## Geometry & Topology

### Rational smoothness, cellular decompositions and GKM theory

Richard Gonzales

#### Abstract

We introduce the notion of $ℚ$–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 $ℚ$–filtrable GKM variety. Applications to the theory of group embeddings are provided.

#### Article information

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

Dates
Revised: 19 August 2013
Accepted: 20 September 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732724

Digital Object Identifier
doi:10.2140/gt.2014.18.291

Mathematical Reviews number (MathSciNet)
MR3159163

Zentralblatt MATH identifier
1284.14060

#### Citation

Gonzales, Richard. Rational smoothness, cellular decompositions and GKM theory. Geom. Topol. 18 (2014), no. 1, 291--326. doi:10.2140/gt.2014.18.291. https://projecteuclid.org/euclid.gt/1513732724

#### References

• 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