Some notes on the Feigin–Losev–Shoikhet integral conjecture

Abstract

Given a holomorphic vector bundle $\mathcal{E}$ on a smooth connected compact complex manifold $X$, Feigin, Losev and Shoikhet [FLS] use a notion of completed Hochschild homology $\widehat{\mathrm{HH}}$ of $\mathcal{D}\mathrm{iff}(\mathcal{E})$ such that ${\widehat{\mathrm{HH}}}_0(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ is isomorphic to $H^{2n}(X,C)$. On the other hand, they construct a linear functional on ${\widehat{\mathrm{HH}}}_0(\mathcal{D}\mathrm{iff}(\mathcal{E}))$. This therefore gives rise to a linear functional $I_{\mathcal{E}}$ on $H^{2n}(X,C)$. 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 $\mathcal{E}$.

In this article it is proved that $I_{\mathcal{E}}=I_{\mathcal{F}}$ for any pair $(\mathcal{E},\mathcal{F})$ of holomorphic vector bundles on $X$. In particular, if $X$ has one vector bundle with non-zero Euler characteristic, then $I_{\mathcal{E}}={\int }_X$ for every vector bundle $\mathcal{E}$ on $X$.

In [FLS] there is also used a notion of completed cyclic homology $\widehat{\mathrm{HC}}$ of $\mathcal{D}\mathrm{iff}(\mathcal{E})$ such that ${\widehat{\mathrm{HC}}}_{-i}(\mathcal{D}\mathrm{iff}(\mathcal{E}))\simeq H^{2n-i}(X,C)\oplus H^{2n-i+2}(X,C)\oplus \cdots$. The construction yielding $I_{\mathcal{E}}$ generalizes to give linear functionals on ${\widehat{\mathrm{HC}}}_{-2i}(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ for each $i\ge 0$. The linear functional thus obtained on ${\widehat{\mathrm{HC}}}_{-2i}(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ yields a linear functional $I_{\mathcal{E},2i,2k}$ on $H^{2n-2k}(X,C)$ for $0\le k\le i$. It is conjectured in [FLS] that $I_{\mathcal{E},2,0}={\int }_X$, and a further conjecture about $I_{\mathcal{E},2,2}$ is made.

In this article we prove that $I_{\mathcal{E},2i,0}=I_{\mathcal{E}}$ for all $i\ge 0$. In particular, if $X$ has at least one vector bundle with non-zero Euler characteristic, then $I_{\mathcal{E},2i,0}={\int }_X$. We also prove that $I_{\mathcal{E},2i,2k}=0$ for $k>0$. The latter is stronger than what is expected in [FLS] when $i=k=1$.