On Morrison’s Cone Conjecture for Klt Surfaces with Numerical Trivial Canonical Divisor

In this paper, we consider the normal projective complex surface X which has at most klt (=Kawamata log terminal singularities) as its singularities and KX ≡ 0. The aim of this paper is to prove that there is a finite rational polyhedral cone which is a fundamental domain for the action Aut(X) on the rational convex hull of its ample cone. 0. Introduction Let X be a normal projective complex surface which has at most klt as its singularities and KX ≡ 0. Then, by the classification of surfaces, X is either an abelian surface, a hyperelliptic surface, a K3 surface with only RDP, an Enriques surface with only RDP, where RDP abbreviates rational double points or a log Enriques surface. Here, a log Enriques surface is a rational normal projective surface which admits at most klt as its singularities and satisfies KX ≡ 0. [Zh] The aim of this paper is to study the 2-dimensional klt analogue of the following Morrison’s Cone Conjecture: Conjecture (Cone Conjecture [M3, Section 4]) Let V be a Calabi-Yau manifold and let A(V ) be an ample cone of V and A′(V ) the convex hull of A(V ) ∩H 2(V,Q) of the ample cone. Then there exists a rational polyhedral cone ∆ ⊂ A′(V ) such that Aut(V )∆ = A′(V ). Our main result is as follows: Theorem 0.1 (Main Theorem). Let X be a normal projective complex surface which has at most klt as its singularities and KX ≡ 0. Then, there exists a rational finite polyhedral cone ∆ which is a fundamental domain for the action of Aut(X) on A′(X) in the sense that 1. A′(X) = ⋃ θ∈Aut(X) θ∗∆, 2. Int∆ ∩ θ∗Int∆ = ∅ unless θ∗ = id. It is known that this conjecture is valid for the smooth abelian case [Kaw] and the smooth K3 surface case [St]. So the rest case is the new results. The outline of the proof is as follows: For X in Theorem 0.1, there exists the smallest positive integer I = I(X) such that IKX is Cartier and is linearly equivalent to zero, and one can obtain the global canonical cover π : Y := Spec(⊕ i=0OX(−iKX)) → X. Here, Y is a projective surface with only rational double points and satisfies OY (KY ) ≃ OY , i.e. Y is either a projective K3 surface with only rational