GKM manifold
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[edit] 1 Definition
Let and be the complex -torus with . A GKM manifold is a complex smooth projective variety with an algebraic action of satisfying the following three conditions;
- the fixed point set is finite,
- the set of the complex one-dimensional orbits is finite,
- the action is equivariantly formal, that is, the Leray-Serre cohomology spectral sequence for the Borel construction collapses at -term.
The letters `G', `K' and `M' correspond to the first initials of the authors of [Goresky&Kottwitz&MacPherson1998] where GKM manifolds were first introduced.
Let . Then it is shown in [Goresky&Kottwitz&MacPherson1998] that forms an -valent simple graph in the sense that the following hold:
- The closure of each orbit is an embedded containing exactly two fixed points at the north and the south pole.
- At any , the closures of orbits meet.
For any edge connecting , let be the weight of the isotropic representation of on , where is the corresponding closure of the one-dimensional -orbit. The graph together with this data often referred to as the GKM graph associated to . The main theorem in [Goresky&Kottwitz&MacPherson1998] states that the rational -equivariant Borel cohomology of , , is described in terms of the GKM graph:
Theorem 1.1 [Goresky&Kottwitz&MacPherson1998].
[edit] 2 Examples
Projective toric manifolds and flag manifolds are examples of GKM manifolds and the above theorem corresponds to the piecewise polynomial description ([Brion&Vergne1997]) and Arabia's description ([Arabia1989]) of the -equivariant cohomology of those manifolds.
[edit] 3 Further discussion
Several generalisation of the theory are known; for example, [Braden&MacPherson2001] for equivariant intersection homology, [Rosu2003] for equivariant -theory, [Harada&Henriques&Holm2005] for complex oriented cohomology theories, [Guillemin&Holm2004] for action with non-isolated fixed points, and [Goertsches&Mare2013] for non-abelian group action. For introductory surveys of GKM manifolds, see [Tymoczko2005][Kuroki2009]. The theory of GKM manifolds has applications also in combinatorics: see, for example, [Guillemin&Zara2001] or [Bolker&Guillemin&Holm2002].
[edit] 4 References
- [Arabia1989] A. Arabia, Cohomologie -équivariante de la variété de drapeaux d'un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), 129–165. MR1015806 (90i:32042) Zbl 0706.57024
- [Bolker&Guillemin&Holm2002] E. Bolker, V. Guillemin, and T. Holm, How is a graph like a manifold?, (2009). Available at the arXiv:0206103.
- [Braden&MacPherson2001] T. Braden and R. MacPherson, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), 533–551. MR1871967 (2003g:14030) Zbl 1077.14522
- [Brion&Vergne1997] M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math. 482 (1997), 67–92. MR1427657 (98a:14067) Zbl 0862.14006
- [Goertsches&Mare2013] O. Goertsches and A.-L. Mare, Non-abelian GKM theory, to appear in Math. Zeit.
- [Goresky&Kottwitz&MacPherson1998] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83. MR1489894 (99c:55009) Zbl 0897.22009
- [Guillemin&Holm2004] V. Guillemin and T. S. Holm, GKM theory for torus actions with nonisolated fixed points, Int. Math. Res. Not. (2004), 2105–2124. MR2064318 (2005d:53136) Zbl 1138.53315
- [Guillemin&Zara2001] V. Guillemin and C. Zara, 1-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J. 107 (2001), 283–349. MR1823050 (2002j:53110) Zbl 1020.57013
- [Harada&Henriques&Holm2005] M. Harada, A. Henriques and T. S. Holm, Computation of generalized equivariant cohomologies of Kac-Moody flag varieties, Adv. Math. 197 (2005), 198–221. MR2166181 (2006h:53086) Zbl 1110.55003
- [Kuroki2009] S. Kuroki, Introduction to GKM theory, Trends in Mathematics (2009).
- [Rosu2003] I. Rosu, Equivariant -theory and equivariant cohomology. With an appendix by Allen Knutson and Rosu, Math. Z. 243 (2003), 423–448. MR1970011 (2004f:19011) Zbl 1019.19003
- [Tymoczko2005] J. S. Tymoczko, An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson, Snowbird lectures in algebraic geometry, Contemp. Math., 388, Amer. Math. Soc., Providence, RI (2005), 169–188. MR2182897 (2006m:55019)