Asymptotics of partial density function vanishing along smooth subvariety

TL;DR

This paper investigates the asymptotic behavior of the partial density function for holomorphic sections of a positive line bundle that vanish along a smooth subvariety, extending previous results under torus-invariance assumptions.

Contribution

It generalizes the understanding of partial density functions' asymptotics by describing the forbidden region for sections vanishing along subvarieties, building on Ross-Singer's work.

Findings

Description of the forbidden region for vanishing sections

Extension of asymptotic analysis to smooth subvarieties

Generalization of previous results under torus-action-invariance

Abstract

We study the asymptotic of the partial density function associated to holomorphic section of a postive line bundle vanishing to high orders along a fixed smooth subvariety. Assuming local torus-action-invariance, we describe the forbidden region, generalizing the result of Ross-Singer.