Log canonical thresholds on group compactifications

TL;DR

This paper calculates the log canonical thresholds for certain line bundles on group compactifications, linking geometric properties to convex functions and polytopes, and providing a formula for the alpha invariant.

Contribution

It introduces a method to compute log canonical thresholds using convex functions and polytopes, advancing understanding of singular hermitian metrics on group compactifications.

Findings

Derived a formula for log canonical thresholds in terms of convex functions.

Connected asymptotic behavior of convex functions to thresholds.

Provided a new expression for the alpha invariant based on polytopes.

Abstract

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical threshold. As a consequence we obtain a formula for the alpha invariant of these line bundles, in terms of the polytope associated to the group compactification.