Geometry & Topology

Rational smoothness, cellular decompositions and GKM theory

Richard Gonzales

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We introduce the notion of –filtrable varieties: projective varieties with a torus action and a finite number of fixed points, such that the cells of the associated Bialynicki-Birula decomposition are all rationally smooth. Our main results develop GKM theory in this setting. We also supply a method for building nice combinatorial bases on the equivariant cohomology of any –filtrable GKM variety. Applications to the theory of group embeddings are provided.

Article information

Geom. Topol., Volume 18, Number 1 (2014), 291-326.

Received: 15 July 2012
Revised: 19 August 2013
Accepted: 20 September 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Secondary: 55N91: Equivariant homology and cohomology [See also 19L47] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]

rational smoothness algebraic torus actions GKM theory equivariant cohomology algebraic monoids group embeddings


Gonzales, Richard. Rational smoothness, cellular decompositions and GKM theory. Geom. Topol. 18 (2014), no. 1, 291--326. doi:10.2140/gt.2014.18.291.

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  • V Alexeev, M Brion, Stable reductive varieties, II: Projective case, Adv. Math. 184 (2004) 380–408
  • A Arabia, Classes d'Euler équivariantes et points rationnellement lisses, Ann. Inst. Fourier 48 (1998) 861–912
  • M F Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics 401, Springer, Berlin (1974)
  • M F Atiyah, G B Segal, The index of elliptic operators, II, Ann. of Math. 87 (1968) 531–545
  • A Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973) 480–497
  • A Białynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976) 667–674
  • A Borel, Seminar on transformation groups, Annals of Mathematics Studies 46, Princeton Univ. Press (1960)
  • L A Borisov, L Chen, G G Smith, The orbifold Chow ring of toric Deligne–Mumford stacks, J. Amer. Math. Soc. 18 (2005) 193–215
  • M Brion, Equivariant Chow groups for torus actions, Transform. Groups 2 (1997) 225–267
  • M Brion, The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998) 137–174
  • M Brion, Equivariant cohomology and equivariant intersection theory, from: “Representation theories and algebraic geometry”, (A Broer, G Sabidussi, editors), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad. Publ., Dordrecht (1998) 1–37
  • M Brion, Rational smoothness and fixed points of torus actions, Transform. Groups 4 (1999) 127–156
  • M Brion, Poincaré duality and equivariant (co)homology, Michigan Math. J. 48 (2000) 77–92
  • M Brion, Local structure of algebraic monoids, Mosc. Math. J. 8 (2008) 647–666, 846
  • J B Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, from: “Algebraic groups and their generalizations: Classical methods”, (W J Haboush, B J Parshall, editors), Proc. Sympos. Pure Math. 56, Amer. Math. Soc. (1994) 53–61
  • T Chang, T Skjelbred, The topological Schur lemma and related results, Ann. of Math. 100 (1974) 307–321
  • N Chriss, V Ginzburg, Representation theory and complex geometry, Birkhäuser, Boston, MA (1997)
  • V I Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978) 85–134 In Russian; translated in Russian Math. Surveys 33 (1978), 97–154
  • A Dimca, Singularities and topology of hypersurfaces, Universitext, Springer, New York (1992)
  • R Gonzales, Equivariant cohomology of rationally smooth group embeddings
  • M Goresky, R Kottwitz, R MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998) 25–83
  • V Guillemin, M Kogan, Morse theory on Hamiltonian $G$–spaces and equivariant $K$–theory, J. Differential Geom. 66 (2004) 345–375
  • M Harada, A Henriques, T S Holm, Computation of generalized equivariant cohomologies of Kac–Moody flag varieties, Adv. Math. 197 (2005) 198–221
  • R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York (1977)
  • W-y Hsiang, Cohomology theory of topological transformation groups, Ergeb. Math. Grenzgeb. 85, Springer, New York (1975)
  • F Kirwan, Intersection homology and torus actions, J. Amer. Math. Soc. 1 (1988) 385–400
  • C McCrory, A characterization of homology manifolds, J. London Math. Soc. 16 (1977) 149–159
  • J Milnor, On the $3$–dimensional Brieskorn manifolds $M(p,q,r)$, from: “Knots, groups, and $3$–manifolds (Papers dedicated to the memory of R H Fox)”, (L P Neuwirth, editor), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 175–225
  • C A M Peters, J H M Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. 52, Springer, Berlin (2008)
  • D Quillen, The spectrum of an equivariant cohomology ring, I, II, Ann. of Math. 94 (1971) 549–572; ibid. (2) 94, 573–602
  • L E Renner, Linear algebraic monoids, Encyclopaedia of Mathematical Sciences 134, Springer, Berlin (2005)
  • L E Renner, The $H$–polynomial of a semisimple monoid, J. Algebra 319 (2008) 360–376
  • L E Renner, Descent systems for Bruhat posets, J. Algebraic Combin. 29 (2009) 413–435
  • L E Renner, Rationally smooth algebraic monoids, Semigroup Forum 78 (2009) 384–395
  • L E Renner, The $H$–polynomial of an irreducible representation, J. Algebra 332 (2011) 159–186
  • L Scull, Equivariant formality for actions of torus groups, Canad. J. Math. 56 (2004) 1290–1307
  • G Segal, Equivariant $K$–theory, Inst. Hautes Études Sci. Publ. Math. (1968) 129–151
  • I R Shafarevich, Basic algebraic geometry, 1: Varieties in projective space, 2nd edition, Springer, Berlin (1994)
  • H Sumihiro, Equivariant completion, Journal Math. Kyoto Univ. 14 (1974) 1–28
  • V Uma, Equivariant $K$–theory of compactifications of algebraic groups, Transform. Groups 12 (2007) 371–406
  • G Vezzosi, A Vistoli, Higher algebraic $K$–theory for actions of diagonalizable groups, Invent. Math. 153 (2003) 1–44
  • A Weber, Formality of equivariant intersection cohomology of algebraic varieties, Proc. Amer. Math. Soc. 131 (2003) 2633–2638