TY - JOUR

T1 - Random Sections of Line Bundles over Real Riemann Surfaces

AU - Ancona, Michele

N1 - Publisher Copyright:
© 2019 The Author(s) 2019. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

PY - 2021/5/1

Y1 - 2021/5/1

N2 - Let L be a positive line bundle over a Riemann surface Σ defined over R. We prove that sections s of Ld, d gg 0, whose number of real zeros \#Z-s deviates from the expected one are rare. We also provide asymptotics of the form E\#Z-s-E}[\# Z-s])k]=Od{k-1-α}) and E}[\#Zk-s]=a-kd k}+b-kd k-1}+Od k-1-α }) for all the (central) moments of the number of real zeros. Here α is any number in (0,1), and a-k and b-k are some explicit and positive constants. Finally, we obtain similar asymptotics for the distribution of complex zeros of random sections. Our proof involves Bergman kernel estimates as well as Olver multispaces.

AB - Let L be a positive line bundle over a Riemann surface Σ defined over R. We prove that sections s of Ld, d gg 0, whose number of real zeros \#Z-s deviates from the expected one are rare. We also provide asymptotics of the form E\#Z-s-E}[\# Z-s])k]=Od{k-1-α}) and E}[\#Zk-s]=a-kd k}+b-kd k-1}+Od k-1-α }) for all the (central) moments of the number of real zeros. Here α is any number in (0,1), and a-k and b-k are some explicit and positive constants. Finally, we obtain similar asymptotics for the distribution of complex zeros of random sections. Our proof involves Bergman kernel estimates as well as Olver multispaces.

UR - http://www.scopus.com/inward/record.url?scp=85108361469&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz051

DO - 10.1093/imrn/rnz051

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AN - SCOPUS:85108361469

SN - 1073-7928

VL - 2021

SP - 7004

EP - 7059

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 9

ER -