BIRS Workshop Lecture Videos
K3 surfaces with symplectic group actions, enumerative geometry, and modular forms Bryan, Jim
The Hilbert scheme parameterizing \(n\) points on a K3 surface \(X\) is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on \(X\). We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group \(G\). Namely, the Euler characteristics of the "\(G\)-fixed Hilbert schemesâ parametrizing \(G\)-invariant collections of points on \(X\) are related to modular forms of level \(|G|\) and the enumerative geometry of rational curves on the stack quotient \([X/G]\) . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as \(\chi_y\) genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.
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