Alpha-invariant of Toric Line Bundles

TL;DR

This paper extends the computation of the alpha invariant to all nef and big toric line bundles using polytope data and analytic methods, providing a unified approach to singular hermitian metrics.

Contribution

It generalizes Jian Song's work by deriving a formula for the alpha invariant of any nef and big toric line bundle via associated polytopes and analytic log canonical threshold calculations.

Findings

Derived a formula for the alpha invariant in terms of polytopes.

Connected the alpha invariant to the log canonical threshold of monomial ideals.

Provided a method to compute the alpha invariant for non-negatively curved singular hermitian metrics.

Abstract

We generalize the work of Jian Song to compute the alpha invariant of any (nef and big) toric line bundle in terms of the associated polytope. We use the analytic version of the computation of the log canonical threshold of monomial ideals to give the log canonical threshold of any non-negatively curved singular hermitian metric on the line bundle, and deduce the alpha invariant from this.