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Birational Kleinian groups

Shengyuan Zhao

Abstract

In this paper we study birational Kleinian groups, i.e.\ groups of birational transformations of complex projective varieties acting in a free, properly discontinuous and cocompact way on an open set of the variety with respect to the usual topology. We obtain a classification in dimension two.

Birational Kleinian groups

Abstract

In this paper we study birational Kleinian groups, i.e.\ groups of birational transformations of complex projective varieties acting in a free, properly discontinuous and cocompact way on an open set of the variety with respect to the usual topology. We obtain a classification in dimension two.

Paper Structure

This paper contains 73 sections, 71 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Birational Kleinian groups
  3. Definitions
  4. Comparison with classical Kleinian groups
  5. Complex projective Kleinian groups and Anosov subgroups
  6. Uniformization and geometric structures
  7. Non compact quotients
  8. Main results
  9. Strategy and plan
  10. Strategy
  11. Plan
  12. Varieties of non-negative Kodaira dimension
  13. Invariant measure
  14. Genus one fibrations.
  15. Complex projective Kleinian groups
  16. ...and 58 more sections

Key Result

Theorem A

There are no birational Kleinian groups acting on projective varieties of Kodaira dimension $\geq 0$.

Theorems & Definitions (127)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1.1
  • proof : Proof of Theorem \ref{['nonnegthm']}
  • Corollary 2.1.2
  • proof
  • Lemma 2.1.3
  • proof
  • Lemma 2.2.1
  • ...and 117 more