Some notes on the Feigin–Losev–Shoikhet integral conjecture

Abstract

Given a holomorphic vector bundle 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 H2n(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 Iℰ on H2n(X, ℂ). They show that this functional is ∫X if ℰ has non-zero Euler characteristic. They conjecture that this functional is ∫X for all ℰ.

In this article it is proved that Iℰ = Iℱ for any pair (ℰ, ℱ) of holomorphic vector bundles on X. In particular, if X has one vector bundle with non-zero Euler characteristic, then Iℰ = ∫X for every vector bundle 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 ${\widehat{\mathrm{HC}}}_{-i}(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ ≃ H2n − i(X, ℂ) ⊕ H2n − i + 2(X, ℂ) ⊕ ⋯. The construction yielding Iℰ generalizes to give linear functionals on ${\widehat{\mathrm{HC}}}_{-2i}(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ for each i ≥ 0. The linear functional thus obtained on ${\widehat{\mathrm{HC}}}_{-2i}(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ yields a linear functional Iℰ,2i,2k on H2n − 2k(X, ℂ) for 0 ≤ k ≤ i. It is conjectured in [FLS] that Iℰ,2,0 = ∫X, and a further conjecture about Iℰ,2,2 is made.

In this article we prove that Iℰ,2i,0 = Iℰ for all i ≥ 0. In particular, if X has at least one vector bundle with non-zero Euler characteristic, then Iℰ,2i,0 = ∫X. We also prove that Iℰ,2i,2k = 0 for k > 0. The latter is stronger than what is expected in [FLS] when i = k = 1.