I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of Catanese, in this situation $\deg \alpha$ is a topological invariant, so all deformations of $S$ will still have Albanese map generically surjective of degree $n$.

Moreover, I know that the following fact is true:

given a general principally polarized abelian surface $A$, I can construct a member $S$ of my family such that $A = \mathrm{Alb}(S).$ $\quad$ $(*)$

Now, let us take the short exact sequence associated with the differential map $d \alpha$, namely $$ 0 \to T_S \stackrel{d \alpha}{\longrightarrow} \alpha^* T_A \longrightarrow \mathcal{N}_{\alpha} \to 0,$$ where the normal sheaf $\mathcal{N}_{\alpha}$ uis supported on the ramification locus of $\alpha$.

Since $T_A = \mathcal{O}_A^{\oplus 2}$ is trivial and the Albanese map induces a isomorphism $H^1(A, \, \mathcal{O}_A) \cong H^1(S, \, \mathcal{O}_S)$, we have $H^1(S, \, \alpha^* T_A) \cong H^1(A, \, T_A)$ so, passing to cohomology in the sequence above, we obtain $$\cdots \to H^1(S, \, T_S) \stackrel{\varepsilon}{\longrightarrow} H^1(S, \, \alpha^*T_A) \cong H^1(A, \, T_A) \to \cdots$$ By standard deformation theory, I can interpret the map $\varepsilon$ as a map $$\varepsilon \colon T_{[S]}\mathrm{Def}(S) \longrightarrow T_{[A]}\mathrm{Def}(A),$$ where $\mathrm{Def}$ denotes as usual the base of the semi-universal deformation.

My geometrical intuition suggests to me that this should be the map that, given any first-order deformation of $S$, associates to it the corresponding (necessarily algebraic) first-order deformation of $A= \mathrm{Alb}(S)$, so in particular condition $(*)$ should imply that the image of $\varepsilon$ is a $3$-dimensional subspace in the $4$-dimensional vector space $H^1(A, \, T_A)$.

Questions. Is my intuition correct, and in particular is $\textrm{Im}(\varepsilon)$ of dimension $3$? If so, what is a rigorous deformation-theoretical argument allowing one to prove this?

Probably the answer is well-known to people who are experts in deforming finite morphisms, and I would also be happy with some standard reference on these topics.


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