UC San Diego

I present formulas for the Euler characteristics of tautological sheaves over the punctual Quot scheme, which parameterizes zero-dimensional quotients of a fixed vector bundle over curves. We observe a striking similarity with the formulas for the Hilbert scheme of points on surfaces. Furthermore, we study the Quot schemes of higher rank quotients for a genus-zero curve. We calculate the holomorphic Euler characteristics of Schur bundles and tautological bundles over Quot schemes. These formulas can be considered a generalization of the formulas for Grassmannians, which were obtained using the Borel-Weil-Bott theorem. Additionally, we show non-trivial vanishing results using these formulas.

The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with a symplectic (or symmetric) bilinear form. I study the intersection theory of the symplectic and orthogonal isotropic Quot schemes. In particular, I construct a virtual fundamental class for these Quot schemes and find explicit formulas for certain intersection numbers. I also calculate the Gromov-Ruan-Witten invariants of the corresponding Grassmannians and compare the answers with those for the isotropic Quot schemes.