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On the Spectrum of the Equivariant Cohomology Ring

Published online by Cambridge University Press:  20 November 2018

Mark Goresky  and
Robert MacPherson
Mark Goresky
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA, e-mail: goresky@ias.edu, rdm@math.ias.edu
Robert MacPherson
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA, e-mail: goresky@ias.edu, rdm@math.ias.edu
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Abstract

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If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\text{Spec}\,H_{T}^{*}\left( X;\,\mathbb{C} \right)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_{2}^{T}\left( X;\,\mathbb{C} \right)$. In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.

Keywords

14L3054H15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 2 , 01 April 2010 , pp. 262 - 283
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Akyildiz, E., Carrell, J., and Lieberman, D., Zeros of holomorphic vector fields on singular spaces and intersection rings of Schubert varieties. Compositio Math. 57(1986), no. 2, 237–248.Google Scholar
[2] Allday, C. and Puppe, V., Cohomological Methods in Transformation Groups. Cambridge Studies in Advanced Mathematics 32. Cambridge University Press, Cambridge, 1993.Google Scholar
[3] Atiyah, M. and Bott, R., The moment map and equivariant cohomology, Topology 23(1984), no. 1, 1–28. doi:10.1016/0040-9383(84)90021-1Google Scholar
[4] Audin, M., The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics 93, Birkhäuser Boston, Boston, MA, 1991.Google Scholar
[5] Braden, T., Licata, T., Phan, C., Proudfoot, N., and Webster, B., Goresky-Mac Pherson duality and deformations of Koszul algebras. ar Xiv:0905.1335v1.Google Scholar
[6] Brion, M., Equivariant cohomology and equivariant intersection theory, Notes de cours, Montréal, 1997. http://www-fourier.ujf-grenoble.fr/-mbrion/notes.html.Google Scholar
[7] Brion, M., Poincaré duality and equivariant (co)homology. Michigan Math. J. 48(2000), 77–92. doi:10.1307/mmj/1030132709Google Scholar
[8] Brion, M. and Carrell, J., The equivariant cohomology ring of regular varieties. Michigan Math. J. 52(2004), no. 1, 189–203. doi:10.1307/mmj/1080837743Google Scholar
[9] Brion, M. and Vergne, M., An equivariant Riemann-Roch theorem for complete, simplicial toric varieties. J. Reine Angew. Math. 482(1997), 67–92.Google Scholar
[10] Carrell, J., Orbits of the Weyl group and a theorem of De Concini and Procesi. Compositio Math. 60(1986), no. 1, 45–52.Google Scholar
[11] Carrell, J., Kaveh, K., and Puppe, V., Vector fields, torus actions and equivariant cohomology. Pacific J. Math. 232(2007), no. 1, 61–76.Google Scholar
[12] Danilov, V. I., The geometry of toric varieties. Russian Math. Surveys 33(1978), no. 2, 97–154.Google Scholar
[13] De Concini, C., Lusztig, G., and Procesi, C., Homology of the zero-set of a nilpotent vector field on a flag manifold. J. Amer. Math. Soc. 1(1988), no. 1, 15–34. doi:10.2307/1990965Google Scholar
[14] De Concini, C., and Procesi, C., Symmetric functions, conjugacy classes, and the flag variety. Invent. Math. 64(1981), no. 2, 203–219. doi:10.1007/BF01389168Google Scholar
[15] Fulton, W., Introduction to Toric Varieties. Annals of Mathematics Studies 131. Princeton University Press, Princeton NJ, 1993.Google Scholar
[16] Goresky, M., Kottwitz, R., and Mac Pherson, R., Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1998), no. 1, 25–83. doi:10.1007/s002220050197Google Scholar
[17] Goresky, M., Homology of affine Springer fibers in the unramified case. Duke Math. J. 121(2004), no. 3, 509–561 doi:10.1215/S0012-7094-04-12135-9Google Scholar
[18] Hsiang, W., Cohomology Theory of Topological Transformation Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 85. Springer Verlag, New York, 1975.Google Scholar
[19] Kaveh, K., Note on the cohomology ring of spherical varieties and volume polynomial. ar Xiv:math/0312503.Google Scholar
[20] Kazhdan, D. and, Lusztig, G., A topological approach to Springer's representations. Adv. in Math. 38(1980), no. 2, 222–228. doi:10.1016/0001-8708(80)90005-5Google Scholar
[21] Lusztig, G., Green polynomials and singularities of unipotent classes. Adv. Math. 42(1981), no. 2, 169–178. doi:10.1016/0001-8708(81)90038-4Google Scholar
[22] Pukhlikov, A. and, Khovanskii, A., The Riemann-Roch theorem for integrals of sums of quasipolynomials on virtual polytopes. Algebra i Analiz 4(1992), no. 4, 188-216 (Russian); St. Petersburg Math. J. 4(1993), no. 4, 789–812 (English translation).Google Scholar
[23] Puppe, V., On a conjecture of Bredon. Manuscripta Math. 12(1974), 11–16. doi:10.1007/BF01166231Google Scholar
[24] Puppe, V., Cohomology of fixed point sets and deformation of algebras. Manuscripta Math. 23(1977/78), no. 4, 343–354. doi:10.1007/BF01167693Google Scholar
[25] Puppe, V., Deformations of algebras and cohomology of fixed point sets. Manuscripta Math. 30(1979/80), no. 2, 119–136. doi:10.1007/BF01300965Google Scholar
[26] Puppe, V., Do manifolds have little symmetry? J. Fixed Point Theory Appl. 2(2007), no. 1, 85–96. doi:10.1007/s11784-007-0021-xGoogle Scholar
[27] Quillen, D., The spectrum of an equivariant cohomology ring. I. Ann. of Math. 94(1971), 549–572. doi:10.2307/1970770Google Scholar
[28] Spaltenstein, N., The fixed point set of a unipotent transformation on the flag manifold. Nederl. Akad. Wetensch. Proc. Ser. A 38(1976), no. 5, 452–456.Google Scholar
[29] Spanier, E., Algebraic Topology. Mc Graw-Hill, New York, 1966.Google Scholar
[30] Springer, T. A., Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36(1976), 173–207. doi:10.1007/BF01390009Google Scholar
[31] Springer, T. A., A purity result for fixed point varieties in flag manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(1984), no. 2, 271–282.Google Scholar
[32] Stroppel, C., Parabolic category O, perverse sheaves on Grassmannians, Springer Fibres and Khovanov homology. Compositio Math. 145(2009), no. 4, 954–992.Google Scholar