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A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type

Eyal Markman, Kota Yoshioka

Abstract

We prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.

A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type

Abstract

We prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.

Paper Structure

This paper contains 3 sections, 15 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Proof of the cone conjecture
  3. A rational polyhedral cone intersects only finitely many walls

Key Result

Theorem 1.3

Assume that Conjecture conj-self-intersection-of-extremal-classes-bounded-below holds for $X$. Then there exists a rational polyhedral cone $D\subset \operatorname{Nef}_X^+$, which is a fundamental domain for the action of the automorphism group of $X$ on $\operatorname{Nef}_X^+$.

Theorems & Definitions (29)

  • Conjecture 1.1
  • Definition 1.2
  • Theorem 1.3: Theorem \ref{['thm-ample-cone-conj']} below
  • Remark 1.4
  • Corollary 1.5: Corollary \ref{['cor-finitely-many-isomorphism-classes']} below
  • Proposition 2.1: hassett-tschinkel-moving-and-ample-cones and BHT
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • ...and 19 more