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Hodge's index theorem: The index ([[signature]]) $\sigma(M)$ of a compact [[Kähler manifold]] $M$ of complex dimension $2n$ can be computed by the formula
$$
\sigma(M) = \sum_{p,q\,:\,p+q\,\text{even}} (-1)^{p+q} h^{p,q}
$$
where $h^{p,q} = \dim H^{p,q}(M)$ is the dimension of the space of harmonic forms of type $(p,q)$ on $M$ (cf. [[Harmonic form]]). This was proved by W.V.D. Hodge .

Hodge's theorem on the decomposition of the space of smooth sections of an elliptic complex on a compact manifold into the orthogonal direct sum of subspaces of harmonic exact and co-exact sections (see [[Laplace operator]]). This was proved by W.V.D. Hodge [[#References|[2]]] for the de Rham complex
$$
E^*(M) = \sum_{p\ge0} E^p(M)
$$
on an orientable compact Riemannian manifold $M$. In this case Hodge's theorem asserts that for any $p\ge0$ the space $H^p(M)$ of harmonic forms on $M$ is finite-dimensional and that there exists a unique operator $G : E^p(M) \rightarrow E^p(M)$ (the Green–de Rham operator) satisfying the conditions
$$
G(H^p(M)) = 0 \ ;\ \ \ Gd = dg\ ;\ \ \ G \delta = \delta G
$$
$$
E^p(M) = H^p(M) \oplus d \delta GE^p(M) \oplus \delta d G E^p(M)
$$
(the Hodge decomposition). In particular, $H^p(M)$ is isomorphic to the real cohomology space $H^p(M,\mathbf{R})$ of $M$. Another important special case is the Hodge theorem for the Dolbeault complex on a compact complex manifold $M$ (see [[Differential form]]) [[#References|[3]]]. These results lead to the classical [[Hodge structure]] in the cohomology spaces of a compact Kähler manifold.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, "The topological invariants of algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''1''' , Amer. Math. Soc. (1952) pp. 182–192 {{MR|0046075}} {{ZBL|0048.41701}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1962) {{MR|1015714}} {{MR|0051571}} {{MR|0003947}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) {{MR|}} {{ZBL|0534.58003}} </TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR></table>

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