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.

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    $\begingroup$ Any regular variety has trivial Albanese, so any flat family of regular varieties will fail- or did I misread the question. Less formally, it seems that except for curves, the Albanese is a very crude invariant and won't detect any part of the structure of a variety that is at all unirational. $\endgroup$ – aginensky Dec 23 '12 at 18:04
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    $\begingroup$ Also, the Albanese map is very often not injective even when it is generically finite. Take a smooth projective variety $X$ of dimension >1 and a double cover $Y\to X$ branched on a smooth ample divisor. Then one can show that the induced map $Alb(Y)\to Alb(X)$ is an isomorphism, so $X\to Alb(X)$ factors through $X\to Y$ and is not injective. You can take for instance $Y$ to be an abelian variety, or a complete intersection of ample divisors in an abelian variety, so that the Albanese map of $Y$ is injective. $\endgroup$ – rita Dec 23 '12 at 18:34

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