Integration over complex manifolds via Hochschild homology

TL;DR

This paper proves that a certain linear functional associated with holomorphic vector bundles on complex manifolds equals the integral over the manifold, extending previous conjectures and constructions using Hochschild homology and geometric arguments.

Contribution

It establishes that the functional $I_{ ext{cale}}$ equals the integral over the manifold for all holomorphic vector bundles, generalizing previous results and conjectures.

Findings

$I_{ ext{cale}}$ equals $oxed{ ext{int}_X}$ for all bundles.

Extension of the functional to arbitrary complex manifolds.

Generalization of cyclic homology results related to $I_{ ext{cale}}$.

Abstract

Given a holomorphic vector bundle $\cale$ on a connected compact complex manifold X, [FLS] construct a $\compl$-linear functional $I_{\cale}$ on $\hh{2n}{\compl}$. This is done by constructing a linear functional on the 0-th completed Hochschild homology $\choch{0}{(\dif(\cale))}$ of the sheaf of holomorphic differential operators on $\cale$ using topological quantum mechanics. They show that this functional is $\int_X$ if $\cale$ has non zero Euler characteristic. They conjecture that this functional is $\int_X$ for all $\cale$. A subsequent work [Ram] by the author proved that the linear functional $I_{\cale}$ is independent of the vector bundle $\cale$. This note builds upon the work in [Ram] to prove that $I_{\cale}=\int_X$ for an arbitrary holomorphic vector bundle $\cale$ on an arbitrary connected compact complex manifold X. This is done using an argument that is very natural from the geometric point of view. This argument enables us to extend the construction in [FLS] to a construction of a linear functional $I_{\cale}$ on $\text{H}^{2n}_{c}(Y,\compl)$ for an arbitrary holomorphic vector bundle $\cale$ on an arbitrary connected complex manifold Y and prove that $I_{\cale} = \int_Y$. We also generalize a result of [Ram] pertaining to "cyclic homology analogs" of $I_{\cale}$.