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On the relative Morrison-Kawamata cone conjecture

Zhan Li, Hang Zhao

TL;DR

The paper links the relative Morrison–Kawamata cone conjecture for Calabi–Yau fiber spaces to the existence of Shokurov polytopes, establishing a polyhedral-tiling framework for movable and ample cones under effective automorphism groups. It proves a weak cone conjecture for movable cones in the K3-fibered case and shows how Shokurov polytopes provide a route to finite-domain, polyhedral descriptions that reflect finiteness of models. A central theme is connecting relative cone conjectures to cone conjectures on generic and geometric fibers, via LMMP assumptions and base-change techniques, including finite étale Galois base changes and spreading-out arguments. The work develops a robust cone-geometry toolkit, clarifying when generic- or geometric-fiber results transfer to the relative setting and highlighting how degeneracies (nonzero W) affect the existence of fundamental domains.

Abstract

We relate the Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces to the existence of Shokurov polytopes. For K3 fibrations, the existence of (weak) fundamental domains for movable cones is established. The relationship between the relative cone conjecture and the cone conjecture for its geometric or generic fibers is studied.

On the relative Morrison-Kawamata cone conjecture

TL;DR

The paper links the relative Morrison–Kawamata cone conjecture for Calabi–Yau fiber spaces to the existence of Shokurov polytopes, establishing a polyhedral-tiling framework for movable and ample cones under effective automorphism groups. It proves a weak cone conjecture for movable cones in the K3-fibered case and shows how Shokurov polytopes provide a route to finite-domain, polyhedral descriptions that reflect finiteness of models. A central theme is connecting relative cone conjectures to cone conjectures on generic and geometric fibers, via LMMP assumptions and base-change techniques, including finite étale Galois base changes and spreading-out arguments. The work develops a robust cone-geometry toolkit, clarifying when generic- or geometric-fiber results transfer to the relative setting and highlighting how degeneracies (nonzero W) affect the existence of fundamental domains.

Abstract

We relate the Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces to the existence of Shokurov polytopes. For K3 fibrations, the existence of (weak) fundamental domains for movable cones is established. The relationship between the relative cone conjecture and the cone conjecture for its geometric or generic fibers is studied.
Paper Structure (11 sections, 26 theorems, 107 equations)

This paper contains 11 sections, 26 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Movable cones and ample cones
  4. Minimal models of varieties
  5. Shokurov polytopes
  6. Geometry of convex cones
  7. Generic properties of fibrations and structures of cones
  8. A variant of the cone conjecture
  9. Generic and Geometric cone conjectures
  10. Generic cone conjecture
  11. Geometric cone conjecture

Key Result

Theorem 1.3

Let $f: (X,\Delta) \to S$ be a klt Calabi-Yau fiber space.

Theorems & Definitions (62)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4: HX13
  • Theorem 2.5: SC11
  • ...and 52 more