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# 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

#### 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