Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 5

Hilbert schemes of points and quasi-modularity

Pages: 1673 – 1706

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a11


Zhongyan Shen (Department of Mathematics, Zhejiang International Studies University, Hangzhou, China)

Zhenbo Qin (Department of Mathematics, University of Missouri, Columbia, Mo., U.S.A.)


We study further connections between Hilbert schemes of points on a smooth projective (complex) surface and quasimodular forms. We prove that the leading terms of certain generating series (with variable $q$) involving intersections with the total Chern classes of the tangent bundles of these Hilbert schemes are quasi-modular forms. The main idea is to link these leading terms with those coming from the equivariant setting for the complex plane $\mathbb{C}^2$.


Hilbert scheme, quasi-modular form, projective surface, multiple zeta value, generalized partition

2010 Mathematics Subject Classification

Primary 14C05. Secondary 11B65, 17B69.

Z. Shen was partially supported by the Natural Science Foundation of Zhejiang Province (No. LY18A010016) and the China Scholarship Council (No. 201608330449).

Z. Qin was partially supported by a grant from the Simons Foundation.

Received 1 July 2019

Accepted 14 October 2019

Published 17 February 2021