hodge theory complex manifolds. by william m. faucette adapted from lectures by mark andrea a....
TRANSCRIPT
Hodge Theory
Complex Manifolds
by William M. Faucette
Adapted from lectures by
Mark Andrea A. Cataldo
Structure of Lecture
Conjugations
Tangent bundles on a complex manifold
Cotangent bundles on a complex manifold
Standard orientation of a complex manifold
Almost complex structure
Complex-valued forms
Dolbeault cohomology
Conjugations
Conjugations
Let us recall the following distinct notions of conjugation.
First, there is of course the usual conjugation in C:
Conjugations
Let V be a real vector space and
be its complexification. There is a natural R-linear isomorphism given by
Tangent Bundles on a Complex Manifold
Tangent Bundles on a Complex Manifold
Let X be a complex manifold of dimension n, x2X and
be a holomorphic chart for X around x. Let zk=xk+iyk for k=1, . . . , N.
Tangent Bundles on a Complex Manifold
TXR) is the real tangent bundle on X. The fiber TX,xR) has real rank 2n and it is the real span
Tangent Bundles on a Complex Manifold
TXC):= TXR)RC is the complex tangent bundle on X. The fiber TX,xC) has complex rank 2n and it is the complex span
Tangent Bundles on a Complex Manifold
Often times it is more convenient to use a basis for the complex tangent space which better reflects the complex structure. Define
Tangent Bundles on a Complex Manifold
With this notation, we have
Tangent Bundles on a Complex Manifold
Clearly we have
Tangent Bundles on a Complex Manifold
In general, a smooth change of coordinates does not leave invariant the two subspaces
Tangent Bundles on a Complex Manifold
However, a holomorphic change of coordinates does leave invariant the two subspaces
Tangent Bundles on a Complex Manifold
TX is the holomorphic tangent bundle on X. The fiber TX,x has complex rank n and it is the complex span
TX is a holomorphic vector bundle.
Tangent Bundles on a Complex Manifold
TX is the anti-holomorphic tangent bundle on X. The fiber TX,x has complex rank n and it is the complex span
TX is an anti-holomorphic vector bundle.
Tangent Bundles on a Complex Manifold
We have a canonical injection and a canonical internal direct sum decomposition into complex sub-bundles:
Tangent Bundles on a Complex Manifold
Composing the injection with the projections we get canonical real isomorphisms
Tangent Bundles on a Complex Manifold
The conjugation map
is a real linear isomorphism which is not complex linear.
Tangent Bundles on a Complex Manifold
The conjugation map induces real linear isomorphism
and a complex linear isomorphism
Cotangent Bundles on Complex Manifolds
Cotangent Bundles on Complex Manifolds
Let {dx1, . . . , dxn, dy1, . . . , dyn} be the dual basis to {x1, . . . , xn, y1, . . . , yn}. Then
Cotangent Bundles on Complex Manifolds
We have the following vector bundles on X:
TX*(R), the real cotangent bundle, with fiber
Cotangent Bundles on Complex Manifolds
TX*(C), the complex cotangent bundle, with fiber
Cotangent Bundles on Complex Manifolds
TX*(C), the holomorphic cotangent bundle, with fiber
Cotangent Bundles on Complex Manifolds
TX*(C), the anti-holomorphic cotangent bundle, with fiber
Cotangent Bundles on Complex Manifolds
We have a canonical injection and a canonical internal direct sum decomposition into complex sub-bundles:
Cotangent Bundles on Complex Manifolds
Composing the injection with the projections we get canonical real isomorphisms
Cotangent Bundles on Complex Manifolds
The conjugation map
is a real linear isomorphism which is not complex linear.
Cotangent Bundles on Complex Manifolds
The conjugation map induces real linear isomorphism
and a complex linear isomorphism
Cotangent Bundles on Complex Manifolds
Let f(x1,y1,…, xn, yn)= u(x1,y1,…, xn, yn)+ i v(x1,y1,…, xn, yn) be a smooth complex-valued function in a neighborhood of x. Then
The Standard Orientation of a Complex Manifold
Standard Orientation
Proposition: A complex manifold X admits a canonical orientation.
