Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Ajay C. Ramadoss

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

Let $\mathcal E$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{\mathbb C}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $\psi_{2n}$ of the algebra $\operatorname{Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{\mathcal E,\psi_{2n}}(\mathcal D)$ from any holomorphic differential operator $\mathcal D$ on $\mathcal E$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405–448; J. Noncommut. Geom. 3 (2009), 27–45] to show that $\int_X f_{\mathcal E,\psi_{2n}}(\mathcal D)$ gives the Lefschetz number of $\mathcal D$ upto a constant independent of $X$ and $\mathcal E$. In addition, we obtain a “local” result generalizing the above statement. When $\psi_{2n}$ 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 $\mathcal D$ defined by B. Shoikhet when $\mathcal E$ 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].

Détail

Keywords: Hochschild homology; Lie algebra homology; Lefschetz number; Fedosov connection; trace density; holomorphic noncommutative residue.

Bibliographie

[1] Adler M., “On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg–de Vries type equations”, Invent. Math., 50 (1979), 219–248 | DOI | MR | Zbl

[2] Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Springer-Verlag, Berlin, 2004 | MR | Zbl

[3] Bott R., Tu L. W., Differential forms in algebraic topology, Springer-Verlag, Berlin, 1995 | MR

[4] Bressler P., Kapranov M., Tsygan B., Vasserot E., “Riemann–Roch for real varieties”, Algebra, Arithmetic and Geometry, in Honor of Yu. I. Manin, Progr. Math., 269, Birkhäuser Boston, Inc., Boston, MA, 2009, 125–164, arXiv: math.DG/0612410 | DOI | MR | Zbl

[5] Brylinski J.-L., “A differential complex for Poisson manifolds”, J. Differential Geom., 28 (1988), 93–114 | MR | Zbl

[6] Calaque D., Dolgushev V., Halbout G., “Formality theorems for Hochschild chains in the Lie algebroid setting”, J. Reine Angew. Math., 612 (2007), 81–127, arXiv: math.KT/0504372 | DOI | MR | Zbl

[7] Calaque D., Rossi C., Lectures on Duflo isomorphisms in Lie algebras and complex geometry, Lecture notes available at http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf

[8] Dolgushev V., “Covariant and equivariant formality theorems”, Adv. Math., 191 (2005), 147–177, arXiv: math.QA/0307212 | DOI | MR | Zbl

[9] Donelly H., “Asymptotic expansions for the compact quotients of properly discontinuous group actions”, Illinois J. Math., 23 (1979), 485–496 | MR

[10] Engeli M., Felder G., “A Riemann–Roch–Hirzebruch formula for traces of differential operators”, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 621–653, arXiv: math.QA/0702461 | MR

[11] Feigin B. L., Felder G., Shoikhet B., “Hochschild cohomology of the Weyl algebra and traces in deformation quantization”, Duke Math. J., 127 (2005), 487–517, arXiv: math.QA/0311303 | DOI | MR | Zbl

[12] Feigin B. L., Tsygan B. L., “Cohomology of the Lie algebra of generalized Jacobian matrices”, Funktsional. Anal. i Prilozhen., 17:2 (1983), 86–87 (in Russian) | DOI | MR | Zbl

[13] Gel'fand I. M., Fuks D. B., “Cohomology of the Lie algebra of tangent vector fields on a smooth manifold. II”, Funktsional. Anal. i Prilozhen., 4:2 (1970), 23–31 (in Russian) | MR

[14] Kashiwara M., Schapira P., Sheaves on manifolds, Springer-Verlag, Berlin, 2002 | MR

[15] Loday J.-L., Cyclic homology, Springer-Verlag, Berlin, 1992 | MR

[16] Nest R., Tsygan B., “Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems”, Asian J. Math., 5 (2001), 599–635, arXiv: math.QA/9906020 | MR | Zbl

[17] Pflaum M. J., Posthuma H. B., Tang X., “An algebraic index theorem for orbifolds”, Adv. Math., 210 (2007), 83–121, arXiv: math.KT/0507546 | DOI | MR | Zbl

[18] Ramadoss A. C., “Some notes on the Feigin–Losev–Shoikhet integral conjecture”, J. Noncommut. Geom., 2 (2008), 405–448, arXiv: math.QA/0612298 | DOI | MR | Zbl

[19] Ramadoss A. C., “Integration over complex manifolds via Hochschild homology”, J. Noncommut. Geom., 3 (2009), 27–45, arXiv: 0707.4528 | DOI | MR | Zbl

[20] Shoikhet B., “Integration of the lifting formulas and the cyclic homology of the algebra of differential operators”, Geom. Funct. Anal., 11 (2001), 1096–1124, arXiv: math.QA/9809037 | DOI | MR | Zbl

[21] Shoikhet B., “Lifting formulas. II”, Math. Res. Lett., 6 (1999), 323–334, arXiv: math.QA/9801116 | MR | Zbl

[22] Willwacher T., Cyclic cohomology of the Weyl algebra, arXiv: 0804.2812

[23] Wodzicki M., “Cyclic homology of differential operators”, Duke Math. J., 54 (1987), 641–647 | DOI | MR | Zbl