JournalsjncgVol. 2 , No. 4DOI 10.4171/jncg/25

Some notes on the Feigin–Losev–Shoikhet integral conjecture

  • Ajay C. Ramadoss

    University of Oklahoma, Norman
Some notes on the Feigin–Losev–Shoikhet integral conjecture cover


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 HH^\widehat{\mathrm{HH}} of Diff(E)\mathcal{D}\mathrm{iff}(ℰ) such that HH^0(Diff(E))\widehat{\mathrm{HH}}_0(\mathcal{D}\mathrm{iff}(ℰ)) is isomorphic to H2n(X, ℂ). On the other hand, they construct a linear functional on HH^0(Diff(E))\widehat{\mathrm{HH}}_0 (\mathcal{D}\mathrm{iff}(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 HC^\widehat{\mathrm{HC}} of Diff(E)\mathcal{D}\mathrm{iff}(ℰ) such that HC^i(Diff(E))\widehat{\mathrm{HC}}_{−i}(\mathcal{D}\mathrm{iff}(ℰ)) ≃ H2n − i(X, ℂ) ⊕ H2n − i + 2(X, ℂ) ⊕ ⋯. The construction yielding Iℰ generalizes to give linear functionals on HC^2i(Diff(E))\widehat{\mathrm{HC}}_{−2i}(\mathcal{D}\mathrm{iff}(ℰ)) for each i ≥ 0. The linear functional thus obtained on HC^2i(Diff(E))\widehat{\mathrm{HC}}_{−2i}(\mathcal{D}\mathrm{iff}(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.