GKM manifold

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1 Definition

Let $\C^* = \C \setminus \{0\}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\C^* = \C \setminus \{0\}$ and $T=(\C^*)^n$$T=(\C^*)^n$ be the complex $n$$n$-torus with $n>1$$n>1$. A GKM manifold $M$$M$ is a complex smooth projective variety with an algebraic action of $T$$T$ satisfying the following three conditions;

1. the fixed point set $M^T$$M^T$ is finite,
2. the set $E$$E$ of the complex one-dimensional orbits is finite,
3. the action is equivariantly formal, that is, the Leray-Serre cohomology spectral sequence for the Borel construction
   $\displaystyle M \hookrightarrow ET\times_T M \to BT$ $$\displaystyle M \hookrightarrow ET\times_T M \to BT$$
   collapses at $E_2$$E_2$-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 $m=\dim_\C(M)$$m=\dim_\C(M)$. Then it is shown in [Goresky&Kottwitz&MacPherson1998] that $(M^T,E)$$(M^T,E)$ forms an $m$$m$-valent simple graph in the sense that the following hold:

1. The closure of each orbit $e\in E$$e\in E$ is an embedded $\C P^1$$\C P^1$ containing exactly two fixed points at the north and the south pole.
2. At any $p\in M^T$$p\in M^T$, the closures of $m$$m$ orbits meet.

For any edge connecting $p,q\in M^T$$p,q\in M^T$, let $\alpha_{pq} \in H^2(BT)$$\alpha_{pq} \in H^2(BT)$ be the weight of the isotropic representation of $T$$T$ on $T_p( \C P^1 )$$T_p( \C P^1 )$, where $\C P^1$$\C P^1$ is the corresponding closure of the one-dimensional $T$$T$-orbit. The graph $(M^T,E)$$(M^T,E)$ together with this data often referred to as the GKM graph associated to $M$$M$. The main theorem in [Goresky&Kottwitz&MacPherson1998] states that the rational $T$$T$-equivariant Borel cohomology of $M$$M$, $H^*_T(M;\Q)$$H^*_T(M;\Q)$, is described in terms of the GKM graph:

Theorem 1.1 [Goresky&Kottwitz&MacPherson1998].

$\displaystyle H^*_T(M;\Q) \simeq \{ (f_p) \in \bigoplus_{p \in M^T} H^*(BT;\Q) \mid f_p - f_q \in (\alpha_{pq}), \text{ for any edge connecting $p$ and $q$ } \},$ $$\displaystyle H^*_T(M;\Q) \simeq \{ (f_p) \in \bigoplus_{p \in M^T} H^*(BT;\Q) \mid f_p - f_q \in (\alpha_{pq}), \text{ for any edge connecting $p$ and $q$ } \},$$

where $(\alpha_{pq})$$(\alpha_{pq})$ is the ideal generated by $\alpha_{pq} \in H^2(BT; \Q)$$\alpha_{pq} \in H^2(BT; \Q)$.

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 $T$$T$-equivariant cohomology of those manifolds.

3 Further discussion

Several generalisation of the theory are known; for example, [Braden&MacPherson2001] for equivariant intersection homology, [Rosu2003] for equivariant $K$$K$-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].

4 References

• [Arabia1989] A. Arabia, Cohomologie $T\!$$T\!$-équivariante de la variété de drapeaux d'un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), 129–165. MR1015806 (90i:32042) Zbl 0706.57024
• [Bolker&Guillemin&Holm2002] E. Bolker, V. Guillemin, and T. Holm, How is a graph like a manifold?, (2009). Available at the arXiv:0206103.
• [Braden&MacPherson2001] T. Braden and R. MacPherson, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), 533–551. MR1871967 (2003g:14030) Zbl 1077.14522
• [Brion&Vergne1997] M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math. 482 (1997), 67–92. MR1427657 (98a:14067) Zbl 0862.14006
• [Goertsches&Mare2013] O. Goertsches and A.-L. Mare, Non-abelian GKM theory, to appear in Math. Zeit.
• [Goresky&Kottwitz&MacPherson1998] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83. MR1489894 (99c:55009) Zbl 0897.22009
• [Guillemin&Holm2004] V. Guillemin and T. S. Holm, GKM theory for torus actions with nonisolated fixed points, Int. Math. Res. Not. (2004), 2105–2124. MR2064318 (2005d:53136) Zbl 1138.53315
• [Guillemin&Zara2001] V. Guillemin and C. Zara, 1-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J. 107 (2001), 283–349. MR1823050 (2002j:53110) Zbl 1020.57013
• [Harada&Henriques&Holm2005] M. Harada, A. Henriques and T. S. Holm, Computation of generalized equivariant cohomologies of Kac-Moody flag varieties, Adv. Math. 197 (2005), 198–221. MR2166181 (2006h:53086) Zbl 1110.55003
• [Kuroki2009] S. Kuroki, Introduction to GKM theory, Trends in Mathematics (2009).
• [Rosu2003] I. Rosu, Equivariant $K$$K$-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)