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 $\hat{\text{HH}}_0(\text{Diff}(\mathcal E))$ is isomorphic to $\text{H}^{2n}(X, \mathbb C)$. On the other hand, they construct a trace on $\hat{\text{HH}}_0(\text{Diff}(\mathcal E))$. 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.