Some notes on the Feigin–Losev–Shoikhet integral conjecture

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