Let $V$ be a rational pure Hodge structure of weight $n$ and assume that $V$ is a Hodge sub-structure of the cohomology of some smooth projective complex algebraic variety $X$, that is

$V \subset H^n(X, \mathbb{Q})$

Is $V$ automatically polarizable?


1 Answer 1


Yes. Choose a projective embedding of $X$; this gives you a Lefschetz decomposition of $H^n(X,\mathbf{Q})$ into polarised pieces. Declaring the pieces to be orthogonal gives you a polarisation of $H^n(X,\mathbf{Q})$; restrict that to $V$.


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