Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

TL;DR

This survey reviews 26 years of research on variations of Hodge structure as exterior differential systems, highlighting key examples, new results on characteristic cohomology, and implications for integrability and the Hodge conjecture.

Contribution

It provides a comprehensive overview of the development in VHS as EDS, introduces new results on characteristic cohomology, and discusses the impact of integrability conditions.

Findings

New results on characteristic cohomology of homogeneous Pfaffian systems

Analysis of how integrability conditions influence the dimension of integral submanifolds

Discussion on potential links between EDS and the Hodge conjecture for Calabi-Yau manifolds

Abstract

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds.