Asymptotics of eigensections on toric varieties

TL;DR

This paper investigates the asymptotic behavior of eigensections on toric varieties, establishing convergence results for normalized sections approaching a semiclassical limit using invariant plurisubharmonic functions and combinatorial data.

Contribution

It introduces new convergence results for eigensections on toric varieties by combining geometric analysis with combinatorial methods.

Findings

Sequences of normalized eigensections converge to a semiclassical ray.

Utilizes exhaustion properties of invariant plurisubharmonic functions.

Provides a framework linking geometric, combinatorial, and asymptotic analysis.

Abstract

Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties convergence results for sequences of distribution functions \phi_n=|s_N| / |s_N|_{L^2} for sections s_N\in \Gamma (X,L^N) approaching a semiclassical ray are proved. Here X is a normal compact toric variety and L is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus.