Hilbert schemes of points and quasi-modularity

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 $\mathbb{C}^2$.

Keywords

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

2010 Mathematics Subject Classification

Primary 14C05. Secondary 11B65, 17B69.

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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