I suspect this is very elementary, but it is not stated anywhere. A Hodge structure of weight $k$ consists of a finite rank lattice $H\_{\mathbb{Z}}$ together with a decomposition of its complexification $H : = H\_{\mathbb{Z}} \otimes \mathbb{C}$, $$H = \bigoplus\_{p+q=k} H^{p,q},$$ with $H^{p,q} = \overline{H^{p,q}}$. Set $F^p : = \bigoplus\_{r \geq p} H^{r,k-r}$.


A blinear form $Q$ on $H$ *polarizes* $H$ if


$Q(u,v) = (-1)^k Q(v,u)$,


$Q(F^p, F^{k-p+1})=0$,


The Hermitian form $Q((\sqrt{-1})^{p-q} \cdot, \overline{\cdot})$ is positive definite.


**Question:** Can every Hodge structure be polarized?



Can every Hodge structure be polarized?
Let me summarize comments.


1. abx points out that $Q$ should be integer valued, otherwise it's not an interesting notion.
2. (Assuming integrality) the answer is no because the categories of Hodge structures of type $\{(1,0), (0,1)\}$ and complex tori are equivalent. The polarizable ones correspond to abelian varieties, which is a proper subset.


