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The Cone Conjecture for Primitive Symplectic Varieties over a Field of Characteristic Zero and an Application

Aurélien Faucher

TL;DR

This work proves the Kawamata–Morrison cone conjecture for projective primitive symplectic varieties over characteristic 0 fields with $b_2\ge5$, including a construction of a rational polyhedral fundamental domain for the Bir(X) action on the movable cone and a parallel domain for Aut(X) on the nef cone. It develops a robust framework using base-change stability, Picard lattices with BBF-type forms, and Coxeter-reflection theory, and then extends these absolute results to the relative setting of fibrations whose very general fiber is a primitive symplectic variety, establishing relative movable and nef cone conjectures and finiteness of birational models over the base. The paper also clarifies how these singular PSV cases behave under field extensions and descent, enabling a seamless transfer from complex geometry to arbitrary characteristic-zero fields. The results have potential implications for understanding the birational geometry of moduli spaces and related singular analogues of holomorphic symplectic manifolds.

Abstract

We prove the Kawamata-Morrison cone conjecture for Q-factorial terminal projective primitive symplectic varieties with second Betti number greater than five defined over a field of characteristic zero. As an application, we prove that the relative movable and the relative nef cone conjectures hold for fibrations whose very general fibre is a projective primitive symplectic varieties under certain assumptions.

The Cone Conjecture for Primitive Symplectic Varieties over a Field of Characteristic Zero and an Application

TL;DR

This work proves the Kawamata–Morrison cone conjecture for projective primitive symplectic varieties over characteristic 0 fields with , including a construction of a rational polyhedral fundamental domain for the Bir(X) action on the movable cone and a parallel domain for Aut(X) on the nef cone. It develops a robust framework using base-change stability, Picard lattices with BBF-type forms, and Coxeter-reflection theory, and then extends these absolute results to the relative setting of fibrations whose very general fiber is a primitive symplectic variety, establishing relative movable and nef cone conjectures and finiteness of birational models over the base. The paper also clarifies how these singular PSV cases behave under field extensions and descent, enabling a seamless transfer from complex geometry to arbitrary characteristic-zero fields. The results have potential implications for understanding the birational geometry of moduli spaces and related singular analogues of holomorphic symplectic manifolds.

Abstract

We prove the Kawamata-Morrison cone conjecture for Q-factorial terminal projective primitive symplectic varieties with second Betti number greater than five defined over a field of characteristic zero. As an application, we prove that the relative movable and the relative nef cone conjectures hold for fibrations whose very general fibre is a projective primitive symplectic varieties under certain assumptions.
Paper Structure (11 sections, 41 theorems, 64 equations)

This paper contains 11 sections, 41 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Definitions and first properties
  3. Primitive symplectic varieties, base change and descent
  4. The Picard scheme and the Picard lattice of a primitive symplectic variety
  5. Absolute cone conjectures for terminal Q-factorial primitive symplectic varieties
  6. Preliminaries
  7. The absolute movable cone conjecture
  8. The absolute nef cone conjecture
  9. Relative cone conjecture for singular primitive symplectic varieties
  10. Preliminaries
  11. Proof of the relative cone conjecture

Key Result

Theorem A

Let $X$ be a projective primitive symplectic variety over a field $F$ of characteristic $0$ such that $b_2(X) \geq 5$ and $X_{\overline{F}}$ has $\mathbb Q$-factorial and terminal singularities.

Theorems & Definitions (102)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Theorem A
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 92 more