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The Morrison Cone Conjecture under Deformation

Wendelin Lutz

TL;DR

The paper proves that the Morrison cone conjecture for a smooth Calabi–Yau threefold is invariant under deformation to any deformation-equivalent smooth CY3-fold, enabling the conjecture to be proved in broader families by passing to convenient degenerations. Central to the method are the big and small Wilson Weyl groups, $W_Y^{big}$ and $W_Y^{sm}$, realized via flops, monodromy, and parallel transport, which control the movable and nef cone structures across the deformation space and relate them to automorphism and pseudo-automorphism groups. The authors apply this framework to give a short proof of the cone conjecture for all smooth $(2,2,2,2)$-divisors in $(\mathbb{P}^1)^4$ and to extend cone-conjecture results to Borcea–Voisin-type families and certain double covers of klt log Calabi–Yau pairs using Xu’s results. Overall, the work provides a deformation-theoretic paradigm for establishing the Morrison cone conjecture in new CY3-fold classes by leveraging Weyl-group actions and MMP along universal deformations.

Abstract

We prove that if the Morrison cone conjecture holds for a smooth Calabi-Yau threefold $Y$, it holds for any smooth Calabi-Yau threefold deformation-equivalent to $Y$. We use this result to prove a new case of the Morrison cone conjecture.

The Morrison Cone Conjecture under Deformation

TL;DR

The paper proves that the Morrison cone conjecture for a smooth Calabi–Yau threefold is invariant under deformation to any deformation-equivalent smooth CY3-fold, enabling the conjecture to be proved in broader families by passing to convenient degenerations. Central to the method are the big and small Wilson Weyl groups, and , realized via flops, monodromy, and parallel transport, which control the movable and nef cone structures across the deformation space and relate them to automorphism and pseudo-automorphism groups. The authors apply this framework to give a short proof of the cone conjecture for all smooth -divisors in and to extend cone-conjecture results to Borcea–Voisin-type families and certain double covers of klt log Calabi–Yau pairs using Xu’s results. Overall, the work provides a deformation-theoretic paradigm for establishing the Morrison cone conjecture in new CY3-fold classes by leveraging Weyl-group actions and MMP along universal deformations.

Abstract

We prove that if the Morrison cone conjecture holds for a smooth Calabi-Yau threefold , it holds for any smooth Calabi-Yau threefold deformation-equivalent to . We use this result to prove a new case of the Morrison cone conjecture.
Paper Structure (22 sections, 47 theorems, 74 equations)

This paper contains 22 sections, 47 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The movable cone under deformation
  4. The behaviour of the movable cone in families
  5. Deformation of divisorial contractions
  6. Proof of Theorem \ref{['thm:main1']}
  7. Roots and the Wilson Weyl group
  8. Generalized Simple Roots
  9. Root systems on Calabi-Yau threefolds
  10. Geometric realization of Weyl groups
  11. The Weyl group action on the movable and the nef cone
  12. The Weyl group and monodromy
  13. The Weyl group action on the deformation space
  14. Brief review of intersection cohomology
  15. The action of $\mathrm{PsAut}(Y)$
  16. ...and 7 more sections

Key Result

Theorem 1.1

(see Theorem thm:MCClocal) If the movable (respectively nef) cone conjecture holds for a smooth Calabi-Yau threefold $Y$, then the movable (respectively nef) cone conjecture holds for any smooth Calabi-Yau threefold deformation-equivalent to $Y$.

Theorems & Definitions (104)

  • Conjecture : Morrison nef cone conjecture
  • Conjecture : KawamataCY movable cone conjecture
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • ...and 94 more