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The relative movable cone conjecture for K-trivial fibrations in varieties with well-clipped movable cones

Aurélien Faucher

Abstract

We prove the weak relative Kawamata-Morrison movable cone conjecture for K-trivial fibrations whose very general fibre is a quotient, by a finite group of automorphisms acting freely in codimension one, of a product of certain Calabi-Yau pairs whose underlying varieties have well-clipped movable cones, a notion recently introduced by Cécile Gachet. Our main result applies in particular when the fibre is a finite product of an abelian variety, smooth rational surfaces underlying klt Calabi-Yau pairs, projective irreducible holomorphic symplectic manifolds and Enriques manifolds, both of a known type. As a consequence, there are only finitely many minimal models over the base, up to isomorphism. When the relative movable cone is non-degenerate, we obtain the full relative movable cone conjecture.

The relative movable cone conjecture for K-trivial fibrations in varieties with well-clipped movable cones

Abstract

We prove the weak relative Kawamata-Morrison movable cone conjecture for K-trivial fibrations whose very general fibre is a quotient, by a finite group of automorphisms acting freely in codimension one, of a product of certain Calabi-Yau pairs whose underlying varieties have well-clipped movable cones, a notion recently introduced by Cécile Gachet. Our main result applies in particular when the fibre is a finite product of an abelian variety, smooth rational surfaces underlying klt Calabi-Yau pairs, projective irreducible holomorphic symplectic manifolds and Enriques manifolds, both of a known type. As a consequence, there are only finitely many minimal models over the base, up to isomorphism. When the relative movable cone is non-degenerate, we obtain the full relative movable cone conjecture.
Paper Structure (8 sections, 36 theorems, 150 equations)

This paper contains 8 sections, 36 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex geometry
  4. Birational geometry
  5. Group schemes associated to K-trivial fiber spaces
  6. From the generic to the relative weak movable cone conjecture
  7. Product of good minimal models
  8. Proof of the main theorem

Key Result

Theorem 1.4

Let $(X,\Delta)$ be a klt Calabi-Yau pair defined over $\mathbb C$ that decomposes as where $A$ is an abelian variety, each $Y_i$ is a primitive symplectic variety with canonical singularities and $b_2(Y_i) \geq 5$, and each $S_j$ is a smooth rational surface underlying a klt Calabi-Yau pair $(S_j,\Delta_j)$. Then the movable cone conjecture holds for every quotient pair $(X/G, \Delt

Theorems & Definitions (86)

  • Definition 1.1
  • Conjecture 1.2: The (weak) Kawamata-Morrison Cone conjecture
  • Definition 1.3
  • Theorem 1.4: Theorem 1.4 - gachet2025well
  • Theorem A
  • Corollary B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 76 more