Note on Quasi-Numerically Positive Log Canonical Divisors

Shigetaka Fukuda

doi:10.1155/2015/841746

Abstract

We propose a subconjecture that implies the semiampleness conjecture for quasi-numerically positive log canonical divisors and prove the ampleness in some elementary cases.

Highlights

In this note, every algebraic variety is defined over the field C of complex numbers
In Fukuda [12], we proved that if the log canonical divisor on a Q-factorial divisorial log terminal variety is nef and log big, it is semiample
In Fukuda [13] (2011), we proved that if the log canonical divisor on a klt variety is numerically equivalent to some semiample Qdivisor, it is semiample

Summary

Introduction

Every algebraic variety is defined over the field C of complex numbers. Ambro [3] and Birkar et al [4] reduced the famous log abundance conjecture to the termination conjecture for log flips and the semiampleness conjecture (Conjecture 4) for quasi-nup log canonical divisors KX + Δ, in the category of Kawamata log terminal (klt, for short) pairs. In Fukuda [12] (base point free theorem of Reid type, 1999), we proved that if the log canonical divisor on a Q-factorial divisorial log terminal variety is nef and log big, it is semiample. In Fukuda [13] (2011), we proved that if the log canonical divisor on a klt variety is numerically equivalent to some semiample Qdivisor, it is semiample There is another approach to the semiampleness Conjecture 4. In Appendix B, we give a straightforward proof to the theorem due to Boucksom et al [16] and Birkar and Hu [17] and Cacciola [18] that, for every divisorial log terminal pair whose log canonical divisor is strongly log big, the log canonical ring is finitely generated

Subconjecture for klt Pairs

Log Canonical Pairs in Some Elementary Cases

Strong Log Bigness

Disclosure