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The Kawamata-Morrison Cone Conjecture for Generalized Hyperelliptic Variety

Martina Monti, Ana Quedo

Abstract

A Generalized Hyperelliptic Variety (GHV) is the quotient of an abelian variety by a free action of a finite group which does not contain any translation. These varieties are natural generalizations of bi-elliptic surfaces. In this paper we prove the Kawamata-Morrison Cone Conjecture for these manifolds using the analogous results established by Prendergast-Smith for abelian varieties.

The Kawamata-Morrison Cone Conjecture for Generalized Hyperelliptic Variety

Abstract

A Generalized Hyperelliptic Variety (GHV) is the quotient of an abelian variety by a free action of a finite group which does not contain any translation. These varieties are natural generalizations of bi-elliptic surfaces. In this paper we prove the Kawamata-Morrison Cone Conjecture for these manifolds using the analogous results established by Prendergast-Smith for abelian varieties.
Paper Structure (14 sections, 23 theorems, 20 equations)

This paper contains 14 sections, 23 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The cone conjecture
  3. The cone conjecture under étale quotients
  4. Generalized Hyperelliptic Varieties
  5. Preliminaries on Abelian varieties
  6. Endomorphism algebra of abelian varieties
  7. Nef cone of an abelian variety
  8. Reduction theory in Convex Geometry
  9. The cone conjecture for GHV
  10. Proof of the main theorem
  11. The $G$-invariant ${\mathbb R}$-algebra $\textup{End}_{{\mathbb R}}(X)^G$
  12. The $G$-invariant ample cone of $X$
  13. Action of the centralizer
  14. Proof of Theorem \ref{['main1']}

Key Result

Theorem 1

Let $Y=X/G$ be a Generalized Hyperelliptic Variety. Then, part (i) of Conjecture conj is verified and $\overline{\textup{Mov}(Y)}^e=\textup{Nef}(Y)^e=\textup{Nef}(Y)^+=\textup{Nef}(Y)$. In particular, also part (ii) of Conjecture conj is verified.

Theorems & Definitions (63)

  • Conjecture 1.1: Kawamata-Morrison
  • Conjecture 1.2
  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 53 more