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Semispecial tensors and quotients of the polydisc

Patrick Graf, Aryaman Patel

Abstract

Let $X$ be a complex-projective variety with klt singularities and ample canonical divisor. We prove that $X$ is a quotient of the polydisc by a group acting properly discontinuously and freely in codimension one if and only if $X$ admits a semispecial tensor with reduced hypersurface. This extends a result of Catanese and Di Scala to singular spaces, and answers a question raised by these authors. As a key step in the proof, we establish the Bochner principle for holomorphic tensors on klt spaces in the negative Kähler--Einstein case.

Semispecial tensors and quotients of the polydisc

Abstract

Let be a complex-projective variety with klt singularities and ample canonical divisor. We prove that is a quotient of the polydisc by a group acting properly discontinuously and freely in codimension one if and only if admits a semispecial tensor with reduced hypersurface. This extends a result of Catanese and Di Scala to singular spaces, and answers a question raised by these authors. As a key step in the proof, we establish the Bochner principle for holomorphic tensors on klt spaces in the negative Kähler--Einstein case.
Paper Structure (7 sections, 13 theorems, 51 equations)

This paper contains 7 sections, 13 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. The Bochner principle for klt spaces
  4. Proof of the Bochner principle
  5. The holonomy of the Kähler--Einstein metric on $X$
  6. Proof of \ref{['main1']}
  7. Proof of corollaries

Key Result

Theorem A

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

Theorems & Definitions (35)

  • Theorem A: Characterization of singular polydisc quotients, \ref{["main1'"]}
  • Remark
  • Theorem B: Bochner principle, \ref{['bochner']}
  • Corollary 1.3: Uniformization of canonical models
  • Remark
  • Corollary 1.4: Conjugates of polydisc quotients
  • Corollary 1.5: Three-dimensional polydisc quotients
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 25 more