Complex Differential Geometry Complex Differential Geometry Roger Bielawski July 27, 2009 Complex...

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Transcript of Complex Differential Geometry Complex Differential Geometry Roger Bielawski July 27, 2009 Complex...

  • Complex Differential Geometry

    Roger Bielawski

    July 27, 2009

    Complex manifolds

    A complex manifold of dimension m is a topological manifold (M,U), such that the transition functions φU ◦φ−1V are holomorphic maps between open subsets of Cm for every intersecting U,V ∈U.

    We have a holomorphic atlas (or “we have local complex coordinates on M.”)

    Remark: Obviously, a complex manifold of dimension m is a smooth (real) manifold of dimension 2m. We will denote the underlying real manifold by MR.

  • Example

    Complex projective space CPm - the set of (complex) lines in Cm+1, i.e. the set of equivalence classes of the relation

    (z0, . . . ,zm)∼ (αz0, . . . ,αzm), ∀α ∈ C∗

    on Cm+1−{0}. In other words CPm =

    ( Cm+1−{0}

    ) /∼.

    The complex charts are defined as for RPm:

    Ui = {[z0, . . . ,zm];zi 6= 0} , i = 1, . . . ,m

    φi : Ui → Cm, φi([z0, . . . ,zm]) = (

    z0 zi

    , . . . , zi−1 zi

    , zi+1 zi

    , . . . , zm zi

    ) .

    Example

    1. Complex Grassmanian Grp(Cm) - the set of all p-dimensional vector subspaces of Cm. 2. The torus T 2 ' S1×S1 is a complex manifold of dimension 1. 3. As for smooth manifolds one gets plenty of examples as level sets of submersions f : Cm+1 → C. If f is holomorphic and df (the holomorphic differential) does not vanish at any point of f−1(c), then f−1(c) is a holomorphic manifold.For example Fermat hypersurfaces:(

    (z0, . . . ,zm); m

    ∑ i=0

    zdii = 1

    ) , d0, . . . ,dm ∈ N.

    4. Similarly, homogeneous f give complex submanifolds of CPm. 5. Complex Lie groups: GL(n,C), O(n,C), etc.

  • Almost complex manifolds

    A complex manifold of (complex) dimension m is also a smooth real manifold of (real) dimension 2m. Obviously, the converse is not true, but it turns out that there is a characterisation of complex manifolds among real ones, which is much simpler than the existence of a holomorphic atlas.

    Identifying R2n with Cn is equivalent to giving a linear map j : R2n → R2n satisfying j2 =−Id. Let x ∈M and let (U,φU) be a holomorphic chart around x . Define an endomorphism JU of TxMR by JU(X) = φ−1U ◦ j ◦φU(X). JU does not depend on U, and so we have an endomorphism Jx : TxMR → TxMR satisfying J2x =−Id. The collection of all Jx , x ∈M, defines a tensor J (of type (1,1)), which satisfies J2 =−Id and which is called an almost complex structure on MR.

    Definition

    An almost complex manifold is a pair (M,J), where M is a smooth real manifold and J : TM → TM is an almost complex structure.

    Thus a complex manifold is an almost complex manifold. The converse is not true, but the existence of complex coordinates follows from vanishing of another tensor. Remark: Obviously, an almost complex manifold has an even dimension, but no every even-dimensional smooth manifold admits an almost complex structure (e.g. S4 does not). Remark: S6 admits an almost complex structure, but it is still an open problem, whether it can be made into a complex manifold.

    Definition

    A smooth map f : (M1,J1)→ (M2,J2) between two complex manifolds is called holomorphic if ψV ◦ f ◦φ−1U is a holomorphic map between open subsets in Cn, for any charts (U,φU) in M1 and (V ,ψV ) in M2. This is equivalent to the differential of f commuting with the complex structures, i.e. f∗ ◦ J1 = J2 ◦ f∗.

  • The complexified tangent bundle

    Let (M,J) be an almost complex manifold. Since J is linear, we can diagonalise it, but only after complexifying the tangent spaces. We define the complexified tangent bundle:

    T CM = TM⊗R C,

    and we extend all linear endomorphisms and linear differential operators from TM to T CM by C-linearity. Let T 1,0M and T 0,1M denote the +i- and the −i-eigenbundle of J. It is easy to verify the following:

    T 1,0M = {X − iJX ; X ∈ TM}, T 0,1M = {X + iJX ; X ∈ TM},

    T CM = T 1,0M⊕T 0,1M.

    Remark −J is also an almost complex structure. T 1,0(M,−J) = T 0,1(M,J).

    Holomorphic tangent vectors

    Let M be a complex manifold. Recall that we have three notions of a tangent space of M at a point p: T Rp M = TM - the real tangent space,

    T Cp M - the complexified tangent space, and T 1,0 p M (resp. T

    0,1 p M) - the

    holomorphic tangent space (resp. antiholomorphic tangent space). Let us choose local complex coordinates z = (z1, . . . ,zn) near z. If we write zi = xi +

    √ −1yi , then:

    TpM = R {

    ∂ ∂xi

    , ∂

    ∂yi

    } ,

    T Cp M = C {

    ∂ ∂xi

    , ∂

    ∂yi

    } = C

    { ∂

    ∂zi ,

    ∂ ∂z̄i

    } ,

    where

    ∂ ∂zi

    = 1 2

    ( ∂

    ∂xi − √ −1 ∂

    ∂yi

    ) ,

    ∂ ∂z̄i

    = 1 2

    ( ∂

    ∂xi + √ −1 ∂

    ∂yi

    ) .

