Almost K ahler 4-manifolds of Constant Holomorphic Sectional - - PowerPoint PPT Presentation

Almost K¨ ahler 4-manifolds of Constant Holomorphic Sectional Curvature are K¨ ahler

• M. Upmeier

JOINT WORK WITH

• M. Lejmi

BASED ON DISCUSSIONS WITH LUIGI VEZZONI

Cogne, January 2018

Preliminaries

Definition

An almost K¨ ahler manifold (M, ω, g, J) is equipped with ω ∈ Ω2(M), J : TM → TM, g metric such that dω = 0, J2 = −1, ω = g(J·, ·).

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Preliminaries

Definition

An almost K¨ ahler manifold (M, ω, g, J) is equipped with ω ∈ Ω2(M), J : TM → TM, g metric such that dω = 0, J2 = −1, ω = g(J·, ·).

Definition

The Hermitian connection (or Chern connection) is ∇XY := Dg

XY −1

2J(Dg

XJ)Y

• AX Y =

.

◮ ∇g = 0, ∇J = 0, but ∇ may have torsion.

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Holomorphic sectional curvature

Hermitian curvature tensor R∇ ∈ Λ2 ⊗ Λ1,1 has fewer symmetries. The Hermitian holomorphic sectional curvature is H(X) := |X|−4 · R∇

X,JX,X,JX,

X ∈ TM. It is called

• 1. constant at p if H(X) = kp for all X ∈ TpM,
• 2. globally constant if H(X) = k for all X ∈ TM.

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Holomorphic sectional curvature

Hermitian curvature tensor R∇ ∈ Λ2 ⊗ Λ1,1 has fewer symmetries. The Hermitian holomorphic sectional curvature is H(X) := |X|−4 · R∇

X,JX,X,JX,

X ∈ TM. It is called

• 1. constant at p if H(X) = kp for all X ∈ TpM,
• 2. globally constant if H(X) = k for all X ∈ TM.

Problem (Gray–Vanhecke 1979)

Classify all manifolds of globally constant holomorphic sectional curvature within your favourite class of almost Hermitian manifolds.

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Statement of Result

Theorem (U.–Lejmi, 2017)

Let M be a closed almost K¨ ahler 4-manifold of globally constant Hermitian holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler–Einstein, holomorphically isometric to: (k > 0) CP2 with the Fubini–Study metric. (k = 0) a complex torus or a hyperelliptic curve with the Ricci-flat K¨ ahler metric. Similar result for k < 0 under assumption that Ricci is J-invariant.

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Statement of Result

Theorem (U.–Lejmi, 2017)

Let M be a closed almost K¨ ahler 4-manifold of globally constant Hermitian holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler–Einstein, holomorphically isometric to: (k > 0) CP2 with the Fubini–Study metric. (k = 0) a complex torus or a hyperelliptic curve with the Ricci-flat K¨ ahler metric. Similar result for k < 0 under assumption that Ricci is J-invariant.

Remark

The above conclusion is known for K¨ ahler manifolds, so we just need to prove integrability.

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Background

Related Work

Balas–Gauduchon 1985 Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is K¨ ahler

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Background

Related Work

Balas–Gauduchon 1985 Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is K¨ ahler Gray–Vanhecke 1979 (Classification of) Almost Hermitian manifolds with constant holomorphic sectional curvature.

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Background

Related Work

Balas–Gauduchon 1985 Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is K¨ ahler Gray–Vanhecke 1979 (Classification of) Almost Hermitian manifolds with constant holomorphic sectional curvature. Sekigawa 1985 On some 4-dimensional compact Einstein almost K¨ ahler manifolds.

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Background

Related Work

Balas–Gauduchon 1985 Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is K¨ ahler Gray–Vanhecke 1979 (Classification of) Almost Hermitian manifolds with constant holomorphic sectional curvature. Sekigawa 1985 On some 4-dimensional compact Einstein almost K¨ ahler manifolds. Armstrong 1997 On four-dimensional almost K¨ ahler manifolds.

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Background

Related Work

Balas–Gauduchon 1985 Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is K¨ ahler Gray–Vanhecke 1979 (Classification of) Almost Hermitian manifolds with constant holomorphic sectional curvature. Sekigawa 1985 On some 4-dimensional compact Einstein almost K¨ ahler manifolds. Armstrong 1997 On four-dimensional almost K¨ ahler manifolds. Lejmi–Vezzoni 2017 Left-invariant structures on almost K¨ ahler 4-dimensional Lie algebras.

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Pointwise Implications (Algebraic Bianchi)

Proposition

Pointwise constant holomorphic sectional curvature H = k is equivalent to

• 1. W − = 0
• 2. ∗ρ = r for two Ricci contractions of R∇:

ρα¯

β = iR∇ γ α¯ βγ ,

rα¯

β = iR∇ γ γ α¯ β,

Moreover, v := Scalg 12 ≤ k 2 with equality if and only if M is K¨ ahler.

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Sketch of Proof for W − = 0

Use R∇

XYZW =Rg XYZW

+ g((∇XAY − ∇Y AX − A[X,Y ])Z, W )

• α∈Λ2⊗Λ2,0+0,2

− g([AX, AY ]Z, W )

• β∈Λ1,1⊗R·ω

. Play off the symmetries of Rg : Λ2 → Λ2 against the assumption

• n R∇ (which gives it a special form).

