A pr 2 00 1 ON MORRISON ’ S CONE CONJECTURE FOR KLT SURFACES WITH K X ≡ 0

This paper considers normal projective complex surfaces X with at worst Kawamata log terminal singularities and KX ≡ 0. The aim is to prove that there is a finite rational polyhedral cone which is a fundamental domain for the action AutX on the convex hull of its rational ample cone. 0. Introduction LetX be a normal projective complex surface with at worst Kawamata log terminal singularities (klt singularities) andKX ≡ 0. By the classification of surfaces, X is either an Abelian surface, a hyperelliptic surface, a K3 surface with only rational double points (RDPs), an Enriques surface with only RDPs or a log Enriques surface. Here a log Enriques surface (compare Zhang [Zh]) is a rational normal projective surface with at worst klt singularities and KX ≡ 0. The aim of this paper is to study the 2-dimensional klt analogue of the following conjecture of Morrison: Conjecture 0.1 (Morrison’s Cone Conjecture [M3, Section 4]). Let V be a Calabi–Yau manifold, A(V ) its ample cone, and A′(V ) the convex hull of its rational ample cone A(V )∩H2(V,Q). Then there exists a rational polyhedral cone ∆ ⊂ A′(V ) such that AutV ·∆ = A′(V ). Our main result is as follows: Theorem 0.2 (Main Theorem). Let X be a normal projective complex surface with at worst klt singularities and KX ≡ 0. Then there exists a rational finite polyhedral cone ∆ which is a fundamental domain for the action of AutX on A′(X) in the sense that 1. A′(X) = ⋃ θ∈AutX θ∗∆, 2. Int∆ ∩ θ∗ Int∆ = ∅ unless θ∗ = id. This conjecture is known for a smooth Abelian variety (Kawamata [Kaw]) or a smooth K3 surface (Sterk [St]). Our new result covers the remaining cases. The plan of the proof is as follows: for X as in Theorem 0.2, write I = I(X) for the (global) index of X, that is, the smallest positive integer such that IKX is Cartier and linearly equivalent to zero and π : Y := Spec( ⊕I−1 i=0 OX(−iKX)) → X for the (global) index one cover of X. Then π : Y → X is a Galois finite cover with Galois group G = 〈g〉 ≃ Z/I, and g(ωY ) = ζIωY , where ωY is a generator of H (OY (KY )) and ζI a primitive Ith root of 1. It is known that Y → X is etale in codimension 1, so that 1