The effective cone conjecture for Calabi--Yau pairs

TL;DR

This paper formulates a conjecture relating the effective cone of divisors on Calabi--Yau pairs to automorphisms, proves its equivalence to a movable cone conjecture under certain conditions, and confirms the conjecture for specific Calabi--Yau threefolds.

Contribution

It introduces an effective cone conjecture for Calabi--Yau pairs, proves its equivalence to the movable cone conjecture assuming minimal models, and verifies the conjecture for certain smooth Calabi--Yau threefolds.

Findings

Effective cone conjecture is equivalent to the movable cone conjecture under certain assumptions.

Unconditional proof of the movable cone conjecture for specific Calabi--Yau threefolds.

All minimal models of these threefolds have rational polyhedral nef cones.

Abstract

We formulate an effective cone conjecture for klt Calabi--Yau pairs $(X,\Delta)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,\Delta)$. 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 $(X,\Delta)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,\Delta)$, 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.