Hodge filtration ($F^p={⨁}_{r\ge p}{H}^{r,w-r}$)

A decreasing filtration defines a Hodge structure when $H_{\mathbb{C}}=F^p\oplus \overline{F^{w-p+1}}$ for every $p$. The decomposition is recovered as $H^{p,q}=F^p\cap \overline{F^q}$. This filtration is the holomorphic datum that varies in a variation of Hodge structure.