Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The effective cone conjecture for Calabi--Yau pairs

Cécile Gachet, Hsueh-Yung Lin, Isabel Stenger, Long Wang

Abstract

We formulate an effective cone conjecture for klt Calabi--Yau pairs $(X,Δ)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,Δ)$. Assuming the existence of good minimal models in dimension $\dim(X)$, known to hold in dimension up to $3$, we prove that the effective cone conjecture for $(X,Δ)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,Δ)$, among other statements. As an application, we show that the movable cone conjecture unconditionally holds for the smooth Calabi--Yau threefolds introduced by Schoen and studied by Namikawa, Grassi and Morrison. We also show that for such a Calabi--Yau threefold $X$, all of its minimal models, apart from $X$ itself, have rational polyhedral nef cones.

The effective cone conjecture for Calabi--Yau pairs

Abstract

We formulate an effective cone conjecture for klt Calabi--Yau pairs , pertaining to the structure of the cone of effective divisors modulo the action of the subgroup of pseudo-automorphisms . Assuming the existence of good minimal models in dimension , known to hold in dimension up to , we prove that the effective cone conjecture for is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for , among other statements. As an application, we show that the movable cone conjecture unconditionally holds for the smooth Calabi--Yau threefolds introduced by Schoen and studied by Namikawa, Grassi and Morrison. We also show that for such a Calabi--Yau threefold , all of its minimal models, apart from itself, have rational polyhedral nef cones.
Paper Structure (35 sections, 33 theorems, 95 equations)

This paper contains 35 sections, 33 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. A chamber decomposition of the effective cone
  3. The effective cone conjecture, and other cone conjectures
  4. A descent result for the nef cone conjecture
  5. Contextualizing Theorem and relation to other works
  6. Cone conjectures for the Calabi--Yau threefolds introduced by Schoen
  7. Notation and preliminaries
  8. Cones of numerical classes of divisors.
  9. Some group actions on cones of divisors
  10. Birational contractions and pseudo-automorphisms
  11. Pushforwards and pullbacks
  12. Pullbacks, pushforwards, and composition.
  13. The $\mathrm{PsAut}(X)$-action on $N^1(X)_{\mathbb{R}}$
  14. Some standard conjectures of the minimal model program
  15. Convex geometry
  16. ...and 20 more sections

Key Result

Proposition 1.1

Let $(X,\Delta)$ be a klt Calabi--Yau pair. Assume the existence of minimal models for $X$ as in Definition (3). We have a chamber decomposition

Theorems & Definitions (74)

  • Conjecture : Kawamata--Morrison cone conjecture
  • Proposition 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 2.1
  • Lemma 2.2
  • ...and 64 more