# Reference for Hodge loci on moduli space of principally polarised abelian varieties

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.

• I am not an expert, (although I play one on television), but p.380 of Beauville's Varie'te's de Prym et jacobiennes interme'diares, suggest that, in the incarnation of the right hand space as a tensor product, the image is the symmetric tensors. – roy smith Sep 7 '16 at 17:31
• Isn't this map really just the Kodaira-Spencer map $H^0(T_b A_g) \to H^1(A_b)$ followed by the cup product ? In which case the isomorphism is pretty well-known. – aginensky Sep 7 '16 at 19:55
• @roy smith - What television show ? – aginensky Sep 7 '16 at 19:56
• I'm shooting from the hip a bit as my recollecction goes back to my (first) thesis days. The map you describe is an isomorphism on the space of complex tori and their deformations. Suppose the complex tori you had was in fact a ppav. Then the polarization amounts to a class in $H^1(\Omega^1)$ and the K-S map (cup product) takes that to $H^2(\mathcal{O})$ which has dimension $\frac{g(g-1)}{2}$ . The condition that the polarization extend, i.e. that the variety stays algebraic is that this cup product class in $H^2(\mathcal{O})$ be zero. That is there are $\frac{g(g-1)}{2}$ conditions. – aginensky Sep 8 '16 at 2:31
• So the moduli space should be described as all elements of $H^1(T_A)$ which 'kill' the polarization and hence the space of deformations of ppav is of dimension $g^2 - \frac{g(g-1)}{2} = \frac{g(g+1)}{2}$ . I hope this a) helps and b) is basically correct. Experts ? – aginensky Sep 8 '16 at 2:34

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.
• A comment, just reformulationg what François wrote. The RHS can be rewritten as $H^1(X,T_X)$, the space of 1st order deformations of $X$. This is because $T_X \cong O_X \otimes H^0(X,\Omega^1_X)^\vee$ (using translation, the tangent bundle of $X$ is the trivial bundle given by the tangent space at the identity). So the map $\nabla$ is including the space of deformations as a ppav into the space of all deformations. – Dan Petersen Sep 8 '16 at 8:35