Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

• Дата: 2011
• Автор: Ramadoss, A.C.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2011
• Назва видання: Symmetry, Integrability and Geometry: Methods and Applications
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/146775
• 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.

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].