Hodge theorem

Hodge's index theorem: The index (signature) of a compact Kähler manifold of complex dimension can be computed by the formula

where is the dimension of the space of harmonic forms of type on (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

on an orientable compact Riemannian manifold . In this case Hodge's theorem asserts that for any the space of harmonic forms on is finite-dimensional and that there exists a unique operator (the Green–de Rham operator) satisfying the conditions

(the Hodge decomposition). In particular, is isomorphic to the real cohomology space of . Another important special case is the Hodge theorem for the Dolbeault complex on a compact complex manifold (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

References

[a1]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004