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Hodge theory
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==Hodge theory for real manifolds== ===De Rham cohomology=== The Hodge theory references the [[de Rham cohomology|de Rham complex]]. Let ''M'' be a [[smooth manifold]]. For a non-negative integer ''k'', let Ξ©<sup>''k''</sup>(''M'') be the [[real number|real]] [[vector space]] of smooth [[differential form]]s of degree ''k'' on ''M''. The de Rham complex is the sequence of [[differential operator]]s :<math>0\to \Omega^0(M) \xrightarrow{d_0} \Omega^1(M)\xrightarrow{d_1} \cdots\xrightarrow{d_{n-1}} \Omega^n(M)\xrightarrow{d_n} 0,</math> where ''d<sub>k</sub>'' denotes the [[exterior derivative]] on Ξ©<sup>''k''</sup>(''M''). This is a [[cochain complex]] in the sense that {{nowrap|1=''d''{{sub|''k''+1}} β ''d''{{sub|''k''}} = 0}} (also written {{nowrap|1=''d''{{i sup|2}} = 0}}). De Rham's theorem says that the [[singular cohomology]] of ''M'' with real coefficients is computed by the de Rham complex: :<math>H^k(M,\mathbf{R})\cong \frac{\ker d_k}{\operatorname{im} d_{k-1}}.</math> ===Operators in Hodge theory=== Choose a Riemannian metric ''g'' on ''M'' and recall that: :<math>\Omega^k(M) = \Gamma \left (\bigwedge\nolimits^k T^*(M) \right ).</math> The metric yields an [[inner product]] on each fiber <math>\bigwedge\nolimits^k(T_p^*(M))</math> by extending (see [[Gramian matrix]]) the inner product induced by ''g'' from each cotangent fiber <math>T_p^*(M)</math> to its <math>k^{th}</math> [[exterior product]]: <math>\bigwedge\nolimits^k(T_p^*(M))</math>. The <math>\Omega^k(M)</math> inner product is then defined as the integral of the pointwise inner product of a given pair of ''k''-forms over ''M'' with respect to the volume form <math>\sigma</math> associated with ''g''. Explicitly, given some <math>\omega,\tau \in \Omega^k(M)</math> we have :<math> (\omega,\tau) \mapsto \langle\omega,\tau\rangle := \int_M \langle \omega(p),\tau(p)\rangle_p \sigma.</math> Naturally the above inner product induces a norm, when that norm is finite on some fixed ''k''-form: :<math>\langle\omega,\omega\rangle = \| \omega\|^2 < \infty,</math> then the integrand is a real valued, square integrable function on ''M'', evaluated at a given point via its point-wise norms, :<math> \|\omega(p)\|_p:M \to \mathbf{R}\in L^2(M).</math> Consider the [[adjoint operator]] of ''d'' with respect to these inner products: :<math>\delta : \Omega^{k+1}(M) \to \Omega^k(M).</math> Then the [[Laplacian]] on forms is defined by :<math>\Delta = d\delta + \delta d.</math> This is a second-order linear differential operator, generalizing the Laplacian for functions on '''R'''<sup>''n''</sup>. By definition, a form on ''M'' is '''harmonic''' if its Laplacian is zero: :<math>\mathcal{H}_\Delta^k(M) = \{\alpha\in\Omega^k(M)\mid\Delta\alpha=0\}.</math> The Laplacian appeared first in [[mathematical physics]]. In particular, [[Differential forms#Applications in physics|Maxwell's equations]] say that the electromagnetic field in a vacuum, i.e. absent any charges, is represented by a 2-form ''F'' such that {{nowrap|1=Ξ''F'' = 0}} on spacetime, viewed as [[Minkowski space]] of dimension 4. Every harmonic form ''Ξ±'' on a [[Closed manifold|closed]] Riemannian manifold is [[Closed and exact differential forms|closed]], meaning that {{nowrap|1=''dΞ±'' = 0}}. As a result, there is a canonical mapping <math>\varphi:\mathcal{H}_\Delta^k(M)\to H^k(M,\mathbf{R})</math>. The '''Hodge theorem''' states that <math>\varphi</math> is an isomorphism of vector spaces.<ref>Warner (1983), Theorem 6.11.