Number variance of random zeros on complex manifolds

Bernard Shiffman; Steve Zelditch

doi:10.1007/s00039-008-0686-3

Number variance of random zeros on complex manifolds

Bernard Shiffman^*, Steve Zelditch

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

48 Scopus citations

Abstract

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set U ⊆C^m with smooth boundary is asymptotic to N^m-1/2ν^mm Vol ∂U , where ν^mm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on C^m . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact Kähler manifold.

Original language	English (US)
Pages (from-to)	1422-1475
Number of pages	54
Journal	Geometric and Functional Analysis
Volume	18
Issue number	4
DOIs	https://doi.org/10.1007/s00039-008-0686-3
State	Published - Dec 2008

Keywords

Holomorphic line bundle
Kähler manifold
Random holomorphic sections
Szego″ kernel
Zeros of random polynomials

ASJC Scopus subject areas

Analysis
Geometry and Topology

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title = "Number variance of random zeros on complex manifolds",

abstract = "We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set U ⊆Cm with smooth boundary is asymptotic to Nm-1/2νmm Vol ∂U , where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on Cm . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact K{\"a}hler manifold.",

keywords = "Holomorphic line bundle, K{\"a}hler manifold, Random holomorphic sections, Szego″ kernel, Zeros of random polynomials",

author = "Bernard Shiffman and Steve Zelditch",

note = "Funding Information: Keywords and phrases: Random holomorphic sections, zeros of random polynomials, holomorphic line bundle, K{\"a}hler manifold, Szego˝ kernel AMS Mathematics Subject Classification: 32L10, 60D05, 32A60 Research of the first author partially supported by NSF grants DMS-0100474 and DMS-0600982; research of the second author partially supported by NSF grants DMS-0302518 and DMS-0603850.",

year = "2008",

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doi = "10.1007/s00039-008-0686-3",

language = "English (US)",

volume = "18",

pages = "1422--1475",

journal = "Geometric and Functional Analysis",

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T1 - Number variance of random zeros on complex manifolds

AU - Shiffman, Bernard

AU - Zelditch, Steve

N1 - Funding Information: Keywords and phrases: Random holomorphic sections, zeros of random polynomials, holomorphic line bundle, Kähler manifold, Szego˝ kernel AMS Mathematics Subject Classification: 32L10, 60D05, 32A60 Research of the first author partially supported by NSF grants DMS-0100474 and DMS-0600982; research of the second author partially supported by NSF grants DMS-0302518 and DMS-0603850.

PY - 2008/12

Y1 - 2008/12

N2 - We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set U ⊆Cm with smooth boundary is asymptotic to Nm-1/2νmm Vol ∂U , where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on Cm . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact Kähler manifold.

AB - We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set U ⊆Cm with smooth boundary is asymptotic to Nm-1/2νmm Vol ∂U , where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on Cm . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact Kähler manifold.

KW - Holomorphic line bundle

KW - Kähler manifold

KW - Random holomorphic sections

KW - Szego″ kernel

KW - Zeros of random polynomials

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