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.

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