Hodge theorem

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 [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) [3]. These results lead to the classical Hodge structure in the cohomology spaces of a compact Kähler manifold.

References

[1]	W.V.D. Hodge, "The topological invariants of algebraic varieties" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 1 , Amer. Math. Soc. (1952) pp. 182–192 MR0046075 Zbl 0048.41701
[2]	W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1962) MR1015714 MR0051571 MR0003947
[3]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[4]	G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) Zbl 0534.58003
[a1]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004