On the infinitesimal Torelli theorem for certain irregular surfaces of general type (Q1821829)

Let S be a minimal surface of general type and \(p: (X,S)\to (B,b_ 0)\) the Kuranishi family of S. There is a naturally associated period map \(\phi _ 2: B\to D_ 2,\) where \(D_ 2\) is the classifying space of Hodge structure of weight 2. The infinitesimal Torelli problem asks if \(d\phi _ 2\), the differential of \(\phi _ 2\), is injective. Following Griffiths this is reduced to the study of the cup-product \(\delta: H^ 1(\Theta _ s)\to Hom(H^ 0(K_ s),H^ 1(\Omega ^ 1_ s))\) where \(\Theta _ s\), \(\Omega ^ 1_ s\), \(K_ s\) are holomorphic tangent, cotangent and canonical bundles of S, respectively. Main results: (1) Some sufficient conditions for \(\delta\) (and hence for \(d\phi _ 2)\) to be injective. - (2) Geometric interpretation of the failure of \(\delta\) to be injective. The main observation is that nonzero elements in ker(\(\delta)\) can be interpreted geometrically provided that S has no irrational pencil and a general 1-form \(\omega\) on S has reduced zero-dimensional locus. This geometric interpretation is the following: Let \(\xi\) be a nonzero element of ker(\(\delta)\), then such \(\xi\) induces a decomposition of \(Z_{\omega}=(\omega =0)\) into subcycles provided the latter is reduced and zero-dimensional. Moreover this decomposition varies holomorphically with \(\omega\). Now assuming that \(\Omega ^ 1_ s\) is very ample in the sense of Hartshorne the above decomposition is impossible and this yields injectivity for \(\delta\) for surfaces with ''large'' irregularity (i.e. \(\delta\) for which \(\Omega ^ 1_ s\) is very ample). The paper also has more detailed study of the geometry of \(Z_{\omega}=(\omega =0)\) (again \(Z_{\omega}\) is reduced and 0-dimensional) under the assumption ker(\(\delta) \neq 0\) as well as the discussion of some examples of failure of \(\delta\) to be injective in the light of the argument outlined above.