Hodge theory, lecture 10 M. Verbitsky Hodge theory lecture 10: Newlander-Nirenberg theorem NRU HSE, Moscow Misha Verbitsky, February 24, 2018 1

Hodge theory, lecture 10 M. Verbitsky Almost complex manifolds (reminder) DEFINITION: Let I : TM − → TM be an endomorphism of a tangent bundle satisfying I 2 = − Id. Then I is called almost complex structure operator , and the pair ( M, I ) an almost complex manifold . EXAMPLE: M = C n , with complex coordinates z i = x i + √− 1 y i , and I ( d/dx i ) = d/dy i , I ( d/dy i ) = − d/dx i . DEFINITION: Let ( V, I ) be a space equipped with a complex structure → V , I 2 = − Id. The Hodge decomposition V ⊗ R C := V 1 , 0 ⊕ V 0 , 1 I : V − is defined in such a way that V 1 , 0 is a √− 1 -eigenspace of I , and V 0 , 1 a −√− 1 -eigenspace. DEFINITION: A function f : → C on an almost complex manifold is M − f ∈ Λ 1 , 0 ( M ). called holomorphic if d REMARK: For some almost complex manifolds, there are no holomorphic functions at all , even locally. 2

Hodge theory, lecture 10 M. Verbitsky Complex manifolds and almost complex manifolds (reminder) DEFINITION: Standard almost complex structure is I ( d/dx i ) = d/dy i , I ( d/dy i ) = − d/dx i on C n with complex coordinates z i = x i + √− 1 y i . DEFINITION: A map Ψ : ( M, I ) − → ( N, J ) from an almost complex mani- fold to an almost complex manifold is called holomorphic if Ψ ∗ (Λ 1 , 0 ( N )) ⊂ Λ 1 , 0 ( M ). REMARK: This is the same as d Ψ being complex linear; for standard almost complex structures, this is the same as the coordinate components of Ψ being holomorphic functions. DEFINITION: A complex manifold is a manifold equipped with an at- las with charts identified with open subsets of C n and transition functions holomorphic. 3

Hodge theory, lecture 10 M. Verbitsky Integrability of almost complex structures (reminder) DEFINITION: An almost complex structure I on a manifold is called inte- grable if any point of M has a neighbourhood U diffeomorphic to an open subset of C n , in such a way that the almost complex structure I is induced by the standard one on U ⊂ C n . CLAIM: Complex structure on a manifold M uniquely determines an integrable almost complex structure, and is determined by it. Proof: Complex structure on a manifold M is determined by the sheaf of holomorphic functions O M , because d O M generates Λ 1 , 0 ( M ), and O M is de- f ∈ Λ 1 , 0 ( M ) } . termined by I , because O M = { f | d 4

Hodge theory, lecture 10 M. Verbitsky Frobenius form (reminder) CLAIM: Let B ⊂ TM be a sub-bundle of a tangent bundle of a smooth manifold. Given vector fiels X, Y ∈ B , consider their commutator [ X, Y ], and lets Ψ( X, Y ) ∈ TM/B be the projection of [ X, Y ] to TM/B . Then Ψ( X, Y ) is C ∞ ( M ) -linear in X , Y : Ψ( fX, Y ) = Ψ( X, fY ) = f Ψ( X, Y ) . Proof: Leibnitz identity gives [ X, fY ] = f [ X, Y ] + X ( f ) Y , and the second term belongs to B , hence does not influence the projection to TM/B . DEFINITION: This form is called the Frobenius form of the sub-bundle B ⊂ TM . This bundle is called involutive , or integrable , or holonomic if Ψ = 0. 5

Hodge theory, lecture 10 M. Verbitsky Formal integrability (reminder) DEFINITION: An almost complex structure I on ( M, I ) is called formally integrable if [ T 1 , 0 M, T 1 , 0 ] ⊂ T 1 , 0 , that is, if T 1 , 0 M is involutive. DEFINITION: The Frobenius form Ψ ∈ Λ 2 , 0 M ⊗ TM is called the Nijenhuis tensor . CLAIM: If a complex structure I on M is integrable, it is formally integrable. Proof: Locally, the bundle T 1 , 0 ( M ) is generated by d/dz i , where z i are com- plex coordinates. These vector fields commute, hence satisfy [ d/dz i , d/dz j ] ∈ T 1 , 0 ( M ). This means that the Frobenius form vanishes. THEOREM: (Newlander-Nirenberg) A complex structure I on M is integrable if and only if it is formally integrable. Proof: (real analytic case) later today. REMARK: In dimension 1, formal integrability is automatic. Indeed, T 1 , 0 M is 1-dimensional, hence all skew-symmetric 2-forms on T 1 , 0 M vanish. 6

