Hodge theory

Summary

Hodge theory studies the cohomology of smooth manifolds through harmonic differential forms. For complex algebraic varieties, it studies the extra structure that harmonic forms put on cohomology. On a compact Riemannian manifold, the Hodge theorem identifies de Rham cohomology with the space of harmonic forms. On a compact Kähler manifold, harmonic forms split by type, which gives the Hodge decomposition and a Hodge structure on each cohomology group. The theory gives strong topological restrictions on algebraic varieties and leads to deep open problems such as the Hodge conjecture.