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On the algebraicity about the Hodge numbers of the Hilbert schemes of algebraic surfaces

Published online by Cambridge University Press:  19 April 2022

Seokho Jin
Affiliation:
Department of Mathematics, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Republic of Korea ([email protected])
Sihun Jo
Affiliation:
Department of Mathematics Education, Woosuk University, 443 Samnye-ro, Samnye-eup, Wanju-Gun, Jeollabuk-do 55338, Republic of Korea ([email protected])

Abstract

Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.

Type
Research Article
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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