Estimates for the Bergman Kernel and the Multidimensional Suita Conjecture

TL;DR

This paper investigates bounds for the Bergman kernel related to the multidimensional Suita conjecture, establishing new inequalities involving volume measures and analyzing convex domains and ellipsoids.

Contribution

It introduces a multidimensional version of the Suita conjecture by relating the Bergman kernel to the Azukawa indicatrix volume, with bounds for convex domains.

Findings

Lower bounds for the Bergman kernel in terms of volume of sublevel sets.

Upper bounds for convex domains and complex ellipsoids.

Connection between the Bergman kernel and the Azukawa indicatrix.

Abstract

We study the lower bound for the Bergman kernel in terms of volume of sublevel sets of the pluricomplex Green function. We show that it implies a bound in terms of volume of the Azukawa indicatrix which can be treated as a multidimensional version of the Suita conjecture. We also prove that the corresponding upper bound holds for convex domains and discuss it in bigger detail on some convex complex ellipsoids.