We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. This encompasses various residue formulas in complex geometry, in particular we shall show that it contains as special cases Carrell-Liebermann’s and Feng-Ma’s residue formulas, and Baum-Bott’s formula for the zeroes of a meromorphic vector field.
@article{bwmeta1.element.doi-10_2478_s11533-012-0054-2,
author = {Ugo Bruzzo and Vladimir Rubtsov},
title = {On localization in holomorphic equivariant cohomology},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1442-1454},
zbl = {1272.32017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0054-2}
}
Ugo Bruzzo; Vladimir Rubtsov. On localization in holomorphic equivariant cohomology. Open Mathematics, Tome 10 (2012) pp. 1442-1454. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0054-2/
[1] Atiyah M.F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 1957, 85, 181–207 http://dx.doi.org/10.1090/S0002-9947-1957-0086359-5 | Zbl 0078.16002
[2] Atiyah M.F., Bott R., The moment map and equivariant cohomology, Topology, 1984, 23(1), 1–28 http://dx.doi.org/10.1016/0040-9383(84)90021-1 | Zbl 0521.58025
[3] Baum P.F., Bott R., On the zeroes of meromorphic vector-fields, In: Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, 29–47 http://dx.doi.org/10.1007/978-3-642-49197-9_4
[4] Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Math. Wiss., 298, Springer, Berlin, 1992 http://dx.doi.org/10.1007/978-3-642-58088-8 | Zbl 0744.58001
[5] Bott R., Vector fields and characteristic numbers, Michigan Math. J., 1967, 14, 231–244 http://dx.doi.org/10.1307/mmj/1028999721 | Zbl 0145.43801
[6] Bruzzo U., Cirio L., Rossi P., Rubtsov V.N., Equivariant cohomology and localization for Lie algebroids, Funct. Anal. Appl., 2009, 43(1), 18–29 http://dx.doi.org/10.1007/s10688-009-0003-4 | Zbl 1271.53073
[7] Carrell J.B., A remark on the Grothendieck residue map, Proc. Amer. Math. Soc., 1978, 70(1), 43–48 http://dx.doi.org/10.1090/S0002-9939-1978-0492408-1 | Zbl 0409.32005
[8] Carrell J.B., Vector fields, residues and cohomology, In: Parameter Spaces, Warsaw, February 1994, Banach Center Publ., 36, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1996, 51–59 | Zbl 0853.32032
[9] Carrell J.B., Lieberman D.I., Vector fields and Chern numbers, Math. Ann., 1977, 225(3), 263–273 http://dx.doi.org/10.1007/BF01425242 | Zbl 0365.32020
[10] Evens S., Lu J.-H., Weinstein A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Q. J. Math., 1999, 50(200), 417–436 http://dx.doi.org/10.1093/qjmath/50.200.417 | Zbl 0968.58014
[11] Feng H., Ma X., Transversal holomorphic sections and localization of analytic torsions, Pacific J. Math., 2005, 219(2), 255–270 http://dx.doi.org/10.2140/pjm.2005.219.255 | Zbl 1098.58015
[12] Griffiths P., Harris J., Principles of Algebraic Geometry, Wiley Classics Lib., John Wiley & Sons, New York, 1994 | Zbl 0836.14001
[13] Hartshorne R., Residues and Duality, Lecture Notes in Math., 20, Springer, Berlin-New York, 1966
[14] Huebschmann J., Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math., 1999, 510, 103–159 | Zbl 1034.53083
[15] Li Y., The equivariant cohomology theory of twisted generalized complex manifolds, Comm. Math. Phys., 2008, 281(2), 469–497 http://dx.doi.org/10.1007/s00220-008-0495-4 | Zbl 1167.53065
[16] Liu K., Holomorphic equivariant cohomology, Math. Ann., 1995, 303(1), 125–148 http://dx.doi.org/10.1007/BF01460983