ORBIFOLD CHERN CLASSES INEQUALITIES AND APPLICATIONS

In this paper we prove that given a pair (X, D) of a threefold X and a boundary divisor D with mild singularities, if (K X + D) is movable, then the orbifold second Chern class c 2 of (X, D) is pseudoeffective. This generalizes the classical result of Miyaoka on the pseudoeffectivity of c 2 for minimal models. As an application, we give a simple solution to Kawamata's effective non-vanishing conjecture in dimension 3, where we prove that H 0 (X, K X + H) = 0, whenever K X + H is nef and H is an ample, effective, reduced Cartier divisor. Furthermore, we study Lang-Vojta's conjecture for codimension one subvarieties and prove that minimal threefolds of general type have only finitely many Fano, Calabi-Yau or Abelian subvarieties of codimension one that are mildly singular and whose numerical classes belong to the movable cone