Pseudoconvexity at infinity in Hodge theory: a codimension one example

TL;DR

This paper investigates pseudoconvexity at infinity in Hodge theory by analyzing a specific example involving codimension one degenerations of a weight two Hodge structure with Hodge number 2, contributing to the understanding of extension problems in compactifications.

Contribution

It provides a detailed example demonstrating pseudoconvexity at infinity in a non-trivial Hodge-theoretic setting, aiding the extension of compactification techniques.

Findings

Establishes pseudoconvexity in a specific Hodge-theoretic example

Supports the extension of Satake--Baily--Borel compactification methods

Clarifies the geometric structure at infinity for certain period maps

Abstract

The generalization of the Satake--Baily--Borel compactification to arbitrary period maps has been reduced to a certain extension problem on certain "neighborhoods at infinity". Extension problems of this type require that the neighborhood be pseudoconvex. The purpose of this note is to establish the desired pseudoconvexity in one relatively simple, but non-trivial, example: codimension one degenerations of a period map of weight two Hodge structures with first Hodge number $h^{2,0}$ equal to 2.