G-fixed Hilbert schemes on K3 surfaces, modular forms, and eta products

Bryan, Jim and Gyenge, Ádám (2022) G-fixed Hilbert schemes on K3 surfaces, modular forms, and eta products. EPIJOURNAL DE GEOMETRIE ALGEBRIQUE, 6. ISSN 2491-6765

Preview

Text 1907.01535.pdf Available under License Creative Commons Attribution. Download (495kB) | Preview

Abstract

Let X be a complex K3 surface with an effective action of a group G which preserves the holomorphic symplectic form. LetZ(X,G)(q) = Sigma(infinity)(n=0)e(Hilb(n)(X)(G))q(n-1)be the generating function for the Euler characteristics of the Hilbert schemes of G-invariant length n subschemes. We show that its reciprocal, Z(X,G)(q)(-1) is the Fourier expansion of a modular cusp form of weight 1/2e(X/G) for the congruence subgroup Gamma(0)(vertical bar G vertical bar). We give an explicit formula for Z(X,G) in terms of the Dedekind eta function for all 82 possible (X, G). We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the chi(y) genus, and the motivic class. As a byproduct of our method, we prove a result which is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system.

Actions (login required)

Edit Item