Geometric and Functional Analysis

, Volume 18, Issue 4, pp 1422–1475

Number Variance of Random Zeros on Complex Manifolds


DOI: 10.1007/s00039-008-0686-3

Cite this article as:
Shiffman, B. & Zelditch, S. GAFA Geom. funct. anal. (2008) 18: 1422. doi:10.1007/s00039-008-0686-3


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 \(U \subset {\mathbb{C}}^m\) with smooth boundary is asymptotic to \(N^{{m-1}/2} \nu_{mm} {\rm Vol}(\partial U)\), where \(\nu_{mm}\) 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 \({\mathbb{C}}^{m}\). 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.

Keywords and phrases:

Random holomorphic sections zeros of random polynomials holomorphic line bundle Kähler manifold Szegő kernel 

AMS Mathematics Subject Classification:

32L10 60D05 32A60 

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

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