Let \(X\) be a smooth algebraic variety over the complex numbers which for simplicity we assume to have ample canonical bundle \(K_ X\). For each \(k\) with \(1 \leq k \leq \dim X\), we have a variation of Hodge structure \((H_ \mathbb{Z} = {\mathcal R}^ k p_ * (\mathbb{Z})_{\text{prim}}\), \(H^{p,q} (y)_{\text{prim}}, Q)\), where \(p + q = k\), \(H_ \mathbb{Z} \otimes \mathbb{C} = \bigoplus_{p + q = k} H^{p,q} (Hy)_{\text{prim}}\), and the polarization \(Q\) is a quadratic form on \(H_ \mathbb{Z}\) for which the subspaces \(H^{p,q}(y)\) are orthogonal to each other. To this variation of Hodge structure there is associated a holomorphic map \(\Phi : Y \to D\), where \(D\) is the classifying space for polarized Hodge structures of type \((h^{k,0}, h^{k-1,1}, \dots, h^{0,k})\) and \(h^{p,q} = \dim H^{p,q} (y)\). The infinitesimal Torelli problem asks whether the differential \(d \Phi\) of \(\Phi\) is injective on \(T_{Y, y_ 0}\). If this is true, then one says that the infinitesimal Torelli theorem holds for \(X\).
The crucial starting point for this study was the following result of P. Griffiths [“Periods of integrals on algebraic manifolds. I. II. III”, Am. J. Math. 90, 568-626 (1968; Zbl 0169.523), ibid. 805-865 (1968; Zbl 0183.255) and Publ. Math., Inst. Hautes Étud. Sci. 38, 125- 180 (1970; Zbl 0212.535)], giving a cohomological description for \(d \Phi\): \(d\Phi\) is induced by the cup-product \(H^ 1(X,T_ X) \otimes H^{p,q} (X) = H^ 1(X,T_ X) \otimes H^ q (X, \Omega^ p_ X) \to H^{q+1} (X, \Omega_ X^{p - 1}) = H^{p-1, q + 1} (X)\), i.e., \(d \Phi : H^ 1(X,T_ X) \to \bigoplus_{p + q = k} \operatorname{Hom} (H^{p,q} (X)\), \(H^{p-1,q-1}(X))\). – Classically, for a curve \(C\) the infinitesimal Torelli theorem holds iff \(g(C) = 1,2\) or iff \(g(C) \geq 3\) and \(C\) is not hyperelliptic. In higher dimensions the situation is not entirely clear. We treat the problem for a class of surfaces of general type, studied by A. Bartalesi and F. Catanese [in “Algebraic varieties of small dimension”, Proc. Int. Conf., Turin 1985, Rend. Semin. Mat., Torino, Fasc. Spec., 91-110 (1986; Zbl 0623.14015)]. They showed that surfaces of general type with \(K^ 2_ S = 6\), \(\chi = 4\) and torsion are essentially obtained as double covers of the projective plane blown up in six special points.
We are able to derive the following result: Let \(S\) be a minimal surface of general type with \(K^ 2_ S = 6\), \(\chi = 4\) and torsion. Moreover, we assume \(K_ S\) to be ample. Then the infinitesimal Torelli map \(d \Phi : H^ 1(S, \Theta_ S) \to \operatorname{Hom} (H^ 0(S, \Omega^ 2_ S)\), \(H^ 1(S, \Omega^ 1_ S))\) is injective.