Hodge theory, lecture 10 M. Verbitsky Real analytic manifolds DEFINITION: Real analytic function on an open set U ⊂ R n is a function which admits a Taylor expansion near each point x ∈ U : a i 1 ,...,i n t i 1 1 t i 2 2 ...t i n � f ( z 1 + t 1 , z 2 + t 2 , ..., z n + t n ) = n . i 1 ,...,i n (here we assume that the real numbers t i satisfy | t i | < ε , where ε depends on f and M ). REMARK: Clearly, real analytic functions constitute a sheaf. DEFINITION: A real analytic manifold is a ringed space which is locally isomorphic to an open ball B ⊂ R n with the sheaf of real analytic functions. 7

Hodge theory, lecture 10 M. Verbitsky Involutions → M such that ι 2 = Id M . DEFINITION: An involution is a map ι : M − EXERCISE: Prove that any linear involution on a real vector space V is diagonalizable, with eigenvalues ± 1. Theorem 1: Let M be a smooth manifold, and ι : M − → M an involutiin. Then the fixed point set N of ι is a smooth submanifold. Proof. Step 1: Inverse function theorem. Let m ∈ M be a point on a smooth k -dimensional manifold and f 1 , ..., f k functions on M such that their differentials d f 1 , ..., d f k are linearly independent in m . Then f 1 , ..., f k define a coordinate system in a neighbourhood of a , giving a diffeomorphism of this neighbourhood to an open ball. Step 2: Assume that dι has k eigenvalues 1 on T m M , and n − k eigenvalues -1. Choose a coordinate system x 1 , ..., x n on M around a point m ∈ N such that dx 1 | m , ..., dx k | m are ι -invariant and dx k +1 | m , ..., dx n | m are ι -anti-invariant. Let y 1 = x 1 + ι ∗ x 1 , y 2 = x 2 + ι ∗ x 2 , ... y k = x k + ι ∗ x k , and y k +1 = x k +1 − ι ∗ x k +1 , y k +2 = x k +2 − ι ∗ x k +2 , ... y n = x n − ι ∗ x n . Since dx i | m = xy i | m , these differentials are linearly independent in m . By Step 1, functions y i define an ι -invariant coordinate system on an open neighbourhood of m , with N given by equations y k +1 = y k +2 = ... = y n = 0 . 8

Hodge theory, lecture 10 M. Verbitsky Real structures → M such that ι 2 = Id M . A DEFINITION: An involution is a map ι : M − real structure on a complex vector space V = C n is an R -linear involution ι : V − → V such that ι ( λx ) = λι ( x ) for any λ ∈ C . DEFINITION: A map Ψ : M − → M on an almost complex manifold ( M, I ) is called antiholomorphic if d Ψ( I ) = − I . A function f is called antiholo- morphic if f is holomorphic. EXERCISE: Prove that antiholomorphic function on M defines an an- tiholomorphic map from M to C . EXERCISE: Let ι be a smooth map from a complex manifold M to itself. Prove that ι is antiholomorphic if and only if ι ∗ ( f ) is antiholomorphic for any holomorphic function f on U ⊂ M . DEFINITION: A real structure on a complex manifold M is an antiholo- morphic involution τ : M − → M . EXAMPLE: Complex conjugation defines a real structure on C n . 9

Hodge theory, lecture 10 M. Verbitsky Real analytic manifolds and real structures PROPOSITION: Let M R ⊂ M C be a fixed point set of an antiholomorphic involution ι , U i a complex analytic atlas, and Ψ ij : U ij − → U ij the gluing functions. Then, for an appropriate choice of coordinate systems all Ψ ij are real analytic on M R , and define a real analytic atlas on the manifold M R . Proof. Step 1: Let z 1 , ..., z n be a holomorphic coordinate system on M C in a neighbourhood of m ∈ M R such that ι ( dz i ) = dz i in T ∗ m M . Such a coordinate system can be chosen by taking linear functions with prescribed differentials Replacing z i by x i := z i + ι ∗ ( z i ) , we obtain another coordinate in m . system x i on M (compare with Theorem 1). Step 2: This new coordinate system satisfies ι ∗ x i = x i , hence M R in these coordinates is giving by equation im x 1 = im x 2 = ... = im x n = 0. All gluing functions from such coordinate system to another one of this type satisfy Ψ ij ( z i ) = Ψ ij ( z i ) , hence they are real on M R . 10

Hodge theory, lecture 10 M. Verbitsky Real analytic manifolds and real structures (2) PROPOSITION: Any real analytic manifold can be obtained from this construction. Proof. Step 1: Let { U i } be a locally finite atlas of a real analytic manifold M , and Ψ ij : U ij − → U ij the gluing map. We realize U i as an open ball with compact closure in Re( C n ) = R n . By local finiteness, there are only finitely many such Ψ ij for any given U i . Denote by B ε an open ball of radius ε in the n -dimensional real space im( C n ). Step 2: Let ε > 0 be a sufficiently small real number such that all Ψ ij can U i := U i × B ε ⊂ C n . be extended to gluing functions ˜ Ψ ij on the open sets ˜ Then (˜ U i , Ψ ij ) is an atlas for a complex manifold M C . Since all Ψ ij are real, they are preserved by natural involution acting on B ε as − 1 and on U i as identity. This involution defines a real structure on M C . Clearly, M is the set of its fixed points. 11

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