Orbifold Chern classes inequalities and applications

TL;DR

This paper proves that the orbifold second Chern class is pseudo-effective for certain threefold pairs with mild singularities, extending classical results and applying to conjectures in algebraic geometry.

Contribution

It generalizes Miyaoka's classical result to pairs with mild singularities and applies this to solve Kawamata's non-vanishing conjecture in dimension three.

Findings

Orbifold second Chern class is pseudo-effective for threefold pairs with mild singularities.

Proves Kawamata's effective non-vanishing conjecture in dimension 3.

Shows finiteness of certain subvarieties in minimal varieties of general type.

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 pseudo-effective. This generalizes the classical result of Miyaoka on the pseudo-effectivity 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 $H^0(X, K_X+H)\neq 0$, 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 varieties of general type have only finitely many Fano, Calabi-Yau or Abelian subvarieties of codimension one, mildly singular, whose classes belong to the movable cone.