Infinitesimal Variations of Hodge Structure at Infinity

TL;DR

This paper introduces the concept of infinitesimal variations of Hodge structure at infinity, showing they can be integrated into polarized variations and are limits of finite IVHS, revealing deep geometric insights.

Contribution

It defines and analyzes infinitesimal variations of Hodge structure at infinity, establishing their integrability and connection to finite IVHS, and explores their maximal dimension properties.

Findings

All infinitesimal variations at infinity can be integrated into polarized variations.

Infinitesimal variations at infinity are limits of finite IVHS.

Examples of maximal dimension problems for these structures are provided.

Abstract

By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure (IVHS) at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the "classical" case of IVHS at finite points.