Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space of principally polarized abelian varieties of dimension $g$.

Is this map locally closed? (That is, do we have Torelli?)

Let $k$ be an algebraically closed field. Is the Albanese map injective on $k$-points?

I'm new to this area and would really appreciate some "less formal" explanations one can't find in the vast literature on this subject.