Variational Torelli implies generic Torelli
The main result of the paper is that if \(f:\quad X\to S\) is a proper smooth family, then the period map \(P:\quad S\to \Gamma \setminus D\) is generically injective modulo polarized equivalence (this is generic Torelli) whenever variational Torelli holds for the family. Recall that variational Torelli for \(f:\quad X\to S\) means that the fibers \(X_ s=f^{-1}(s)\) are determined up to polarized equivalence by the real part \((H_{{\mathbb{R}}},F^.,Q,T,\delta)\) of the infinitesimal variation of Hodge structure (IVHS). In practice, \(\Gamma\) is usually either \(\Gamma_ 0\), the monodromy group of the family, or \(G_{{\mathbb{Z}}}\), the integral automorphisms of the Hodge structure. However, it turns out that \(\Gamma\) may be taken to be any discrete subgroup of \(G_{{\mathbb{R}}}\) that contains the monodromy group \(\Gamma_ 0.\) Previous treatments of this result always needed extra hypotheses, including infinitesimal Torelli, the existence of a suitable quotient of S, and the existence of a regular value of the period map P. This paper shows that none of these are needed. - The two main tools used in the proof are the principle of prolongation and the existence of generic quotients. Both of these results are part of the mathematical ''folklore'', and are given precise formulations and proofs. - The one limitation of the proof is that ''generic'' has to be interpreted as the complement of an analytic subvariety. It would be preferable to know that the subvariety were algebraic. This is related to the question of putting an algebraic structure on the period map.
infinitesimal variation of Hodge structure
scientific article; zbMATH DE number 3955092
scientific article