Ajay C.Ramadoss

Let X be a smooth scheme over a field of characteristic 0. It is well known (from a paper of M. Kapranov) that the Atiyah class of the tangent bundle T_X of X equips T_X[-1] with the structure of a Lie algebra object in the derived category D⁺(X) of bounded below complexes of O_X-modules with coherent cohomology. We realize this structure as the Lie bracket of an "almost free" Lie algebra L generated over O_X such that L is a bounded below complex of O_X-modules. We then give a theorem for L paralleling the computation of the differential of the inverse exponential map of a Lie algebra. We also show that the complex of polydifferential operators with Hochschild coboundary is the universal enveloping algebra of L, and hence T_X[-1], in D⁺(X). This enables us to interpret the Chern character of a vector bundle on X as the "character of a representation". The results obtained are then used to give an explicit formula for the big Chern classes of a vector bundle E on X in terms of the Chern character of E and the Atiyah class of T_X.

This paper expands upon the core computations in a well known preprint of Markarian to compute Caldararu's Mukai pairing on Hochschild homology at the level of Hodge cohomology. It turns out that the Hochschild-Kostant-Rosenberg map twisted by the square-root of the Todd genus "almost preserves" the Mukai pairing, thus verifying a conjecture of Caldararu.

Given a vector bundle E on a compact complex manifold X of complex dimension n, B. Feigin, A. Losev and B. Shoikhet use "topological quantum mechanics" to construct a linear functional on the 0-th completed Hochschild homology of the sheaf Diff(E) of holomorphic differential operators on E. This gives a linear`functional I_E on the top cohomology of X with complex coefficients (see their preprint math.QA/0401400). They conjecture that I_E is precisely integration over X . This paper shows that I_E = I_F for any two vector bundles E and F. Similar linear functionals can be constructed on the 2i-th completed cyclic homology of Diff(E) for each i. This yields linear functionals I_E;2i;2k on H^2n-2k(X) for 0 <= k <= i. We show that I_E;2i;0 = I_E and that I_E;2i;2k vanishes for k > 0.

The Mukai pairing and integral transforms in Hochschild homology. Available on the Arxiv at arxiv:0805.1760. Submitted for publication.

The Hochschild homology HH_*(X) of a smooth proper scheme X over a field of characteristic 0 comes with two pairings: a "categorical" or "natural" pairing constructed by D. Shklyarov, and the Mukai pairing of Caldararu. This paper proves a theorem computing the first pairing in terms of the second. Further, if X and Y are smooth and proper, an element of perf(X x Y ) gives rise to an integral transform from HH_*(X) to HH_*(Y ) via two "a priori different" constructions: a natural construction of Shklyarov and a construction of Caldararu. This paper proves that these two constructions are equivalent. These results give rise to a Hirzebruch Riemann-Roch theorem for the sheafification of the Dennis trace map.

This short note proves a generalization of the Hirzebruch Riemann-Roch theorem equivalent to the Cardy condition described by A. Caldararu. This is done using a result appearing in an earlier paper of mine that describes what the Mukai pairing in Hochschild homology descends to in Hodge cohomology via the Hochschild-Kostant-Rosenberg map twisted by the root Todd genus.