Can someone suggest a reference to study Hodge locus, period mappings and period domains on moduli space of principally polarised abelian varieties?


More precisely, consider the moduli space of principally polarized abelian varieties of dimension $g$, denoted $A\_g$. Using the Gauss-Manin connection, one defines a natural map 
$$\nabla:T\_bA\_g \to \mbox{Hom}(H^{1,0}(X\_b,\mathbb{C}),H^{0,1}(X\_b,\mathbb{C})),$$ where $b \in A\_g, X\_b$ the corresponding abelian variety and $T\_bA\_g$ the tangent space at $b$. I am interested in the image of the map $\nabla$. Is it surjective? Is it injective? I have read the simple description of the Hodge decomposition as given in the "Abelian varieties" book by Mumford.


Any reference/idea will be most welcome.



Reference for Hodge loci on moduli space of principally polarised abelian varieties
It is NOT an isomorphism, by dimension count: the LHS has dimension $g(g+1)/2$, whereas the RHS has dimension $g^2$. 


It is injective though. A (polarized) abelian variety is the same as a polarized weight 1 Hodge structure. Such a Hodge structure on $H^1(X\_b,\mathbb C)$ is determined by the subspace $H^{1,0}(X\_b)$, and the RHS is the tangent space to the Grassmannian of $g$-subspaces of $H^1(X\_b,\mathbb C)$. The LHS is smaller because $H^{1,0}(X\_b)$ must be Lagrangian with respect to the alternating form on $H^1(X\_b,\mathbb C)$ for it to yield a polarized Hodge structure.