If one looks at the determinant of the transition matrix of the tangent bundle of X, the Cauchy-Riemann equations immediately imply that this determinant must be positive.
Standard Orientation
If (U,{z1,…,zn}) with zj=xj+i yj, the real 2n-form
is nowhere vanishing in U.
Standard Orientation
Since the holomorphic change of coordinates is orientation preserving, these non-vanishing differential forms patch together using a partition of unity argument to give a global non-vanish- ing differential form.
This differential form is the standard orientation of X.
The Almost Complex Structure
Almost Complex Structure
The holomorphic tangent bundle TX of a complex manifold X admits the complex linear automorphism given by multiplication by i.
Almost Complex Structure
By the isomorphism
We get an automorphism J of the real tangent bundle TX(R) such that J2=-Id. The same is true for TX* using the dual map J*.
Almost Complex Structure
An almost complex structure on a real vector space VR of finite even dimension 2n is a R-linear automorphism
Almost Complex Structure
An almost complex structure is equivalent to endowing VR with a structure of a complex vector space of dimension n.
Almost Complex Structure
Let (VR, JR) be an almost complex structure. Let VC:= VRRC and JC:= JRIdC: VC VC be the complexification of JR.
The automorphism JC of VC has eigenvalues i and -i.
Almost Complex StructureThere are a natural inclusion and a natural
direct sum decomposition
where the subspace VRVC is the fixed locus of the
conjugation map associated with the complexification.
Almost Complex Structure V and V are the JCeigenspaces
corresponding to the eigenvalues i and -i, respectively,
since JC is real, that is, it fixes VRVC, JC
commutes with the natural conjugation map and V and V are exchanged by this conjugation map,
Almost Complex Structure there are natural R-linear isomorphisms
coming from the inclusion and the projections to the direct summands
and complex linear isomorphisms
Almost Complex Structure The complex vector space defined by
the complex structure is C-linearly isomorphic to V.
Almost Complex Structure
The same considerations are true for the almost complex structure (VR*, JR*). We have
Complex-Valued Forms
Complex-Valued Forms
Let M be a smooth manifold. Define the complex valued smooth p-forms as
Complex-Valued Forms
The notion of exterior differentiation extends to complex-valued differential forms:
Complex-Valued Forms
Let X be a complex manifold of dimension n, x2X, (p,q) be a pair of non-negative integers and define the complex vector spaces
Complex-Valued Forms
There is a canonical internal direct sum decomposition of complex vector spaces
Complex-Valued Forms
Definition: The space of (p,q)-forms on X
is the complex vector space of smooth sections of the smooth complex vector bundle p,q(TX*).
Complex-Valued Forms
There is a canonical direct sum decomposition
and
Complex-Valued Forms
Let l=p+q and consider the natural projections
Define operators
Complex-Valued Forms
Note that
Also,
Dolbeault Cohomology
Dolbeault Cohomology
Definition: Fix p and q. The Dolbeault complex is the complex of vector spaces
Dolbeault Cohomology
The Dolbeault cohomology groups are the cohomology groups of the complex
Dolbeault Cohomology
That is,
Dolbeault CohomologyTheorem: (Grothendieck-Dolbeault Lemma)
Let q>0. Let X be a complex manifold and u2Ap,q(X) be such that du=0. Then, for every point x2X, there is an open neighborhood U of x in X and a form v2Ap,q-1(U) such that
Dolbeault Cohomology
The Grothendieck-Dolbeault Lemma guarantees that Dolbeault cohomology is locally trivial.
Dolbeault CohomologyFor those familiar with sheaves and sheaf
cohomology, the Dolbeault Lemma tells us that the fine sheaves Ap,q
X of germs of C (p,q)-forms give a fine resolution of the sheaf p
X of germs of holomorphic p-forms on X. Hence, by the abstract deRham theorem