# Some notes on the Feigin–Losev–Shoikhet integral conjecture

### Ajay C. Ramadoss

University of Oklahoma, Norman

## 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 $HH$ of $Diff(E)$ such that $HH_{0}(Diff(E))$ is isomorphic to $H_{2n}(X, C)$. On the other hand, they construct a linear functional on $HH_{0}(Diff(E))$. This therefore gives rise to a linear functional $I_{E}$ on $H_{2n}(X, C)$. 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 $I_{E}=I_{F}$ 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 $I_{E}=∫_{X}$ for every vector bundle $E$ on $X$.

In [FLS] there is also used a notion of completed cyclic homology $HC$ of $Diff(E)$ such that $HC_{−i}(Diff(E))≃H_{2n − i}(X, C)⊕H_{2n − i +2}(X, C)⊕⋯$. The construction yielding $I_{E}$ generalizes to give linear functionals on $HC_{−2i}(Diff(E))$ for each $i≥0$. The linear functional thus obtained on $HC_{−2i}(Diff(E))$ yields a linear functional $I_{E,2i,2k}$ on $H_{2n −2k}(X, C)$ for $0≤k≤i$. It is conjectured in [FLS] that $I_{E,2,0}=∫_{X}$, and a further conjecture about $I_{E,2,2}$ is made.

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

## Cite this article

Ajay C. Ramadoss, Some notes on the Feigin–Losev–Shoikhet integral conjecture. J. Noncommut. Geom. 2 (2008), no. 4, pp. 405–448

DOI 10.4171/JNCG/25