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Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting

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Abstract

We consider tensor powers L N of a positive Hermitian line bundle (L,h L) over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed as N→∞ with respect to the natural measure coming from the curvature of L. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL 2(ℤ) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.

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Authors and Affiliations

  1. UMR 7586, Institut de Mathématiques de Jussieu, UPMC Univ. Paris 06, 4 place Jussieu, 75005, Paris, France

    Tien-Cuong Dinh

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Cologne, Germany

    George Marinescu & Viktoria Schmidt

Authors
  1. Tien-Cuong Dinh
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  2. George Marinescu
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  3. Viktoria Schmidt
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Correspondence to George Marinescu.

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Supported by a DFG funded Mercator Professorship, SFB/TR 12 and Graduiertenkolleg 1269 “Globale Strukturen in Geometrie und Analysis”

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Dinh, TC., Marinescu, G. & Schmidt, V. Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting. J Stat Phys 148, 113–136 (2012). https://doi.org/10.1007/s10955-012-0526-6

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  • DOI: https://doi.org/10.1007/s10955-012-0526-6

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