2009-04-10
Ajay Ramadoss, Cornell University
Integration of Hochschild cocycles and Lefschetz number formulae for
differential operators
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
$E$.