The effective cone conjecture for Calabi--Yau pairs

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.

arXiv commentv2: Theorem 6.1 is strengthened: The existence of good minimal models is now assumed only for specific varieties (see new Assumption 2.5) instead of every variety of a given dimension. The definition of the group $\mathrm{PsAut}(X,\Delta ;f)$ is corrected, and the proofs of Lemmas 5.1, 5.2 are modified accordingly. Exposition is shortened and improved following the referees' feedback