Analyticity theorems for parameter-dependent plurisubharmonic functions

TL;DR

This paper establishes that certain unions of level sets related to fiberwise Lelong numbers and complex singularity exponents are generally pluripolar or analytic, with counterexamples showing limits of analyticity.

Contribution

It proves analyticity theorems for unions of sub-level sets of fiberwise invariants of plurisubharmonic functions, extending understanding of their geometric structure.

Findings

Union of upper-level sets of fiberwise Lelong numbers is pluripolar.

Under certain conditions, unions of sub-level sets of complex singularity exponents are analytic.

Counterexamples show these sets can be non-analytic even for continuous plurisubharmonic functions.

Abstract

In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous.