Spectral measures on toric varieties and the asymptotic expansion of Tian-Yau-Zelditch

TL;DR

This paper extends spectral measure asymptotics for toric varieties and uses them with Tian-Yau-Zelditch expansion to derive Abreu's scalar curvature formula from polytope data.

Contribution

It generalizes spectral measure asymptotics to any toric metric and connects these results with scalar curvature formulas in toric geometry.

Findings

Extended spectral measure asymptotics to all toric metrics.

Derived Abreu's scalar curvature formula from polytope data.

Connected spectral analysis with geometric curvature computations.

Abstract

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the scalar curvature of a toric metric on a toric variety in terms of polytope data.