Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Abstract

Let ε be a holomorphic vector bundle on a complex manifold X such that dimCX = n. Given any continuous, basic Hochschild 2n-cocycle ψ2 of the algebra Diffn of formal holomorphic differential operators, one obtains a 2n-form fε; ψ2n(D) from any holomorphic differential operator D on ε. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that ∫Xfε; ψ2n(D) gives the Lefschetz number of D upto a constant independent of X and ε. In addition, we obtain a \local result generalizing the above statement. When ψ2nis the cocycle from [Duke Math. J. 127 (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 D defined by B. Shoikhet when ε 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 [Geom. Funct. Anal. 11 (2001), 1096-1124].