# Variation of the Albanese map

Let $S$ be an irregular surface of general type over $\mathbb{C}$ and $a \colon S \to A:=\textrm{Alb}(S)$ be its Albanese map. Let Def($S$) and Def($A$) be the bases of the Kuranishi family of $S$ and $A$, respectively. Then $a$ induces a map $f \colon \textrm{Def}(S) \to \textrm{Def}(A)$, whose differential is $f_* \colon H^1(S, T_S) \to H^1(A, T_A)$.

Since every small deformation of $S$ is again a surface of general type, if Def(S) is generically smooth then the image of $f_*$ is contained in the subspace of dimension $\frac{g(g+1)}{2}$ of $H^1(A, T_A)$ corresponding to the algebraic deformations of $A$ (here $g := \dim(A)$ ).

Is this still true when Def($S$) is not generically smooth, i.e. everywhere non-reduced? I suspect that the answer should be "yes", but I would like to see a rigorous proof (or a counterexample, if my guess is wrong).

• Ehm, I'm new in MathOverflow and unintentionally I deleted a comment trying to answer it. Sorry! Anyway, the answer was the following. Yes you are right, I've been sloppy in defining $f$ . Anyway, you can also define $f_*$ directly, considering the map induced in cohomology by the pullback, namely $H^1(S, T_S) \to H^1(S, a^*T_A)$, and then using the fact that $T_A$ is trivial and that $H^1(S, \mathcal{O}_S)=H^1(A, \mathcal{O}_A)$. Your translation seems to me correct; moreover, since S$is assumed to be a surface of general type, its first-order analytic and algebraic deformations coincide Jul 9, 2010 at 15:01 • @Francesco: you didn't delete my comment, I deleted it upon seeing that Angelo had given an answer ("projective" seems avoidable, and in general settings one shouldn't expect all first-order deformations to be projective). Actually, for the equality of first-order analytic and algebraic deformations of a smooth proper$\mathbf{C}\$-scheme, can't we just make an application of GAGA, and so not require mention of "general type", etc.? Jul 9, 2010 at 15:15
• You are right, I wrote "first-order deformations" but I was actually thinking of the semiuniversal formal deformation, that can be actually non algebraizable (e.g. for K3 surfaces or complex tori). Jul 9, 2010 at 15:35