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Hodge Theory on Differentiable Manifolds

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Hodge Theory on Differentiable Manifolds Hodge Theory on Differentiable Manifolds This appendix is meant to remind the reader of a few basic definitions and facts from differential geometry, but it cannot replace an introduction to the sub- ject. We use the opportunity to introduce the related notations used through- out the text. No proofs are given, the material is far from being complete and the reader is advised to go back to any of the standard textbooks for details. Definition A.0.1 An m-dimensional Cfc-manifold is a topological space M together with an open covering M = [jUi and homeomorphisms <pi : Ui == V, onto open subsets V* C Rm such that i) M is Hausdorff. ii) The topology of M admits a countable basis. 1 hi) The transition functions <pj o ipT \ <pt(Ui D fj) —+ <Pj{JJ% n Uj) are Cfc-maps. A differentiable manifold is a C°°-manifold and only those will be con- sidered. The datum {(Ui,ipi)} is called an atlas and each tuple (Ui,ipi) is a chart. We say that two atlases define the same manifold if the transition functions ^ o tp~l are differentiable. If M is a differentiable manifold, then one can introduce differentiable functions on M. By CM we denote the sheaf of differentiable functions, i.e. for any open subset U C M the value of CM on U is the space of differentiable functions / : U —> H., i.e. functions such that / o (p~l : <pi(U n Ui) —> M. is differentiable for any chart (Ui,<f*i). Analogously, one introduces differentiable maps between differentiable manifolds. In particular, there is the stalk CM,X of the sheaf of differentiable functions at every point x € M. The tangent space TXM of M at the point x (E M can be denned as TxM:=DerR(CM,x,K), the vector space of derivations D : CM,X —» K, i-e. of M-linear maps satisfying D{f • g) = f(x) • D(g) + D(f) • g{x).'E.g. any curve 7 : (-e,e) -» M with 7(0) = x defines a tangent vector D1 by -D7(/) = (d(f o j)/dt)(O). 282 A Hodge Theory on Differentiable Manifolds All the tangent spaces TXM glue to the tangent bundle TM = {JxeM TXM which is an example of a differentiable real vector bundle on M. Definition A.0.2 Let M be a differentiable manifold. A differentiable vector bundle of rank r on M consists of a differentiable manifold E, a differen- tiable map •K : E —> M, and the structure of a real vector space on any fibre E(x) := TT~1(X), such that there exists an open covering M = (J Ui and dif- feomorphisms ipi '• n~l(Ui) —* Ui xRr with pr^. o^ = n and such that for all x £ Ui the map ipi(x) : E(x) —> M1" is an isomorphism of real vector spaces. A real vector bundle can also be described in terms of the cocycle 4>ij : Ui n Uj —> Gl(r, M). For line bundles, i.e. vector bundles of rank one, this leads to a complete parametrization of isomorphism classes by the sheaf X cohomology H (M,CM) of the multiplicative sheaf of nowhere vanishing dif- ferentiable functions. Similarly, isomorphism classes of complex line bundles 1 are in bijection with the elements of H (M,CM c), where CM,C is the sheaf of complex valued differentiable functions. See also Appendix B. In fact, all this works also for higher rank vector bundles, which are parametrized by H1(M, Gl(r, CM)) with the difference that the latter coho- mology group needs an extra definition, for the sheaf of functions with values in the group of invertible matrices is not abelian for r > 1. In the cocycle language the tangent bundle TM corresponds to {D(tpi o fJl)OiPj}, where DfajOtpJ1) is the total differential of the transition function. To any vector bundle TT : E —> M one associates its sheaf of differentiable sections, also denoted E, by E(U) := r(U, E) := {s : U -• E TT O S = (This might lead to confusion: In the main body of the text we often speak about holomorphic vector bundles E, which in particular are differentiable vector bundles. Thus, there are two sheaves associated with it: the sheaf of holomorphic sections and the sheaf of differentiable sections, both denoted E. It should be clear from the context which one is meant.) Recall that vector bundles E, F on a manifold M give rise to new vector bundles (still on M) by taking direct sums E © F, tensor products E ® F, homomorphisms Hom(E,F), etc. Moreover, any