 Hodge structure

A pure integral Hodge structure is a finite-rank free abelian group with a complex decomposition satisfying $\overline{H^{p,q}}=H^{q,p}$ and $p+q=w$. The integer $w$ is its weight. Its equivalent Hodge filtration is $F^p={⨁}_{r\ge p}{H}^{r,w-r}$.

A Hodge structure is a concept in algebraic geometry and differential geometry that is used to study the relationships between algebraic and topological properties of complex manifolds. It provides a bridge between the geometric structure of a manifold and its algebraic properties. A Hodge structure on a vector space \( V \) over the complex numbers can be described as a decomposition of the space into subspaces that reflect the complex geometry of the underlying manifold.