Polarizations of Hodge structures

Question

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?

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 $V$.