  • Consequently:

    T 1,0p M = C {

    ∂ ∂zi

    } , T 0,1p M = C

    { ∂

    ∂z̄i

    } ,

    and T 1,0p M (resp. T 0,1 p M) is the space of derivations which vanish on

    anti-holomorphic functions (resp. holomorphic functions). Now observe that:

    V ,W ∈ Γ ( T 1,0M

    ) =⇒ [V ,W ] ∈ Γ

    ( T 1,0M

    ) ,

    and V ,W ∈ Γ

    ( T 0,1M

    ) =⇒ [V ,W ] ∈ Γ

    ( T 0,1M

    ) .

    The Newlander-Nirenberg Theorem

    Theorem (Newlander-Nirenberg)

    Let (M,J) be an almost complex manifold. The almost complex structure J comes from a holomorphic atlas if and only if

    V ,W ∈ Γ ( T 0,1M

    ) =⇒ [V ,W ] ∈ Γ

    ( T 0,1M

    ) .

    An almost complex structure, which comes from complex coordinates is called a complex structure. Remark: An equivalent condition is the vanishing of the Nijenhuis tensor:

    N(X ,Y ) = [JX ,JY ]− [X ,Y ]− J[X ,JY ]− J[JX ,Y ].

  • As for the proof: we just have seen the ”only if” part. The ”if” part is very hard. See Kobayashi & Nomizu for a proof under an additional assumption that M and J are real-analytic, and Hörmander’s ”Introduction to Complex Analysis in Several Variables” for a proof in full generality.

    Definition

    A section Z of T 1,0M is a holomorphic vector field if Z (f ) is holomorphic for every locally defined holomorphic function f .

    In local coordinates Z = ∑ni=1 gi ∂∂zi , where all gi are holomorphic.

    Definition

    A real vector field X ∈ Γ(TM) is called real holomorphic if its (1,0)-component X − iJX is a holomorphic vector field.

    Lemma The following conditions are equivalent:

    (1) X is real holomorphic.

    (2) X is an infinitesimal automorphism of the complex structure J, i.e. LX J = 0 (LX - the Lie derivative).

    (3) The flow of X consists of holomorphic transformations of M.

  • The dual picture:

    We can decompose the complexified cotangent bundle Λ1M⊗C into

    Λ1,0M = {

    ω ∈ Λ1M⊗C; ω(Z ) = 0 ∀Z ∈ T 0,1M }

    and Λ0,1M =

    { ω ∈ Λ1M⊗C; ω(Z ) = 0 ∀Z ∈ T 1,0M

    } .

    We have Λ1M⊗C = Λ1,0M⊕Λ0,1M.

    and, consequently, we can decompose the k -th exterior power of Λ1M⊗C as

    Λk M⊕C = Λk ( Λ1,0M⊕Λ0,1M

    ) =

    M p+q=k

    Λp ( Λ1,0M

    ) ⊗Λq

    ( Λ0,1M

    ) .

    We write Λp,qM = Λp

    ( Λ1,0M

    ) ⊗Λq

    ( Λ0,1M

    ) ,

    so that Λk M⊕C =

    M p+q=k

    Λp,qM.

    Sections of Λp,qM are called forms of type (p,q) and their space is denoted by Ωp,qM. In local coordinates, forms of type (p,q) are generated by

    dzi1 ∧·· ·∧dzip ∧dz̄j1 ∧dz̄jq .

    Remark: The above decomposition of Λk M⊗C is valid on any almost complex manifold. The difference between ”complex” and ”almost complex” lies in the behaviour of the exterior derivative d .

    Theorem

    Let (M,J) be an almost complex manifold of dimension 2n. The following conditions are equivalent:

    (i) J is a complex structure.

    (ii) dΩ1,0M ⊂ Ω2,0M⊕Ω1,1M. (iii) dΩp,qM ⊂ Ωp+1,qM⊕Ωp,q+1M for all 0≤ p,q ≤m.

  • Proof. The only non-trivial bit is (ii) =⇒ (i). Use the following elementary formula for the exterior derivative of a 1-form:

    2dω(Z ,W ) = Z ( ω(W )

    ) −W

    ( ω(Z )

    ) −ω([Z ,W ]).

    Using statement (iii), we decompose the exterior derivative d : Ωk M → Ωk+1M as d = ∂+ ∂̄, where ∂ : Ωp,qM → Ωp+1,qM, ∂̄ : Ωp,qM → Ωp,q+1M. Lemma

    ∂2 = 0, , ∂̄2 = 0, ∂∂̄+ ∂̄∂ = 0.

    Proof.

    0 = d2 = (∂+ ∂̄)2 = ∂2 + ∂̄2 +(∂∂̄+ ∂̄∂) and the three operators in the last term take values in different subbundles of Λ∗M⊗C.

    The Dolbeault operator

    Definition

    The operator ∂̄ : Ωp,qM → Ωp,q+1M is called the Dolbeault operator. A p-form ω of type (p,0) is called holomorphic if ∂̄ω = 0.

    One uses ∂̄ to define Dolbeault cohomology groups of a complex manifold, analogously to the de Rham cohomology:

    Z p,q ∂̄

    (M) = {ω ∈ Ωp,q(M); ∂̄ω = 0} − ∂̄-closed forms.

    Hp,q ∂̄

    (M) = Z p,q

    ∂̄ (M)

    ∂̄Ωp,q−1(M) .

    A holomorphic map f : M → N between complex manifolds induces a map

    f ∗ : Hp,q ∂̄

    (N)→ Hp,q ∂̄

    (M).

    Warning: Dolbeault cohomology is not a topological invariant: it depends on the complex struct