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

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

r = λh

for some constant λ ∈ R.

r = λh

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

— A. Einstein, to G. Gamow

r = λh

As punishment . . .

r = λh

λ called Einstein constant.

r = λh

Has same sign as the scalar curvature

s = rjj = Rijij.

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:

Given a smooth compact manifold Mn, can one

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

When n = 3, answer is “Yes.”

Proof by Ricci flow. Perelman et al.

Perhaps reasonable in other dimensions?

Recognition Problem:

Suppose Mn admits Einstein metric h.

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

Can we recognize M by looking at h?

When n = 3, h has constant sectional curvature!

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

But when n ≥ 5, situation seems hopeless.

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

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

Moduli Spaces of Einstein metrics

E (M) = {Einstein h}/(Diffeos× R+)

Known to be connected for certain 4-manifolds:

M = T 4, K3, H4/Γ, CH2/Γ.

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

M = T 4, K3, H4/Γ, CH2/Γ.

Four Dimensions is Exceptional

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

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

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

One key question:

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

Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

Kahler geometry is a rich source of examples.

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

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

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?

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Conventions:

CP2 = reverse oriented CP2.

Conventions:

Connected sum #:

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Conventions:

Connected sum #:

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Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Mdiff≈

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,

K3/Z2,

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.

But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

Definitive list . . .

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

T 4/Z2, T4/Z3, T

4/Z4, T4/Z6,

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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:

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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.

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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+)

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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.

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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!

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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:

Above the line:

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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:

Above the line:

Know an Einstein metric on each manifold.

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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:

Above the line:

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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:

Above the line:

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

CP2#kCP2, 0 ≤ k ≤ 8,

S2 × S2,K3,K3/Z2,

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:

Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will define explicit open

U ⊂ {Riemannian metrics on M}

such that

• Every known Einstein metric belongs to U ;

such that

• These form a connected family, mod diffeos; and

such that

• These form a connected family, mod diffeos; and

•No other Einstein metrics belong to U !

Formulation will depend on. . .

Special character of dimension 4:

The Lie group SO(4) is not simple:

so(4) ∼= so(3)⊕ so(3).

On oriented (M4, g), =⇒

so(4) ∼= so(3)⊕ so(3).

On oriented (M4, g), =⇒Λ2 = Λ+ ⊕ Λ−

so(4) ∼= so(3)⊕ so(3).

where Λ± are (±1)-eigenspaces of

? : Λ2→ Λ2,

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.

Riemann curvature of g

R : Λ2→ Λ2

splits into 4 irreducible pieces:

R : Λ2→ Λ2

Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

R : Λ2→ Λ2

Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

s = scalar curvature

r = trace-free Ricci curvature

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

W− = anti-self-dual Weyl curvature

R : Λ2→ Λ2

Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

s = scalar curvature

r = trace-free Ricci curvature

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

W− = anti-self-dual Weyl curvature ′′

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

H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

Hodge theory:

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

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 ,

Hodge theory:

H2(M,R) = H+g ⊕H−g ,

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

Hodge theory:

H2(M,R) = H+g ⊕H−g ,

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

b±(M) = dimH±g .

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{a | a · a = 0} ⊂H2(M,R)

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{a | a · a = 0} ⊂ H2(M,R)

H+g′

H−g′

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{a | a · a = 0} ⊂ H2(M,R)

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b±(M) = dimH±g .

H+g′

H−g′

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b±(M) = dimH±g .

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b±(M) = dimH±g .

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b±(M) = dimH±g .

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

h(·, ·) = h(J ·, J ·)?

Kahler if the 2-form

ω = h(J ·, ·)is closed:

dω = 0.

But we do not assume this!

h(·, ·) = h(J ·, J ·)?

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

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

c1(M4, J) = λ[ω].

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

c1(M4, J) = λ[ω].

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

c1(M4, J) = λ[ω].

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

c1(M4, J) = λ[ω].

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

Only two metrics arise in non-Kahler case!

c1(M4, J) = λ[ω].

Key Point. Metrics are actually conformally Kahler.

c1(M4, J) = λ[ω].

h = s−2g

c1(M4, J) = λ[ω].

Strictly 4-dimensional phenomenon!

c1(M4, J) = λ[ω].

Strictly 4-dimensional phenomenon!

“Riemannian Goldberg-Sachs Theorem.”

So, which complex surfaces admit

Einstein Hermitian metrics with λ > 0?

Del Pezzo surfaces:

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

Del Pezzo surfaces:

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

Del Pezzo surfaces:

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

Del Pezzo surfaces:

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Blowing up:

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

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Blowing up:

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Blowing up:

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Blowing up:

M ≈ N#CP2

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

M............................................................................................................... ..........

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Blowing up:

M ≈ N#CP2

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Blowing up:

M ≈ N#CP2

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Blowing up:

M ≈ N#CP2

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Del Pezzo surfaces:

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Del Pezzo surfaces:

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

Del Pezzo surfaces:

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Del Pezzo surfaces:

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Del Pezzo surfaces:

For each topological type:

Del Pezzo surfaces:

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

Del Pezzo surfaces:

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

Just a point if b2(M) ≤ 5.

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

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

dω = 0, ?ω = ω.

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

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.

dω = 0, ?ω = ω.

• open condition;

dω = 0, ?ω = ω.

• open condition;

• conformally invariant; and

dω = 0, ?ω = ω.

