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. 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.

However, actually checking when surjectivity holds 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 that I'm not aware of?