Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 4

Special issue celebrating the work of Herb Clemens

Guest Editor: Ron Donagi

Differential of a period mapping at a singularity

Pages: 1421 – 1484

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a6

Authors

Mark Green (Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

Phillip Griffiths (Phillip Griffiths Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Abstract

The study of the variation of Hodge structure in a family of algebraic varieties is an important topic in algebraic geometry. Of special interest is analysis of the Hodge structure in a family of varieties $X_t, t \in \Delta = \textrm{unit disc}$, where $X_t$ is smooth for $t \neq 0$ while $X_0$ may be singular. It is known that if the monodromy is finite, then the Hodge structure fills in at $t = 0$. When the monodromy is infinite there is a well-developed understanding of how the Hodge structure degenerates. In this paper, we shall define and study properties of the first order variation of the Hodge structure at $t = 0$, both when the monodromy is finite, e.g., for a family of smooth surfaces acquiring a normal non-Gorenstein singulariy, and for the case when the monodromy is infinite. In the latter case, we shall give a Torelli type result that may be used to infer Torelli properties in the interior of a moduli space from these properties on the boundary.

Received 3 October 2021

Received revised 6 May 2022

Accepted 23 May 2022

Published 25 October 2022