sub-bundle F C E is a di- rect summand of E by writing E — F © F1-, where F1- is the orthogonal complement of F with respect to a metric on E. In this vain, one defines the cotangent bundle f\M as the dual (TM)* = Hom(TM, Mxl) and the bundles of /b-forms k Their sheaves of sections are given special names: By A M one denotes the sheaf of sections of /\ M, the sheaf of k-forms. In particular, A°M — CM, which is A Hodge Theory on Differentiable Manifolds 283 a sheaf of real algebras, and all the higher AM are sheaves of modules over •AM-A° If / : M —> N is a differentiable map, then there exists a natural pull-back 1 map f~ A% —> AM for any k. Using the local description of Ak{U) for an open subset U C Mm as the space of differential forms ]T) f^.^dx^ A ... A dxik with f^...^ 6 C(U), one defines the exterior product which is a map of CM sheaves, and the exterior differential d : AM which is a map of sheaves of M-vector spaces only. Using vector fields, i.e. sections of TM, one can invariantly define the exterior differential by fc+i (da)(vi,..., vk+i) := Here, the Lie bracket is understood as the Lie bracket of derivations. Since d2 = 0 locally, one obtains a complex of sheaves, the de Rham complex 2 CM ^AM *A M -••• Proposition A.0.3 (Poincare lemma) The de Rham complex of sheaves is a resolution of the sheaf of locally constant functions M C CM- Definition A.0.4 The de Rham cohomology of a differentiable manifold M is defined as Rk(M R) _ [ ' j Im(d:Ak1(M)^Ak(M)) Due to the Poincare lemma and the fact that the sheaves AM are acyclic, because they are soft (cf. Definition B.0.38), the de Rham cohomology co- incides with the sheaf cohomology of the sheaf R of locally constant real functions. k Definition A.0.5 The k-th Betti number of M is bk(M) := dimR H (M,R). fc The Euler number is e(M) := £(-l) 6fc(M). Of course, this definition makes only sense if the cohomology is finite di- mensional, which is often the case in particular for compact manifolds. 284 A Hodge Theory on Differentiable Manifolds The exterior product yields a multiplicative structure on the de Rham cohomology H*(M,R) = <$Hk(M,R): Hk{M,R) x He(M,R) ^Hk+e(M,R) . If M is compact and oriented, i.e. /\m M is trivial and a trivializing section has been chosen up to scaling by positive functions, then integration yields a linear map m fM:H (M,R) *R, [a] I *fMa. If M is in addition connected, then the integral is an isomorphism and one obtains a pairing k m k H {M,R)xH - {M,R) ^K, ([a],[/3])i ^/M«A/3. Proposition A.0.6 (Poincare duality) Under the above assumptions the pairing is non-degenerate. In particular, &fe(M) = 6m_fc(M). One possible approach to prove the proposition is via harmonic forms (see Corollary A.0.15). Examples A.0.7Using the Mayer-Vietoris sequence one can easily compute the Betti-numbers of some of the basic examples: i) Hk(Sm, R) = R for k = 0, m and trivial otherwise. ii) tf*(Pn,M) = H2*{Pn,R) = R[t]/{tn+1) with deg(t) = 2. (Here, as always for us, P" is the complex projective space.) Let us now turn to Riemannian manifolds. A Riemannian manifold is a differentiable manifold M with the additional structure of a Riemannian metric g, i.e. a section g of f\M®f\M inducing a positive definite symmetric bilinear form, i.e. a scalar product, on each TXM. By standard linear algebra, the metric g also endows /\x M and all f\x M with a natural scalar product. If M is in addition oriented, then one has a unique m-form, the volume form, vol = vol(M,s) which is of norm one and positive oriented at every point x £ M. The volume of the Riemannian manifold (M,g) is Jx vol(jw,g)- Let (M, g) be an oriented Riemannian manifold of dimension m. Using the metric and the orientation one introduces the Hodge ^-operator *: Ak(M) ^Am~k(M) (cf. Section 1.2). The form *1 is the volume form vol(^]9). The adjoint d* of the exterior differential d is given by m fc 1 1 k d* :=(-l) ( + )+ *d* on A {M) A Hodge Theory on Differentiable Manifolds 285 and the Laplace operator is A := d*d + dd*. Clearly, d* is of degree —1, whereas A is of degree zero, i.e. A induces an endomorphism of each Ak(M). Since the metric g induces a natural scalar product on any fibre /\x M for all x € M, one can introduce a scalar product on the space of global k-iorms whenever M is compact.
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