Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

[摘要] Let E be a holomorphic vector bundle on a complex manifold X such that dim C X = n . Given any continuous, basic Hochschild 2 n -cocycle ψ 2 n of the algebra Diff n of formal holomorphic differential operators, one obtains a 2 n -form f E ,ψ 2 n ( 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 f E ,ψ 2 n ( 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 ψ 2 n 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].

[关键词] Hochschild homology;Lie algebra homology;Lefschetz number;Fedosov connection;trace density;holomorphic noncommutative residue