Integration over complex manifolds via Hochschild homology

Abstract

Given a holomorphic vector bundle $\mathcal{E}$ on a connected compact complex manifold $X$, in [FLS] a $C$-linear functional $I_{\mathcal{E}}$ on $H^{2n}(X,C)$ is constructed. This is done by producing a linear functional on the 0-th completed Hochschild homology ${\widehat{\mathrm{HH}}}_0(\mathcal{D}\mathrm{iff}(\mathcal{E}))$ of the sheaf of holomorphic differential operators on $\mathcal{E}$ using topological quantum mechanics. It is shown in [FLS] that this functional is ${\int }_X$ if $\mathcal{E}$ has non-zero Euler characteristic, and the conjecture is that it is ${\int }_X$ for all $\mathcal{E}$.

In a subsequent work [Ram] the author proved that the linear functional $I_{\mathcal{E}}$ is independent of the vector bundle $\mathcal{E}$. This article builds upon the work in [Ram] to prove that $I_{\mathcal{E}}={\int }_X$ for an arbitrary holomorphic vector bundle $\mathcal{E}$ on an arbitrary connected compact complex manifold $X$. This is done using an argument that is very natural from the geometric point of view. Moreover, this argument enables one to make the approach to this conjecture developed first in [FLS] and subsequently in [Ram] independent of the Riemann–Roch–Hirzebruch theorem. This argument allows us to extend the construction in [FLS] to a construction of a linear functional $I_{\mathcal{E}}$ on $H_c^{2n}(Y,C)$ for a holomorphic vector bundle $\mathcal{E}$ with bounded geometry on an arbitrary connected complex manifold $Y$ with bounded geometry, and to prove that $I_{\mathcal{E}}={\int }_Y$. We also generalize a result of [Ram] pertaining to “cyclic homology analogs” of $I_{\mathcal{E}}$.