Let $X$ be a complex projective manifold of dimension $m$ with positive closed $(1,1)$ form $\omega$ induced by a projective embedding and let $Pic^0(X)$ be the associated Picard torus. This is a polarized abelian variety with polarization induced by the hermitian form $H$ on $H^0(X,\Omega^1_X)$ given by $$H(\alpha,\beta)=-2i\int_X\omega^{m-1} \wedge \alpha \wedge \bar{\beta}$$
I do not manage to prove that this is positive definite:tried to write $\omega$ explicitly locally using Kahler property but I'm not getting anything. I would not know how to use that $\omega$ comes from the Chern class of a positive line bundle