Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
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 fε,ψ2n(D) from any holomorphic differential operator D on E. We apply our ear...
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| Zitieren: | Integration of Cocycles and Lefschetz Number Formulae for Differential Operators / A.C. Ramadoss // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 23 назв. — англ. |
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Ramadoss, A.C. 2019-02-11T14:54:27Z 2019-02-11T14:54:27Z 2011 Integration of Cocycles and Lefschetz Number Formulae for Differential Operators / A.C. Ramadoss // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16E40; 32L05; 32C38; 58J42 DOI:10.3842/SIGMA.2011.010 https://nasplib.isofts.kiev.ua/handle/123456789/146775 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 fε,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 fε,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 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]. I am grateful to Giovanni Felder and Thomas Willwacher for some very useful discussions. This work would not have reached its current form without their pointing out important shortcomings in earlier versions. I am also grateful to Boris Shoikhet for useful discussions. I thank the referees of this article for their constructive suggestions. This work was done (prior to my joining my current position) partly at Cornell University and partly at IHES. I am grateful to both these institutions for providing me with a congenial work atmosphere. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Integration of Cocycles and Lefschetz Number Formulae for Differential Operators Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators |
| spellingShingle |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators Ramadoss, A.C. |
| title_short |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators |
| title_full |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators |
| title_fullStr |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators |
| title_full_unstemmed |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators |
| title_sort |
integration of cocycles and lefschetz number formulae for differential operators |
| author |
Ramadoss, A.C. |
| author_facet |
Ramadoss, A.C. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 fε,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 fε,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 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].
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146775 |
| citation_txt |
Integration of Cocycles and Lefschetz Number Formulae for Differential Operators / A.C. Ramadoss // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 23 назв. — англ. |
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2025-12-07T19:22:21Z |
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2025-12-07T19:22:21Z |
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1850878555985018880 |