I found several examples where the infinitesimal Torelli theorem (e.g. as stated by Carlson, Green, Griffiths, and Harris on p. 144 of ``Infinitesimal variations of Hodge structure I") holds in the literature, but it was strange that I couldn't find anything about Jacobians or abelian varieties (which is the specific case I was reading about).

In particular, I tried looking at the cup product map associated with the theta divisor of the Jacobian. More specifically, we consider how $H^{1, 1}$ varies when $J(C)$ is deformed as $C$ varies in a family of curves with a fixed automorphism group $G$. If the infinitesimal Torelli theorem holds, then this map must be injective (i.e. the dual map is surjective). At first, this seemed like something doable since the evaluation of the dual map at a fixed point is something we can write down easily in terms of linear algebra (which is known very explicitly in specific cases).

However, actually checking when surjectivity holds in general even for this specific component seemed a lot more complicated than I expected without making assumptions on the point (which is actually a matrix) such as diagonalizability. Is there something I'm missing in my (attempted) computations or is there some deeper problem (e.g. with the $H^{p, q}$'s) that I'm not aware of?