Integration of cocycles and Lefschetz number formulae for differential operators

DOI10.3842/SIGMA.2011.010zbMATH Open1227.32027arXiv0904.1891OpenAlexW2134202054WikidataQ115219672 ScholiaQ115219672MaRDI QIDQ542753

Ajay Ramadoss

Publication date: 17 June 2011

Published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications (Search for Journal in Brave)

Abstract: Let $mathcalE$ be a holomorphic vector bundle on a complex manifold $X$ such that $di{m}_{mathbbC}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $ps{i}_{2n}$ of the algebra ${mDiff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{mathcalE,ps{i}_{2n}}(mathcalD)$ from any holomorphic differential operator $mathcalD$ on $mathcalE$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that $in{t}_X{f}_{mathcalE,ps{i}_{2n}}(mathcalD)$ gives the Lefschetz number of $mathcalD$ upto a constant independent of $X$ and $mathcalE$. In addition, we obtain a "local" result generalizing the above statement. When $ps{i}_{2n}$ 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 $mathcalD$ defined by B. Shoikhet when $mathcalE$ 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].

Full work available at URL: https://arxiv.org/abs/0904.1891

File on IPFS (Hint: this is only the Hash - if you get a timeout, this file is not available on our server.)

zbMATH Keywords

Hochschild homologyLefschetz numberLie algebra homologyFedosov connectionholomorphic non-commutative residue

Mathematics Subject Classification ID

Sheaves of differential operators and their modules, (D)-modules (32C38) Holomorphic bundles and generalizations (32L05) Noncommutative global analysis, noncommutative residues (58J42)

Related Items (1)

Title not available (Why is that?)

Recommendations

• Higher integrability of iterated operators on differential forms 👍 👎
• Differential cocycles and Dixmier-Douady bundles 👍 👎
• Algebraic structure and integration maps in cocycle models for differential cohomology 👍 👎
• Integration of the lifting formulas and the cyclic homology of the algebras of differential operators 👍 👎
• Equivariant Lefschetz number of differential operators 👍 👎
• Title not available (Why is that?) 👍 👎
• Title not available (Why is that?) 👍 👎
• Title not available (Why is that?) 👍 👎
• Title not available (Why is that?) 👍 👎
• Title not available (Why is that?) 👍 👎

This page was built for publication: Integration of cocycles and Lefschetz number formulae for differential operators

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q542753)