Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Let $\mathcal E$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{\mathbb C}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $\psi_{2n}$ of the algebra $\operatorname{Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{\mathcal E,\psi_{2n}}(\mathcal D)$ from any holomorphic differential operator $\mathcal D$ on $\mathcal E$. We apply our earlier results [<i>J. Noncommut. Geom.</i> <b>2</b> (2008), 405–448; <i>J. Noncommut. Geom.</i> <b>3</b> (2009), 27–45] to show that $\int_X f_{\mathcal E,\psi_{2n}}(\mathcal D)$ gives the Lefschetz number of $\mathcal D$ upto a constant independent of $X$ and $\mathcal E$. In addition, we obtain a “local” result generalizing the above statement. When $\psi_{2n}$ is the cocycle from [<i>Duke Math. J.</i> <b>127</b> (2005), 487–517], we obtain a new proof as well as a generalization of the Lefschetz number theorem of Engeli–Felder. We also obtain an analogous “local” result pertaining to B. Shoikhet's construction of the holomorphic noncommutative residue of a differential operator for trivial vector bundles on complex parallelizable manifolds. This enables us to give a rigorous construction of the holomorphic noncommutative residue of $\mathcal D$ defined by B. Shoikhet when $\mathcal E$ is an arbitrary vector bundle on an arbitrary compact complex manifold $X$. Our local result immediately yields a proof of a generalization of Conjecture 3.3 of [<i>Geom. Funct. Anal.</i> <b>11</b> (2001), 1096–1124].