Holomorphic Bisectional Curvature and Applications to Deformations and Rigidity for Variations of Mixed Hodge Structure

TL;DR

This paper establishes a rigidity criterion for period maps associated with variations of mixed Hodge structures, with applications to families of algebraic varieties and their fundamental groups, advancing understanding of deformation and rigidity phenomena.

Contribution

It introduces a new rigidity criterion for period maps of mixed Hodge structures and demonstrates its applicability across various geometric contexts.

Findings

Rigidity criterion for period maps proved.

Rigidity established for families of curves and surfaces.

Applications to fundamental groups and normal functions.

Abstract

In this article, we prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective curves with ordinary double points, the complement of the canonical curve in families of Kynev--Todorov surfaces, period maps attached to the fundamental groups of smooth varieties and normal functions.