Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question

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 $V$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.