Eventual positivity of Hermitian algebraic functions and associated integral operators

TL;DR

This paper provides a geometric proof demonstrating that certain integral operators associated with Hermitian algebraic functions become positive over time, extending previous results on positivity in complex geometry.

Contribution

It offers an elementary, geometric proof of the eventual positivity of integral operators linked to Hermitian algebraic functions, simplifying prior asymptotic approaches.

Findings

Elementary geometric proof of positivity

Extension of positivstellensatz to Hermitian algebraic functions

Simplification of asymptotic estimates

Abstract

Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator. The arguments of Catlin-D'Angelo, as well as that of a subsequent refinement by Varolin, involve subtle asymptotic estimates of the Bergman kernel. In this article, we give an elementary and geometric proof of the eventual positivity of this integral operator, thereby yielding another proof of the corresponding positivstellensatz.