Semiampleness for generalized pairs

TL;DR

This paper proves that for generalized klt pairs with nef and abundant canonical divisors, the divisors are semiample, extending the basepoint free theorem to this broader context.

Contribution

It establishes semi-ampleness for generalized klt pairs with nef and abundant canonical divisors, generalizing the basepoint free theorem.

Findings

Proves semi-ampleness of $K_X+B+ extbf{M}_X$ for generalized klt pairs.

Extends the basepoint free theorem to generalized pairs.

Provides conditions under which the canonical divisor is semiample.

Abstract

We prove that if $(X, B+\mathbf{M})$ is a generalized klt pair with $K_X+B+\mathbf{M}_X$ nef and abundant, then $K_X+B+\mathbf{M}_X$ is semiample. More generally, we prove a generalized basepoint free theorem for generalized klt pairs.