Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 010, 26 pages      arXiv:0904.1891      https://doi.org/10.3842/SIGMA.2011.010

Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Ajay C. Ramadoss
Department Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

Received August 12, 2010, in final form January 07, 2011; Published online January 18, 2011

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

Key words: Hochschild homology; Lie algebra homology; Lefschetz number; Fedosov connection; trace density; holomorphic noncommutative residue.

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