In "$C$ is not algebraically equivalent to $C^{-}$ in its Jacobian" by Ceresa and then in "On the periods of certain rational integrals, II" by Griffiths (which is quoted as a reference in the first paper) I found that infinitesimal deformations of a principally polarized abelian variety $\left( A , \theta \right)$ are the ones in $H^{1} \left( A , T_{A} \right)$ whose cup product with $\omega$ is zero in $H^{2}\left(A, \mathcal{O}_{A}\right)$, where $\omega$ is the class of $\theta$ in $H^{1} \left( A, \Omega^{1}_{A}\right)$.
I have some references both for deformation theory and for abelian varieties, but no one dealing with both of them... I have no clue why these particular elements in $H^{1}\left(A,T_{A}\right)$ are the right ones...
Thanks in advance for any good reference, hint or answer!