Some notes on the Feigin Losev Shoikhet integral conjecture

Abstract

Given a vector bundle $\mathcal{E}$ on a connected compact complex manifold $X$, [FLS] use a notion of completed Hochschild homology $\widehat{\text{HH}}$ of $\text{Diff}(\mathcal{E})$ such that ${\widehat{\text{HH}}}_0(\text{Diff}(\mathcal{E}))$ is isomorphic to ${\text{H}}^{2n}(X,\mathbb{C})$. On the other hand, they construct a trace on ${\widehat{\text{HH}}}_0(\text{Diff}(\mathcal{E}))$. This therefore gives to a linear functional on ${\text{H}}^{2n}(X,\mathbb{C})$. They show that this functional is ${\int }_X$ if $\mathcal{E}$ has non zero Euler characteristic. They conjecture that this functional is ${\int }_X$ for all $\mathcal{E}$. These notes prove the integral conjecture in [FLS] for compact complex manifolds having at least one vector bundle with non zero Euler characteristic.

Final version. A very crucial correction was made to the previous version. To appear in Journal of Noncommutative Geometry

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