Partially ample line bundles on toric varieties

TL;DR

This paper investigates the structure and properties of partially ample line bundles on simplicial projective toric varieties, including cone descriptions, restriction theorems, and vanishing results, revealing new geometric insights.

Contribution

It characterizes the cone of q-ample line bundles as a union of rational polyhedral cones and establishes a Kodaira-type vanishing theorem for these bundles.

Findings

The cone of q-ample line bundles is a union of rational polyhedral cones.

q-ampleness of the anticanonical bundle is not preserved under flips.

A Kodaira-type vanishing theorem for q-ample line bundles is proved.

Abstract

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.