Anticanonical minimal models and Zariski decomposition

TL;DR

This paper extends the theory of minimal models to the anticanonical setting, showing that certain positivity conditions imply the existence of anticanonical minimal models for pairs with specific singularities.

Contribution

It proves that pairs with pklt singularities and a birational Zariski decomposition of the negative canonical divisor have anticanonical minimal models, complementing previous results for canonical models.

Findings

Establishes the existence of anticanonical minimal models under new conditions.

Extends the framework of Zariski decompositions to the anticanonical case.

Provides new tools for studying the birational geometry of pairs with pklt singularities.

Abstract

Birkar and Hu showed that if a pair $(X,\Delta)$ is lc and $K_{X}+\Delta$ admits a birational Zariski decomposition, then $(X,\Delta)$ has a minimal model. Analogously, we prove that if a pair $(X,\Delta)$ is pklt and $-(K_{X}+\Delta)$ admits a birational Zariski decomposition, then $(X,\Delta)$ has an anticanonical minimal