Almost K¨ ahler 4-manifolds of Constant Holomorphic Sectional Curvature are K¨ ahler M. Upmeier JOINT WORK WITH M. Lejmi BASED ON DISCUSSIONS WITH L UIGI V EZZONI Cogne, January 2018

Preliminaries Definition An almost K¨ ahler manifold ( M , ω, g , J ) is equipped with ω ∈ Ω 2 ( M ) , J : TM → TM , g metric such that J 2 = − 1 , d ω = 0 , ω = g ( J · , · ) . 2 / 11

Preliminaries Definition An almost K¨ ahler manifold ( M , ω, g , J ) is equipped with ω ∈ Ω 2 ( M ) , J : TM → TM , g metric such that J 2 = − 1 , d ω = 0 , ω = g ( J · , · ) . Definition The Hermitian connection (or Chern connection) is X Y − 1 ∇ X Y := D g 2 J ( D g X J ) Y . � �� � A X Y = ◮ ∇ g = 0, ∇ J = 0, but ∇ may have torsion. 2 / 11

Holomorphic sectional curvature Hermitian curvature tensor R ∇ ∈ Λ 2 ⊗ Λ 1 , 1 has fewer symmetries. The Hermitian holomorphic sectional curvature is H ( X ) := | X | − 4 · R ∇ X ∈ TM . X , JX , X , JX , It is called 1. constant at p if H ( X ) = k p for all X ∈ T p M , 2. globally constant if H ( X ) = k for all X ∈ TM . 3 / 11

Holomorphic sectional curvature Hermitian curvature tensor R ∇ ∈ Λ 2 ⊗ Λ 1 , 1 has fewer symmetries. The Hermitian holomorphic sectional curvature is H ( X ) := | X | − 4 · R ∇ X ∈ TM . X , JX , X , JX , It is called 1. constant at p if H ( X ) = k p for all X ∈ T p M , 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. 3 / 11

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) C P 2 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. 4 / 11

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) C P 2 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. 4 / 11

Background Related Work Balas–Gauduchon 1985 Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is K¨ ahler 5 / 11

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. 5 / 11

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. 5 / 11

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. 5 / 11

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. 5 / 11

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 ∇ : γ β = iR ∇ γ β = iR ∇ r α ¯ ρ α ¯ βγ , β , α ¯ γ α ¯ Moreover, v := Scal g ≤ k 12 2 with equality if and only if M is K¨ ahler. 6 / 11

Sketch of Proof for W − = 0 Use XYZW = R g R ∇ XYZW + g (( ∇ X A Y − ∇ Y A X − A [ X , Y ] ) Z , W ) − g ([ A X , A Y ] Z , W ) . � �� � � �� � β ∈ Λ 1 , 1 ⊗ R · ω α ∈ Λ 2 ⊗ Λ 2 , 0+0 , 2 Play off the symmetries of R g : Λ 2 → Λ 2 against the assumption on R ∇ (which gives it a special form). Λ + Λ − � W + + Scal g � 12 g R 0 R g = Λ 2 = Λ + ⊕ Λ − , W − + Scal g R T 12 g 0 7 / 11

Sketch of Proof for W − = 0 Use XYZW = R g R ∇ XYZW + g (( ∇ X A Y − ∇ Y A X − A [ X , Y ] ) Z , W ) − g ([ A X , A Y ] Z , W ) . � �� � � �� � β ∈ Λ 1 , 1 ⊗ R · ω α ∈ Λ 2 ⊗ Λ 2 , 0+0 , 2 Play off the symmetries of R g : Λ 2 → Λ 2 against the assumption on R ∇ (which gives it a special form). Λ 1 , 1 Λ (2 , 0)+(0 , 2) R ω 0 W + d · g R F F R g = ( W + W + 00 + c F ) T 2 g R 00 W − + Scal g R T R T 12 g 00 F 7 / 11

