On localization in holomorphic equivariant cohomology

Ugo Bruzzo; Vladimir Rubtsov

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1442-1454
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Ugo Bruzzo, and Vladimir Rubtsov. "On localization in holomorphic equivariant cohomology." Open Mathematics 10.4 (2012): 1442-1454. <http://eudml.org/doc/269781>.

@article{UgoBruzzo2012,
abstract = {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.},
author = {Ugo Bruzzo, Vladimir Rubtsov},
journal = {Open Mathematics},
keywords = {Holomorphic equivariant cohomology; Atiyah algebroid; Vector bundle; Residue formula; Meromorphic vector field; holomorphic equivariant cohomology; vector bundle; residue formula; meromorphic vector field},
language = {eng},
number = {4},
pages = {1442-1454},
title = {On localization in holomorphic equivariant cohomology},
url = {http://eudml.org/doc/269781},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Ugo Bruzzo
AU - Vladimir Rubtsov
TI - On localization in holomorphic equivariant cohomology
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1442
EP - 1454
AB - 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.
LA - eng
KW - Holomorphic equivariant cohomology; Atiyah algebroid; Vector bundle; Residue formula; Meromorphic vector field; holomorphic equivariant cohomology; vector bundle; residue formula; meromorphic vector field
UR - http://eudml.org/doc/269781
ER -

References

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  1. [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 Zbl0078.16002
  2. [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 Zbl0521.58025
  3. [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 
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  5. [5] Bott R., Vector fields and characteristic numbers, Michigan Math. J., 1967, 14, 231–244 http://dx.doi.org/10.1307/mmj/1028999721 Zbl0145.43801
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  7. [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 Zbl0409.32005
  8. [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 Zbl0853.32032
  9. [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 Zbl0365.32020
  10. [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 Zbl0968.58014
  11. [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 Zbl1098.58015
  12. [12] Griffiths P., Harris J., Principles of Algebraic Geometry, Wiley Classics Lib., John Wiley & Sons, New York, 1994 Zbl0836.14001
  13. [13] Hartshorne R., Residues and Duality, Lecture Notes in Math., 20, Springer, Berlin-New York, 1966 
  14. [14] Huebschmann J., Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math., 1999, 510, 103–159 Zbl1034.53083
  15. [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 Zbl1167.53065
  16. [16] Liu K., Holomorphic equivariant cohomology, Math. Ann., 1995, 303(1), 125–148 http://dx.doi.org/10.1007/BF01460983 

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