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Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains

Patrick Graf, Aryaman Patel

Abstract

We prove an equivalence between two approaches to characterizing complex-projective varieties $X$ with klt singularities and ample canonical divisor that are uniformized by bounded symmetric domains. In order to do so, we show how to construct a uniformizing variation of Hodge structure from a slope zero tensor and vice versa. As a consequence, we generalize various uniformization results of Catanese and Di Scala to the singular setting. For example, we prove that $X$ is a quotient of a bounded symmetric domain of tube type by a group acting properly discontinuously and freely in codimension one if and only if $X$ admits a slope zero tensor. As a key step in the proof, we establish the compactness of the holonomy group of the singular Kähler--Einstein metric on $X_{\mathrm{reg}}$.

Slope zero tensors, uniformizing variations of Hodge structure and quotients of tube domains

Abstract

We prove an equivalence between two approaches to characterizing complex-projective varieties with klt singularities and ample canonical divisor that are uniformized by bounded symmetric domains. In order to do so, we show how to construct a uniformizing variation of Hodge structure from a slope zero tensor and vice versa. As a consequence, we generalize various uniformization results of Catanese and Di Scala to the singular setting. For example, we prove that is a quotient of a bounded symmetric domain of tube type by a group acting properly discontinuously and freely in codimension one if and only if admits a slope zero tensor. As a key step in the proof, we establish the compactness of the holonomy group of the singular Kähler--Einstein metric on .
Paper Structure (9 sections, 19 theorems, 30 equations, 2 tables)

This paper contains 9 sections, 19 theorems, 30 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hodge groups and Hermitian symmetric spaces
  4. Classification of irreducible Hermitian symmetric spaces
  5. Compactness of the holonomy group
  6. The restricted holonomy group
  7. Local isometries
  8. Proof of \ref{['main5']}
  9. Proof of \ref{['main2']}, \ref{['main3']} and \ref{['main4']}

Key Result

Theorem 1.1

Let $X$ be an $n$-dimensional normal projective variety with klt singularities and such that $K_X$ is ample. The following are equivalent:

Theorems & Definitions (59)

  • Theorem 1.1: cf. CataneseDiScala13
  • Theorem 1.2: cf. CataneseDiScala13
  • Theorem 1.3: cf. CataneseDiScala14
  • Theorem 1.4
  • Theorem 1.5: Holonomy cover
  • Remark
  • Definition 2.1: Hodge group of Hermitian type
  • Definition 2.3: Uniformizing system of Hodge bundles
  • Definition 2.4: Metric
  • Proposition 2.5
  • ...and 49 more