• open condition;

• holds in Kahler case.

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

W+(ω, ω) > 0

at every point of M .

W+(ω, ω) > 0

Notice that

W+(ω, ω) > 0

Notice that

• =⇒ ω is symplectic form.

W+(ω, ω) > 0

Notice that

• =⇒ ω is symplectic form.

• =⇒ W+ is everywhere non-zero.

W+(ω, ω) > 0

However,

W+(ω, ω) > 0

However,

• recall that W+ is trace-free; so

W+(ω, ω) > 0

However,

• recall that W+ is trace-free; so

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

W+(ω, ω) > 0

Thus, we are really just demanding that

W+(ω, ω) > 0

• ω 6= 0 everywhere;

W+(ω, ω) > 0

•W+ 6= everywhere; and

W+(ω, ω) > 0

•W+ 6= everywhere; and

•W+ and ω are everywhere roughly aligned.

W+(ω, ω) > 0

• open condition;

W+(ω, ω) > 0

• open condition;

W+(ω, ω) > 0

• open condition;

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

W+(ω, ω) > 0

• open condition;

Kahler =⇒ W+ =

− s12− s

W+(ω, ω) > 0

• open condition;

W+(ω, ω) > 0

• open condition;

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).

In other words, h is known, and is either

•Kahler-Einstein, with λ > 0; or

• Page metric on CP2#CP2; or

• CLW metric on CP2#2CP2.

Conversely, all these are of positive symplectic type.

For M4 a Del Pezzo surface, set

considered as smooth compact oriented 4-manifold,

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

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

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

b2(M) ≤ 5, then E +ω (M) = {point}.

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

ponent of E (M).

Method of Proof.

Key Observation:

By second Bianchi identity,

Key Observation:

h Einstein =⇒ δW+ = (δW )+ = 0.

Key Observation:

(δW )bcd := −∇aW abcd = −∇[crd]b +

6hb[c∇d]s

Key Observation:

6hb[c∇d]s

Our strategy:

Key Observation:

6hb[c∇d]s

Our strategy:

study weaker equation

Key Observation:

6hb[c∇d]s

Our strategy:

δW+ = 0

Key Observation:

6hb[c∇d]s

Our strategy:

δW+ = 0

as proxy for Einstein equation.

Now suppose that ω 6= 0 everywhere.

Rescale h to obtain g with |ω| ≡√

g =1√2|ω|h.

This g is almost-Kahler:

g =1√2|ω|h.

related to symplectic form ω by

g =1√2|ω|h.

g = ω(·, J ·)

g =1√2|ω|h.

g = ω(·, J ·)

for some g-preserving almost-complex structure J .

Weitzenbock for harmonic self-dual 2-form ω:

0 = ∇∗∇ω − 2W+(ω, ·) +s

Taking inner product with ω:

0 = ∇∗∇ω − 2W+(ω, ·) +s

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

3|ω|2

0 = ∇∗∇ω − 2W+(ω, ·) +s

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

3|ω|2

In almost-Kahler case, |ω|2 ≡ 2, so

0 = ∇∗∇ω − 2W+(ω, ·) +s

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

3|ω|2

2|∇ω|2 = W+(ω, ω)− s

0 = ∇∗∇ω − 2W+(ω, ·) +s

2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +

3|ω|2

2|∇ω|2 = W+(ω, ω)− s

W+(ω, ω) ≥ s

Equation δW+ = 0? not conformally invariant!

Equation δW+ = 0 not conformally invariant!

If h = f2g satisfies

δW+ = 0

then g instead satisfies

δW+ = 0

δ(fW+) = 0

δW+ = 0

δ(fW+) = 0

which in turn implies the Weitzenbock formula

δW+ = 0

δ(fW+) = 0

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

δW+ = 0

δ(fW+) = 0

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

for fW+ ∈ End(Λ+).

Now take inner product of Weitzenbock formula

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

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

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

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

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

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

0 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

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

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 = ∇∗∇(fW+) +s

2fW+ − 6fW+ ◦W+ + 2f |W+|2I

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

This yields

(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2

)f dµ,

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

(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2

)f dµ,

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

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

(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2

)f dµ

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

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

)f dµ

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)

2|∇ω|2

)f dµ

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

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

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

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

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.

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

Kahler =⇒ W+ =

− s12− s

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

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

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

∇(fW+) = 0 =⇒ δ(fW+) = 0.

0 ≥∫MW+(ω, ω)|∇ω|2f dµ

δ(fW+) = 0

∇(fW+) = 0 =⇒ δ(fW+) = 0.

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.

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

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

W+(ω, ω) > 0

h = s−2g

W+(ω, ω) > 0

h = s−2g

Theorem A follows by restricting to Einstein case.

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

ponent of E (M).

Application to Almost-Kahler Geometry:

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

s ≥ 0, δW+ = 0.

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

s ≥ 0, δW+ = 0.

In particular, gives a new proof of the following:

s ≥ 0, δW+ = 0.

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

s ≥ 0, δW+ = 0.

(Special case of “Goldberg conjecture.”)

s ≥ 0, δW+ = 0.

Helped motivate the discovery of Theorem A. . .

Tanti auguri, Stefano!

E grazie agli organizzatori

Tanti auguri, Stefano!

E grazie agli organizzatori

per un convegno cosı bello!

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