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

Gregory Pearlstein, Chris Peters

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.

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

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.

Paper Structure

This paper contains 44 sections, 54 theorems, 199 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Historical background
  3. Main results on deformations of mixed period maps
  4. Boundedness results
  5. Geometric applications
  6. Structure of the paper
  7. The pure case
  8. Basics on period domains and period maps
  9. Curvature properties
  10. Deformations of period maps
  11. Examples of rigid period maps
  12. Mixed period domains and Hodge metrics
  13. Basics on mixed Hodge structure
  14. Period maps for variations of mixed Hodge structure
  15. Mixed Hodge metrics
  16. ...and 29 more sections

Key Result

Theorem 1

The space of infinitesimal deformations of a period map $F: S \to {\mathcal{A}}_g$ over a curve $S$ can be canonically identified with the direct summand of $\mathop{\rm End}\nolimits^\Gamma(H_{{\mathbb{C}}},Q)$ of Hodge type $(-1,1)$. Consequently, if $F$ is not constant, $F$ is rigid if and only

Figures (1)

  • Figure 1: Decomposition of ${\mathfrak{g}}_{{\mathbb{C}}}$

Theorems & Definitions (105)

  • Theorem : arafal
  • Theorem : hodge4
  • Theorem : =Theorem \ref{['thm:Main']}
  • Remark 1.2.1
  • Theorem : =\ref{['hodge-tate-estimate']}
  • Theorem : =Theorems \ref{['unpotent-estimate']}, \ref{['thm:weight-minus-2-case']}, Corollaries \ref{['cor:IandII']},\ref{['corr:height-pairing']}
  • Remark
  • Proposition 2.2.1: schmid
  • Lemma 2.2.2
  • Proposition 2.2.3
  • ...and 95 more