Symmetry and variation of Hodge structures (Q1769524)

Let \(X\) be a smooth algebraic variety over \(\mathbb C\) whose canonical bundle \(K_X\) is ample. Then \(K_X\) determines a natural polarization on \(X\), and there is a semiuniversal deformation \(p: \mathcal X \to (Y, y_0)\) of \(X\). In particular, the tangent space \(T_{Y, y_0}\) of \(Y\) at \(y_0\) is naturally isomorphic to \(H^1 (X, T_X)\), and the dimension of \(Y\) at \(y_0\) is at least \(\dim H^1 (X, T_X) - \dim H^2 (X, T_X)\). To each positive integer \(k \leq \dim X\) there corresponds a variation of Hodge structure \((R^k p_* (\mathbb Z), H^{p,q} (y), Q)\) with \(p+q=k\) and \(R^k p_* (\mathbb Z) \otimes \mathbb C = \bigoplus_{p+q =k} H^{p,q} (y)\), where the polarization \(Q\) is a quadratic form on \(R^k p_* (\mathbb Z)\) for which the subspaces \(H^{p,q} (y)\) are mutually orthogonal. There is a holomorphic map \(\Phi: Y \to D\) associated to this variation of Hodge structure, where \(D\) is a classifying domain of polarized Hodge structures of type \((h^{k,0}, h^{k-1,1}, \ldots, h^{0,k})\) with \(h^{p,q} = \dim H^{p,q} (y)\). The infinitesimal Torelli theorem is said to hold for the variety \(X\) if the differential \(d\Phi\) of \(\Phi\) is injective. In this paper the authors provide a series of examples of surfaces of general type for which the infinitesimal Torelli theorem fails to hold on the whole moduli space despite the fact that the canonical system is generally quasi very ample.