Rg =

• Λ+

Λ−

W + + Scalg

12 g

R0 RT W − + Scalg

12 g

• ,

Λ2 = Λ+ ⊕ Λ−

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Sketch of Proof for W − = 0

Use R∇

XYZW =Rg XYZW

+ g((∇XAY − ∇Y AX − A[X,Y ])Z, W )

• α∈Λ2⊗Λ2,0+0,2

− g([AX, AY ]Z, W )

• β∈Λ1,1⊗R·ω

. Play off the symmetries of Rg : Λ2 → Λ2 against the assumption

• n R∇ (which gives it a special form).

Rg =  

Rω Λ(2,0)+(0,2) Λ1,1

d · g W +

F

RF (W +

F )T

W +

00 + c 2g

R00 RT

F

RT

00

W − + Scalg

12 g

 

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Sketch of Proof for W − = 0

Use R∇

XYZW =Rg XYZW

+ g((∇XAY − ∇Y AX − A[X,Y ])Z, W )

• α∈Λ2⊗Λ2,0+0,2

− g([AX, AY ]Z, W )

• β∈Λ1,1⊗R·ω

. Play off the symmetries of Rg : Λ2 → Λ2 against the assumption

• n R∇ (which gives it a special form).

R∇ =  

Rω Λ(2,0)+(0,2) Λ1,1 sC 2 g

∗ ??? ??? ∗ ∗  

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Sketch of Proof for W − = 0

Use R∇

XYZW =Rg XYZW

+ g((∇XAY − ∇Y AX − A[X,Y ])Z, W )

• α∈Λ2⊗Λ2,0+0,2

− g([AX, AY ]Z, W )

• β∈Λ1,1⊗R·ω

. Play off the symmetries of Rg : Λ2 → Λ2 against the assumption

• n R∇ (which gives it a special form).

R∇ =  

Rω Λ(2,0)+(0,2) Λ1,1 sC 2 g

RF (W +

F )T

R00 −RT

F sg 12g

 

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Global Impliciations

From the differential Bianchi identity:

Proposition

Let M be a closed almost K¨ ahler 4-manifold of pointwise constant holomorphic sectional curvature k. Then

• M

|R00|2 =

• M

|W +

F |2 + |W + 00|2 + 4(5k − 7v)(k − 2v)

(1) χ = −1 8π2

• M

|W +

00|2 + (60v2 − 72kv + 18k2)

(2) 3 2σ = 1 8π2

• M

2|W +

F |2 + |W + 00|2 + 6(2k − 3v)2

≥ 0 (3) Recall: v := Scalg

12

= k

2 implies K¨

ahler.

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Corollary (Signature zero case)

Let M be closed almost K¨ ahler 4-manifold of pointwise constant holomorphic sectional curvature k. Suppose σ = 0. Then k = 0 and M is K¨ ahler, with a Ricci-flat metric.

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Corollary (Signature zero case)

Let M be closed almost K¨ ahler 4-manifold of pointwise constant holomorphic sectional curvature k. Suppose σ = 0. Then k = 0 and M is K¨ ahler, with a Ricci-flat metric.

Proof.

• 1. From 0 = 3

2σ = 1 8π2

• M 2|W +

F |2 + |W + 00|2 + 6(2k − 3v)2 we

get W +

F = 0, W00 = 0, 2k = 3v.

• 2. Put this into
• M

|R00|2 =

• M

|W +

F |2+|W + 00|2+4(5k−7v)(k−2v) = −4

9k2 Vol(M) to get k = v = 0.

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Corollary (‘Reverse’ Bogomolov–Miyaoka–Yau inequality)

If M is closed almost K¨ ahler of globally constant holomorphic sectional curvature k ≥ 0, then for the Euler characteristic 3σ ≥ χ. Equality holds if and only if M is K¨ ahler–Einstein.

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

Proof.

◮ Suppose that M is not K¨

ahler: v < k

2 somewhere.

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

Proof.

◮ Suppose that M is not K¨

ahler: v < k

2 somewhere. ◮ M c1(TM) ∪ ω =

• M

sC 2π =

• M

3k 2π

≥0

+

• M

k−2v 2π

> 0.

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

Proof.

◮ Suppose that M is not K¨

ahler: v < k

2 somewhere. ◮ M c1(TM) ∪ ω =

• M

sC 2π =

• M

3k 2π

≥0

+

• M

k−2v 2π

> 0.

◮ SW-theory =

⇒ M symplectom. to ruled surface or CP2

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

Proof.

◮ Suppose that M is not K¨

ahler: v < k

2 somewhere. ◮ M c1(TM) ∪ ω =

• M

sC 2π =

• M

3k 2π

≥0

+

• M

k−2v 2π

> 0.

◮ SW-theory =

⇒ M symplectom. to ruled surface or CP2

◮ M = CP2 has 3σ = χ.

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

Proof.

◮ Suppose that M is not K¨

ahler: v < k

2 somewhere. ◮ M c1(TM) ∪ ω =

• M

sC 2π =

• M

3k 2π

≥0

+

• M

k−2v 2π

> 0.

◮ SW-theory =

⇒ M symplectom. to ruled surface or CP2

◮ M = CP2 has 3σ = χ. ◮ M rational =

⇒ σ ≤ 0 = ⇒ σ = 0.

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End of the proof

Theorem

M4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0. Then M is K¨ ahler.

Proof.

◮ Suppose that M is not K¨

ahler: v < k

2 somewhere. ◮ M c1(TM) ∪ ω =

• M

sC 2π =

• M

3k 2π

≥0

+

• M

k−2v 2π

> 0.

◮ SW-theory =

⇒ M symplectom. to ruled surface or CP2

◮ M = CP2 has 3σ = χ. ◮ M rational =

⇒ σ ≤ 0 = ⇒ σ = 0.

◮ By previous propositions, ‘=’ implies K¨

ahler, contradiction!

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