The effective cone conjecture for Calabi--Yau pairs

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