Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators

Chinese Annals of Mathematics, Series B volume 41, pages967–988(2020)Cite this article

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.

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Author information

Author notes

  1. Colin Tan

    Present address: Current address: Science, Mathematics and Technology, Singapore University of Technology and Design, 8 Somapah Road, Singapore, 487372, Singapore

Affiliations

  1. Department of Statistics & Applied Probability, National University of Singapore, Singapore, Singapore

    Colin Tan

  2. Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Wing-Keung To

Authors
  1. Colin Tan
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  2. Wing-Keung To
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Corresponding authors

Correspondence to Colin Tan or Wing-Keung To.

Additional information

This work was partially supported by the Singapore Ministry of Education Academic Research Fund Tier 1 grant R-146-000-142-112.

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Tan, C., To, WK. Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators. Chin. Ann. Math. Ser. B 41, 967–988 (2020). https://doi.org/10.1007/s11401-020-0241-6

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Keywords

  • Hermitian algebraic functions
  • Integral operators
  • Positivity

2010 MR Subject Classification

  • 32L05
  • 32A26
  • 32H02