# A Lefschetz number theorem for differential operators

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Ping Xu, Nigel Higson, Mathieu Stiénon.

**Speaker:** Ajay Ramadoss, Cornell University

**Abstract:** Let $X$ be a compact, complex manifold and let $E$ be a holomorphic vector bundle on $X$. Let $ Diff^{\bullet}(E)$ be the Dolbeault resolution of the sheaf of holomorphic differential operators on $E$. A construction due to Feigin, et. al , gives a linear functional on the $0$-th Hochschild homology of $Diff^{\bullet}(E)$. This functional "extends" to a linear functional on the "completed" $0$-th Hochschild homology of $Diff^{\bullet}(E)$ ,thereby giving a linear functional $I_E$ on the top cohomology $\text{H}^{2n} (X,\mathbb C)$ of $X$ with complex coefficients. The main result is that $I_E$ is just integration over $X$. As a consequence, one obtains a Lefschetz number formula for a global holomorphic differential operator on $E$.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 12/01/2009

**Time:** 2:30pm - 3:30pm