Hodge structure

Remarks

Equivalently, the data can be given by the Hodge filtration $F^p = \bigoplus_{p' \ge p} H^{p', n - p'}$, a decreasing filtration of $H_\C$ with $H_\C = F^p \oplus \overline{F^{n-p+1}}$ for all $p$; then $H^{p,q} = F^p \cap \overline{F^q}$.

Example: let $X$ be a compact Kähler manifold, for instance a smooth projective complex variety. Then $H^k(X, \Z)$ modulo torsion, with the Hodge decomposition $H^k(X, \C) = \bigoplus H^{p,q}(X)$, is a Hodge structure of weight $k$: clauses 2 and 3 are parts 1 and 2 of the Hodge decomposition theorem. The Tate structure $\Z(1)$ is $H_\Z = 2\pi i \Z \subseteq \C$ with $H_\C = H^{-1,-1}$, a Hodge structure of weight $-2$.

Non-example: let $H_\Z = \Z^2$ and $n = 1$, with $H^{1,0} = H_\C$ and $H^{0,1} = 0$. Clause 2 holds, but clause 3 fails, since $\overline{H^{1,0}} = H_\C \ne 0 = H^{0,1}$.