A note on $\xi-$Bergman kernels

TL;DR

This paper introduces $\xi$-complex singularity exponents derived from the asymptotic behavior of $\xi$-Bergman kernels, establishing their relations with existing invariants and generalizing key properties of complex singularity exponents.

Contribution

It defines $\xi$-complex singularity exponents, explores their properties, and extends classical results to this new setting, enriching the understanding of singularity invariants.

Findings

Established relations among $\xi$-complex singularity exponents, complex singularity exponents, and jumping numbers.

Proved a closedness property for $\xi$-complex singularity exponents.

Generalized restriction formula and subadditivity property to $\xi$-complex singularity exponents.

Abstract

In the present note, we introduce the $\xi-$complex singularity exponents, which come from the asymptotic property of $\xi-$Bergman kernels on sub-level sets of plurisubharmonic functions; give some relations (including a closedness property) among $\xi-$complex singularity exponents, complex singularity exponents, and jumping numbers; generalize some properties of complex singularity exponents (such as the restriction formula and subadditivity property) to $\xi-$complex singularity exponents.