Sketch of Proof for W − = 0 Use XYZW = R g R ∇ XYZW + g (( ∇ X A Y − ∇ Y A X − A [ X , Y ] ) Z , W ) − g ([ A X , A Y ] Z , W ) . � �� � � �� � β ∈ Λ 1 , 1 ⊗ R · ω α ∈ Λ 2 ⊗ Λ 2 , 0+0 , 2 Play off the symmetries of R g : Λ 2 → Λ 2 against the assumption on R ∇ (which gives it a special form). Λ 1 , 1 Λ (2 , 0)+(0 , 2) R ω 0 s C 2 g 0 ∗ R ∇ = ??? 0 ??? ∗ 0 ∗ 7 / 11

Sketch of Proof for W − = 0 Use XYZW = R g R ∇ XYZW + g (( ∇ X A Y − ∇ Y A X − A [ X , Y ] ) Z , W ) − g ([ A X , A Y ] Z , W ) . � �� � � �� � β ∈ Λ 1 , 1 ⊗ R · ω α ∈ Λ 2 ⊗ Λ 2 , 0+0 , 2 Play off the symmetries of R g : Λ 2 → Λ 2 against the assumption on R ∇ (which gives it a special form). Λ 1 , 1 Λ (2 , 0)+(0 , 2) R ω 0 s C 2 g 0 R F R ∇ = ( W + F ) T 0 R 00 s g − R T 0 12 g F 7 / 11

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 � � | R 00 | 2 = F | 2 + | W + 00 | 2 + 4(5 k − 7 v )( k − 2 v ) | W + (1) M M � χ = − 1 00 | 2 + (60 v 2 − 72 kv + 18 k 2 ) | W + (2) 8 π 2 M � 3 1 F | 2 + | W + 00 | 2 + 6(2 k − 3 v ) 2 2 | W + 2 σ = ≥ 0 (3) 8 π 2 M Recall: v := Scal g = k 2 implies K¨ ahler. 12 8 / 11

Corollary (Signature zero case) ahler 4 -manifold of pointwise constant Let M be closed almost K¨ holomorphic sectional curvature k. Suppose σ = 0 . Then k = 0 and M is K¨ ahler, with a Ricci-flat metric. 9 / 11

Corollary (Signature zero case) ahler 4 -manifold of pointwise constant Let M be closed almost K¨ holomorphic sectional curvature k. Suppose σ = 0 . Then k = 0 and M is K¨ ahler, with a Ricci-flat metric. Proof. � F | 2 + | W + 00 | 2 + 6(2 k − 3 v ) 2 we 1. From 0 = 3 1 M 2 | W + 2 σ = 8 π 2 get W + F = 0 , W 00 = 0 , 2 k = 3 v . 2. Put this into � � 00 | 2 +4(5 k − 7 v )( k − 2 v ) = − 4 | R 00 | 2 = 9 k 2 Vol( M ) | W + F | 2 + | W + M M to get k = v = 0. 9 / 11

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 3 σ ≥ χ. Equality holds if and only if M is K¨ ahler–Einstein. 10 / 11

End of the proof Theorem M 4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0 . Then M is K¨ ahler. 11 / 11

End of the proof Theorem M 4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0 . Then M is K¨ ahler. Proof. ahler: v < k ◮ Suppose that M is not K¨ 2 somewhere. 11 / 11

End of the proof Theorem M 4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0 . Then M is K¨ ahler. Proof. ahler: v < k ◮ Suppose that M is not K¨ 2 somewhere. � 3 k ◮ � � � s C k − 2 v M c 1 ( TM ) ∪ ω = 2 π = + > 0 . M M 2 π 2 π M � �� � ≥ 0 11 / 11

End of the proof Theorem M 4 closed almost K¨ ahler of constant holomorphic sectional curvature k ≥ 0 . Then M is K¨ ahler. Proof. ahler: v < k ◮ Suppose that M is not K¨ 2 somewhere. � 3 k ◮ � � � s C k − 2 v M c 1 ( TM ) ∪ ω = 2 π = + > 0 . M M 2 π 2 π M � �� � ≥ 0 ◮ SW-theory = ⇒ M symplectom. to ruled surface or C P 2 11 / 11

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