Editing Hodge theory (section)

===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&nbsp;''dV''. Suppose that

:<math>L_i:\Gamma(E_i)\to\Gamma(E_{i+1})</math>

are linear [[differential operators]] acting on [[smoothness|C<sup>∞</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|∗}} be the adjoint of ''L''. Define the elliptic operator {{nowrap|1=Δ = ''LL''{{sup|∗}} + ''L''{{sup|∗}}''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 Δ. The '''Hodge theorem''' then asserts the following:<ref>Wells (2008), Theorem IV.5.2.</ref>

#''H'' and ''G'' are well-defined.
#Id = ''H'' + Δ''G'' = ''H'' + ''G''Δ
#''LG'' = ''GL'', ''L''{{sup|∗}}''G'' = ''GL''{{sup|∗}}
#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.