Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

Dan Coman; Wen Lu; Xiaonan Ma; George Marinescu

doi:10.1016/j.aim.2022.108854

Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

Dan Coman, Wen Lu, Xiaonan Ma, George Marinescu

Department of Mathematics

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

Abstract

Given a sequence of positive Hermitian holomorphic line bundles (L_p,h_p) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of L_p, under a natural convergence assumption on the sequence of curvatures c₁(L_p,h_p). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of L_p as p→+∞.

Original language	English (US)
Article number	108854
Journal	Advances in Mathematics
Volume	414
DOIs	https://doi.org/10.1016/j.aim.2022.108854
State	Published - Feb 1 2023

Keywords

Approximation of currents by analytic sets
Bergman kernel
Non-integral Kähler metric
Zeros of random holomorphic sections

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2022.108854

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title = "Bergman kernels and equidistribution for sequences of line bundles on K{\"a}hler manifolds",

abstract = "Given a sequence of positive Hermitian holomorphic line bundles (Lp,hp) on a K{\"a}hler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of Lp, under a natural convergence assumption on the sequence of curvatures c1(Lp,hp). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of Lp as p→+∞.",

keywords = "Approximation of currents by analytic sets, Bergman kernel, Non-integral K{\"a}hler metric, Zeros of random holomorphic sections",

author = "Dan Coman and Wen Lu and Xiaonan Ma and George Marinescu",

note = "Funding Information: D. C. is partially supported by the Simons Foundation Grant 853088 and the NSF Grant DMS-2154273.W. L. supported by NSFC Nos. 11401232, 11871233.X. M. partially supported by NSFC No. 11829102, ANR-21-CE40-0016 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.G. M. partially supported by the DFG funded projects SFB TRR 191 {\textquoteleft}Symplectic Structures in Geometry, Algebra and Dynamics{\textquoteright} (Project-ID 281071066 – TRR 191) and SPP 2265 {\textquoteleft}Random Geometric Systems{\textquoteright} (Project-ID 422743078). Publisher Copyright: {\textcopyright} 2023 Elsevier Inc.",

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doi = "10.1016/j.aim.2022.108854",

language = "English (US)",

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T1 - Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

AU - Coman, Dan

AU - Lu, Wen

AU - Ma, Xiaonan

AU - Marinescu, George

N1 - Funding Information: D. C. is partially supported by the Simons Foundation Grant 853088 and the NSF Grant DMS-2154273.W. L. supported by NSFC Nos. 11401232, 11871233.X. M. partially supported by NSFC No. 11829102, ANR-21-CE40-0016 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.G. M. partially supported by the DFG funded projects SFB TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’ (Project-ID 281071066 – TRR 191) and SPP 2265 ‘Random Geometric Systems’ (Project-ID 422743078). Publisher Copyright: © 2023 Elsevier Inc.

PY - 2023/2/1

Y1 - 2023/2/1

N2 - Given a sequence of positive Hermitian holomorphic line bundles (Lp,hp) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of Lp, under a natural convergence assumption on the sequence of curvatures c1(Lp,hp). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of Lp as p→+∞.

AB - Given a sequence of positive Hermitian holomorphic line bundles (Lp,hp) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of Lp, under a natural convergence assumption on the sequence of curvatures c1(Lp,hp). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of Lp as p→+∞.

KW - Approximation of currents by analytic sets

KW - Bergman kernel

KW - Non-integral Kähler metric

KW - Zeros of random holomorphic sections

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Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

Abstract

Keywords

ASJC Scopus subject areas

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