s−2g



308


W+(ω, ω) > 0


h = s−2g



309


W+(ω, ω) > 0


h = s−2g



Remark. If such metrics exist, b+(M) = 1.

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W+(ω, ω) > 0


h = s−2g



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W+(ω, ω) > 0


h = s−2g



Theorem A follows by restricting to Einstein case.

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313




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Corollary.E +ω (M) is exactly one connected com-

ponent of E (M).

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Application to Almost-Kahler Geometry:

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Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with

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318


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s ≥ 0, δW+ = 0.

Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.

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s ≥ 0, δW+ = 0.


321


s ≥ 0, δW+ = 0.


322


s ≥ 0, δW+ = 0.


323


s ≥ 0, δW+ = 0.


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s ≥ 0, δW+ = 0.


In particular, gives a new proof of the following:

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s ≥ 0, δW+ = 0.



Corollary (Sekigawa). Every compact almost-KahlerEinstein 4-manifold with non-negative Einsteinconstant is Kahler-Einstein.

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s ≥ 0, δW+ = 0.




(Special case of “Goldberg conjecture.”)

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s ≥ 0, δW+ = 0.




Helped motivate the discovery of Theorem A. . .

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Tanti auguri, Stefano!

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E grazie agli organizzatori

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E grazie agli organizzatori

per un convegno cosı bello!

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