Ampleness of canonical divisors of hyperbolic normal projective varieties

TL;DR

This paper proves that for algebraic Lang hyperbolic projective varieties with certain conditions, the canonical divisor is ample at smooth and Kawamata log terminal points, supporting parts of Lang's conjecture.

Contribution

It establishes the ampleness of the canonical divisor for hyperbolic varieties under specific conditions, advancing understanding of their geometric properties.

Findings

X and its subvarieties are of general type

K_X is ample at smooth points and KLT points

Supports Lang's conjecture in certain cases

Abstract

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety is algebraic Lang hyperbolic and a weak abundance conjecture holds.