## Tautological sheaves on Hilbert schemes of points

**Speaker(s): ** 王芝兰 (Tsinghua University)

**Time: ** 00:00-00:00 March 26, 2014

**Venue: ** Room 29 at Quan Zhai, BICMR

Speaker: 王芝兰 (Tsinghua University)

Time: Wednesday Mar. 26, 13:30-15:30.

Venue: Room 29 at Quan Zhai, BICMR

Abstract: It is an interesting fact that many invariants of the Hilbert schemes of points on a projective surface can be determined explicitly by the corresponding invariants of the surface. These include the Betti numbers (Gottsche), Hodge numbers (Gottsche-Soergel), cobordism classes (Ellisngsrud-Gottsche-Lehn), and elliptic genus (Borisov-Libgober). Professor Jian Zhou and I extend such results to the (equivariant) Euler characteristics of some naturally defined vector bundles related to the tautological vector bundles on the Hilbert schemes X^{[n]} of points in a projective or quasi-projective variety X. They are related to the Macdonald polynomials. And Using these formulas, we calculate the integrals of some chern classes on the Hilbert schemes of points on surfaces. In this talk, I will begin with the basic facts on Hilbert schemes. Then I will present some examples of the above generating series and briefly explain our strategy to computing this kind of generating series.