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Random zeros on complex manifolds: conditional expectations

Part of: Holomorphic fiber spaces

Published online by Cambridge University Press:  11 March 2011

Bernard Shiffman ,
Steve Zelditch  and
Qi Zhong
Bernard Shiffman
Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA (shiffman@math.jhu.edu)
Steve Zelditch
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA (zelditch@math.northwestern.edu)
Qi Zhong
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA (qi.zhong@vanderbilt.edu)
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Abstract

We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros but we show that it has quite a different small distance behaviour. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension 1. To prove this, we give universal scaling asymptotics for around p. The key tool is the conditional Szegő kernel and its scaling asymptotics.

Keywords

random holomorphic sectionszeros of random polynomialsholomorphic line bundleKähler manifoldSzegő kernel

MSC classification

Secondary: 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results

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