2008-03-07
Ajay Ramadoss, University of Oklahoma
Hochschild homology of sheaves of differential operators and integration
over complex manifolds
Let X be a compact complex manifold and let E be a holomorphic vector bundle
on X. Any global holomorphic differential operator D on E induces an
endomorphism of $\text{H}^{\bullet}(X,E)$. The super-trace of this
endomorphism is the super-trace of D. This is a linear functional on the
0-th Hochschild homology of Diff(E), the algebra of global holomorphic
differential operators on E.
While the Hochschild homology in the usual sense of Diff(E) is "too big" for
explicit computation, there is a notion of completed Hochschild homology of
Diff(E) with a very nice property: If HH_i(Diff(E)) denotes the i-th
completed Hochschild homology, then HH_i(Diff(E)) is isomorphic to
\text{H}^{2n-i}(X), the 2n-i th cohomology of X with complex coefficients.
We shall attempt to outline how the supertrace mentioned above extends to a
linear functional on the 0-th completed Hochschild
homology of Diff(E), and thus, on H^{2n}(X). A priori, this linear
functional depends on E. It however, can be shown that it is precisely the
integral over X. This fact also helps one connect the "local Riemann-Roch
theorems" of Nest-Tsygan to the Hirzebruch Riemann-Roch theorem. Analogous
results about similar constructions using cyclic homology instead of
Hochschild homology are also available.