The effective cone conjecture for Calabi--Yau pairs

We formulate an { effective cone conjecture} for klt Calabi--Yau pairs (X,Δ), pertaining to the structure of the cone of effective divisors Eff(X) modulo the action of the subgroup of pseudo-automorphisms PsAut(X,Δ). Assuming the existence of good minimal models in dimension (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). 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.