Hodge structure

Summary

A Hodge structure of weight $n$ is a lattice, or a rational vector space, with extra data on its complexification. The complexification splits into pieces $H^{p,q}$ with $p + q = n$, and complex conjugation exchanges $H^{p,q}$ and $H^{q,p}$. It abstracts the structure that the Hodge decomposition puts on the cohomology of a compact Kähler manifold. Hodge structures form an abelian category with tensor products. They are the linear algebra data behind period maps, variations of Hodge structure and the Hodge conjecture.