Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators

Citation:

Colin TAN,Wing-Keung TO.Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators[J].Chinese Annals of Mathematics B,2020,41(6):967~988
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Authors:

Colin TAN; Wing-Keung TO

Foundation:

This work was partially supported by the Singapore Ministry of Education Academic Research Fund Tier 1 grant R-146-000-142-112.
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 involve subtle asymptotic estimates of the Bergman kernel. In this article, the authors give an elementary and geometric proof of the eventual positivity of this integral operator, thereby yielding another proof of the corresponding positivstellensatz.

Keywords:

Hermitian algebraic functions, Integral operators, Positivity

Classification:

32L05, 32A26, 32H02
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