A variational mixed Torelli theorem (Q1331697)

The Torelli problem for a family of varieties deals with the question of whether the period map from the base space of this family to a period matrix space is injective. Strong Torelli-type theorems have been proved using the infinitesimal variation of Hodge structure (IVHS) techniques of \textit{J. A. Carlson} and \textit{P. A. Griffiths} [in Journées géométrie algébrique, Angers 1979, 51-76 (1980; Zbl 0479.14007)] and of \textit{J. Carlson}, \textit{M. Green}, \textit{P. Griffiths} and \textit{J. Harris} [Compos. Math. 50, 109-205 (1983; Zbl 0531.14006)]. The idea is to prove a generic Torelli theorem which states that the period map for the varieties in question has degree 1 onto its image. A result of \textit{D. Cox}, \textit{R. Donagi}, and \textit{L. Tu} [Invent. Math. 88, 439-446 (1987; Zbl 0606.14005)] allows us to reduce it to the variational Torelli problem. In a Torelli problem, one seeks to recover a variety from the algebraic data of its period map, but in a variational Torelli problem, the algebraic data comprise not only the period map but also its derivative. By adapting \textit{R. Donagi}'s symmetrizer technique [Compos. Math. 50, 325-353 (1983; Zbl 0598.14007)] as used by \textit{M. L. Green} [ibid. 55, 135-156 (1985; Zbl 0588.14004)] and by applying a new way of recovering the variety in question from its IVHS, we show in this article a variational mixed Torelli theorem.