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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2011 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
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Інститут математики НАН України
2011
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146775 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862730811029061632 |
|---|---|
| author | Ramadoss, A.C. |
| author_facet | Ramadoss, A.C. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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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| first_indexed | 2025-12-07T19:22:21Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146775 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:22:21Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | 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 |
| spellingShingle | Integration of Cocycles and Lefschetz Number Formulae for Differential Operators Ramadoss, A.C. |
| title | 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_short | Integration of Cocycles and Lefschetz Number Formulae for Differential Operators |
| title_sort | integration of cocycles and lefschetz number formulae for differential operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146775 |
| work_keys_str_mv | AT ramadossac integrationofcocyclesandlefschetznumberformulaefordifferentialoperators |