</ref> In other words, each real cohomology class on ''M'' has a unique harmonic representative. Concretely, the harmonic representative is the unique closed form of minimum ''L''<sup>2</sup> norm that represents a given cohomology class. The Hodge theorem was proved using the theory of [[elliptic operator|elliptic]] partial differential equations, with Hodge's initial arguments completed by [[Kunihiko Kodaira|Kodaira]] and others in the 1940s. For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are [[finite-dimensional]]. (Admittedly, there are other ways to prove this.) Indeed, the operators Ξ are elliptic, and the [[kernel (algebra)|kernel]] of an elliptic operator on a closed manifold is always a finite-dimensional vector space. Another consequence of the Hodge theorem is that a Riemannian metric on a closed manifold ''M'' determines a real-valued [[inner product]] on the integral cohomology of ''M'' modulo [[torsion subgroup|torsion]]. It follows, for example, that the image of the [[isometry group]] of ''M'' in the [[general linear group]] {{nowrap|GL(''H''{{sup|β}}(''M'', '''Z'''))}} is finite (because the group of isometries of a [[lattice (group)|lattice]] is finite). A variant of the Hodge theorem is the '''Hodge decomposition'''. This says that there is a unique decomposition of any differential form ''Ο'' on a closed Riemannian manifold as a sum of three parts in the form :<math>\omega = d \alpha +\delta \beta + \gamma,</math> in which ''Ξ³'' is harmonic: {{nowrap|1=Ξ''Ξ³'' = 0}}.<ref>Warner (1983), Theorem 6.8.</ref> In terms of the ''L''<sup>2</sup> metric on differential forms, this gives an orthogonal [[direct sum]] decomposition: :<math> \Omega^k(M) \cong \operatorname{im} d_{k-1} \oplus \operatorname{im} \delta_{k+1} \oplus \mathcal H_\Delta^k(M).</math> The Hodge decomposition is a generalization of the [[Helmholtz decomposition]] for the de Rham complex. ===Hodge theory of elliptic complexes=== [[Michael Atiyah|Atiyah]] and [[Raoul Bott|Bott]] defined [[elliptic complex]]es as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let <math>E_0,E_1,\ldots,E_N</math> be [[vector bundles]], equipped with metrics, on a closed smooth manifold ''M'' with a volume form ''dV''. Suppose that :<math>L_i:\Gamma(E_i)\to\Gamma(E_{i+1})</math> are linear [[differential operators]] acting on [[smoothness|C<sup>β</sup>]] sections of these vector bundles, and that the induced sequence :<math> 0\to\Gamma(E_0)\to \Gamma(E_1) \to \cdots \to \Gamma(E_N) \to 0</math> is an elliptic complex. Introduce the direct sums: : <math>\begin{align} \mathcal E^\bullet &= \bigoplus\nolimits_i \Gamma(E_i) \\ L &= \bigoplus\nolimits_i L_i:\mathcal E^\bullet\to\mathcal E^\bullet \end{align}</math> and let ''L''{{sup|β}} be the adjoint of ''L''. Define the elliptic operator {{nowrap|1=Ξ = ''LL''{{sup|β}} + ''L''{{sup|β}}''L''}}. As in the de Rham case, this yields the vector space of harmonic sections :<math>\mathcal H=\{e\in\mathcal E^\bullet\mid\Delta e=0\}.</math> Let <math>H:\mathcal E^\bullet\to\mathcal H</math> be the orthogonal projection, and let ''G'' be the [[Green's function|Green's operator]] for Ξ. The '''Hodge theorem''' then asserts the following:<ref>Wells (2008), Theorem IV.5.2.</ref> #''H'' and ''G'' are well-defined. #Id = ''H'' + Ξ''G'' = ''H'' + ''G''Ξ #''LG'' = ''GL'', ''L''{{sup|β}}''G'' = ''GL''{{sup|β}} #The cohomology of the complex is canonically isomorphic to the space of harmonic sections, <math>H(E_j)\cong\mathcal H(E_j)</math>, in the sense that each cohomology class has a unique harmonic representative. There is also a Hodge decomposition in this situation, generalizing the statement above for the de Rham complex.
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