GKM manifold

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[edit] 1 Definition

Let \C^* = \C \setminus \{0\} and T=(\C^*)^n be the complex n-torus with n>1. A GKM manifold M is a complex smooth projective variety with an algebraic action of T satisfying the following three conditions;

  1. the fixed point set M^T is finite,
  2. the set 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
    collapses at 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). Then it is shown in [Goresky&Kottwitz&MacPherson1998] that (M^T,E) forms an m-valent simple graph in the sense that the following hold:

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

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

[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 T-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 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].

[edit] 4 References

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