Orbifold Chern classes inequalities and applications

Abstract

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.