Number variance of random zeros on complex manifolds

Bernard Shiffman*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-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 ⊆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.

Original languageEnglish (US)
Pages (from-to)1422-1475
Number of pages54
JournalGeometric and Functional Analysis
Volume18
Issue number4
DOIs
StatePublished - 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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