hodge theory
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/1.jpg)
Hodge Theory
Complex Manifolds
![Page 2: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/2.jpg)
by William M. Faucette
Adapted from lectures by
Mark Andrea A. Cataldo
![Page 3: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/3.jpg)
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
![Page 4: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/4.jpg)
Conjugations
![Page 5: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/5.jpg)
Conjugations
Let us recall the following distinct notions of conjugation.
First, there is of course the usual conjugation in C:
![Page 6: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/6.jpg)
Conjugations
Let V be a real vector space and
be its complexification. There is a natural R-linear isomorphism given by
![Page 7: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/7.jpg)
Tangent Bundles on a Complex Manifold
![Page 8: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/8.jpg)
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.
![Page 9: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/9.jpg)
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
![Page 10: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/10.jpg)
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
![Page 11: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/11.jpg)
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
![Page 12: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/12.jpg)
Tangent Bundles on a Complex Manifold
With this notation, we have
![Page 13: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/13.jpg)
Tangent Bundles on a Complex Manifold
Clearly we have
![Page 14: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/14.jpg)
Tangent Bundles on a Complex Manifold
In general, a smooth change of coordinates does not leave invariant the two subspaces
![Page 15: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/15.jpg)
Tangent Bundles on a Complex Manifold
However, a holomorphic change of coordinates does leave invariant the two subspaces
![Page 16: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/16.jpg)
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.
![Page 17: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/17.jpg)
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.
![Page 18: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/18.jpg)
Tangent Bundles on a Complex Manifold
We have a canonical injection and a canonical internal direct sum decomposition into complex sub-bundles:
![Page 19: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/19.jpg)
Tangent Bundles on a Complex Manifold
Composing the injection with the projections we get canonical real isomorphisms
![Page 20: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/20.jpg)
Tangent Bundles on a Complex Manifold
The conjugation map
is a real linear isomorphism which is not complex linear.
![Page 21: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/21.jpg)
Tangent Bundles on a Complex Manifold
The conjugation map induces real linear isomorphism
and a complex linear isomorphism
![Page 22: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/22.jpg)
Cotangent Bundles on Complex Manifolds
![Page 23: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/23.jpg)
Cotangent Bundles on Complex Manifolds
Let {dx1, . . . , dxn, dy1, . . . , dyn} be the dual basis to {x1, . . . , xn, y1, . . . , yn}. Then
![Page 24: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/24.jpg)
Cotangent Bundles on Complex Manifolds
We have the following vector bundles on X:
TX*(R), the real cotangent bundle, with fiber
![Page 25: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/25.jpg)
Cotangent Bundles on Complex Manifolds
TX*(C), the complex cotangent bundle, with fiber
![Page 26: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/26.jpg)
Cotangent Bundles on Complex Manifolds
TX*(C), the holomorphic cotangent bundle, with fiber
![Page 27: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/27.jpg)
Cotangent Bundles on Complex Manifolds
TX*(C), the anti-holomorphic cotangent bundle, with fiber
![Page 28: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/28.jpg)
Cotangent Bundles on Complex Manifolds
We have a canonical injection and a canonical internal direct sum decomposition into complex sub-bundles:
![Page 29: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/29.jpg)
Cotangent Bundles on Complex Manifolds
Composing the injection with the projections we get canonical real isomorphisms
![Page 30: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/30.jpg)
Cotangent Bundles on Complex Manifolds
The conjugation map
is a real linear isomorphism which is not complex linear.
![Page 31: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/31.jpg)
Cotangent Bundles on Complex Manifolds
The conjugation map induces real linear isomorphism
and a complex linear isomorphism
![Page 32: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/32.jpg)
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
![Page 33: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/33.jpg)
The Standard Orientation of a Complex Manifold
![Page 34: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/34.jpg)
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.
![Page 35: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/35.jpg)
Standard Orientation
If (U,{z1,…,zn}) with zj=xj+i yj, the real 2n-form
is nowhere vanishing in U.
![Page 36: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/36.jpg)
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.
![Page 37: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/37.jpg)
The Almost Complex Structure
![Page 38: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/38.jpg)
Almost Complex Structure
The holomorphic tangent bundle TX of a complex manifold X admits the complex linear automorphism given by multiplication by i.
![Page 39: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/39.jpg)
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*.
![Page 40: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/40.jpg)
Almost Complex Structure
An almost complex structure on a real vector space VR of finite even dimension 2n is a R-linear automorphism
![Page 41: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/41.jpg)
Almost Complex Structure
An almost complex structure is equivalent to endowing VR with a structure of a complex vector space of dimension n.
![Page 42: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/42.jpg)
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.
![Page 43: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/43.jpg)
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.
![Page 44: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/44.jpg)
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,
![Page 45: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/45.jpg)
Almost Complex Structure there are natural R-linear isomorphisms
coming from the inclusion and the projections to the direct summands
and complex linear isomorphisms
![Page 46: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/46.jpg)
Almost Complex Structure The complex vector space defined by
the complex structure is C-linearly isomorphic to V.
![Page 47: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/47.jpg)
Almost Complex Structure
The same considerations are true for the almost complex structure (VR*, JR*). We have
![Page 48: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/48.jpg)
Complex-Valued Forms
![Page 49: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/49.jpg)
Complex-Valued Forms
Let M be a smooth manifold. Define the complex valued smooth p-forms as
![Page 50: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/50.jpg)
Complex-Valued Forms
The notion of exterior differentiation extends to complex-valued differential forms:
![Page 51: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/51.jpg)
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
![Page 52: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/52.jpg)
Complex-Valued Forms
There is a canonical internal direct sum decomposition of complex vector spaces
![Page 53: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/53.jpg)
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*).
![Page 54: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/54.jpg)
Complex-Valued Forms
There is a canonical direct sum decomposition
and
![Page 55: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/55.jpg)
Complex-Valued Forms
Let l=p+q and consider the natural projections
Define operators
![Page 56: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/56.jpg)
Complex-Valued Forms
Note that
Also,
![Page 57: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/57.jpg)
Dolbeault Cohomology
![Page 58: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/58.jpg)
Dolbeault Cohomology
Definition: Fix p and q. The Dolbeault complex is the complex of vector spaces
![Page 59: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/59.jpg)
Dolbeault Cohomology
The Dolbeault cohomology groups are the cohomology groups of the complex
![Page 60: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/60.jpg)
Dolbeault Cohomology
That is,
![Page 61: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/61.jpg)
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
![Page 62: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/62.jpg)
Dolbeault Cohomology
The Grothendieck-Dolbeault Lemma guarantees that Dolbeault cohomology is locally trivial.
![Page 63: Hodge Theory](https://reader036.vdocuments.us/reader036/viewer/2022062322/568143a6550346895db02bad/html5/thumbnails/63.jpg)
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