Random Sections of Line Bundles over Real Riemann Surfaces

Michele Ancona*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)7004-7059
Number of pages56
JournalInternational Mathematics Research Notices
Issue number9
StatePublished - 1 May 2021
Externally publishedYes


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