Degenerations of Hodge structure

TL;DR

This paper surveys recent advances in understanding degenerations of Hodge structures, providing a comprehensive overview of the Hodge-theoretic answers to classical questions in algebraic geometry about degenerations of varieties.

Contribution

It offers an expository overview of the complete solution to key Hodge-theoretic questions about degenerations, connecting them to algebraic geometry and moduli space applications.

Findings

Hodge-theoretic analogs of degeneration questions are fully answered.

Schmid's Nilpotent Orbit Theorem is central to understanding degenerations.

Insights gained have implications for the study of moduli spaces.

Abstract

Two interesting questions in algebraic geometry are: (i) how can a smooth projective varieties degenerate? and (ii) given two such degenerations, when can we say that one is "more singular/degenerate" than the other? Schmid's Nilpotent Orbit Theorem yields Hodge-theoretic analogs of these questions, and the Hodge-theoretic answers in turn provide insight into the motivating algebro-geometric questions, sometimes with applications to the study of moduli. Recently the Hodge-theoretic questions have been completely answered. This is an expository survey of that work.