Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators∗ Colin TAN1
Wing-Keung TO2
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 2010 MR Subject Classification 32L05, 32A26, 32H02
1 Introduction A central topic in real algebraic geometry is Hilbert’s seventeenth problem of representing a nonnegative form on Rn as a sum of squares of rational functions. An affirmative solution was first provided by Artin’s positivstellensatz in 1927 [1]. Since then, related topics have continued to be widely studied from different viewpoints (see [6–8, 10, 11–12, 15–16] and the references therein). For the corresponding problem in the Hermitian case, Quillen [11] and Catlin-D’Angelo [6] proved independently the following positivstellensatz: for any Hermitian bihomogeneous polynomial f on Cn which is positive on Cn \ {0}, there exists ℓo > 0 such that for any ℓ ≥ ℓo ,
there are homogeneous holomorphic polynomials g1 , · · · , gN on Cn satisfying n X i=1
|zi |2
ℓ
· f (z) =
N X j=1
|gj (z)|2 ,
Manuscript received February 25, 2018. Revised June 7, 2019. of Statistics & Applied Probability, National University of Singapore. 1 Current address: Science, Mathematics and Technology, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372. E-mail: colin [email protected] 2 Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076. E-mail: [email protected] ∗ This work was partially supported by the Singapore Ministry of Education Academic Research Fund Tier 1 grant R-146-000-142-112. 1 Department
(1.1)
968
C. Tan and W.-K. To
where z = (z1 , · · · , zn ). Later, Catlin-D’Angelo generalized this positivstellensatz to the case of positive Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds, and this was formulated as an isometric embedding theorem of the associated Hermitian metrics on the line bundles in [7] (see Section 2 for the two equivalent definitions of Hermitian algebraic functions and Catlin-D’Angelo’s result stated as Theorem 2.1). For
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