Geodesic
Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 262-294 
Shintarô Kuroki. Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 262-294. http://geodesic.mathdoc.fr/item/TM_2011_275_a17/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian $T$-isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology $H_T^*(M;\mathbb Z)$ with a point $\hat a$ in $H^2(M;\mathbb R)\setminus\{0\}$ up to weak $H^*(BT;\mathbb Z)$-algebra isomorphism preserving $\hat a$.

[1] Atiyah M.F., Bott R., “The moment map and equivariant cohomology”, Topology, 23 (1984), 1–28 | DOI | MR | Zbl

[2] Bielawski R., “Complete hyperkähler $4n$-manifolds with a local tri-Hamiltonian $\mathbb R^n$-action”, Math. Ann., 314:3 (1999), 505–528 | DOI | MR | Zbl

[3] Bielawski R., Dancer A.S., “The geometry and topology of toric hyperkähler manifolds”, Commun. Anal. and Geom., 8:4 (2000), 727–760 | MR | Zbl

[4] Bredon G.E., Introduction to compact transformation groups, Acad. Press, New York, 1972 | MR | Zbl

[5] Delzant T., “Hamiltoniens périodiques et images convexes de l'application moment”, Bull. Soc. math. France, 116:3 (1988), 315–339 | MR | Zbl

[6] Fulton W., Introduction to toric varieties, Ann. Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[7] Goto R., “On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method”, Intern. J. Mod. Phys. A, 7, Suppl. 1A (1992), 317–338 | DOI | MR | Zbl

[8] Guillemin V.W., Sternberg S., Supersymmetry and equivariant de Rham theory, Springer, Berlin, 1999 | MR

[9] Harada M., Holm T.S., “The equivariant cohomology of hypertoric varieties and their real loci”, Commun. Anal. and Geom., 13:3 (2005), 527–559 | DOI | MR | Zbl

[10] Hattori A., Masuda M., “Theory of multi-fans”, Osaka J. Math., 40:1 (2003), 1–68 | MR | Zbl

[11] Harada M., Proudfoot N., “Properties of the residual circle action on a hypertoric variety”, Pac. J. Math., 214:2 (2004), 263–284 | DOI | MR | Zbl

[12] Hausel T., Sturmfels B., “Toric hyperkähler varieties”, Doc. math., 7 (2002), 495–534 | MR | Zbl

[13] Hsiang W.Y., Cohomology theory of topological transformation groups, Ergebn. Math., 85, Springer, Berlin, 1975 | MR | Zbl

[14] Kawakubo K., The theory of transformation groups, Oxford Univ. Press, London, 1991 | MR | Zbl

[15] Konno H., “Equivariant cohomology rings of toric hyperkähler manifolds”, Quaternionic structures in mathematics and physics, Rome, 1999, Univ. Roma “La Sapienza”, Rome, 2001, 231–240, electronic | MR | Zbl

[16] Konno H., “Cohomology rings of toric hyperkähler manifolds”, Intern. J. Math., 11:8 (2000), 1001–1026 | DOI | MR | Zbl

[17] Konno H., “The geometry of toric hyperkähler varieties”, Toric topology, Proc. Intern. Conf., Osaka, 2006, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 241–260 | DOI | MR | Zbl

[18] Masuda M., “Unitary toric manifolds, multi-fans and equivariant index”, Tohoku Math. J., 51:2 (1999), 237–265 | DOI | MR | Zbl

[19] Masuda M., “Equivariant cohomology distinguishes toric manifolds”, Adv. Math., 218:6 (2008), 2005–2012 | DOI | MR | Zbl

[20] Mimura M., Toda H., Topology of Lie groups, I and II, Amer. Math. Soc., Providence, RI, 1991 | MR | Zbl

[21] Oda T., Convex bodies and algebraic geometry: An introduction to the theory of toric varieties, Ergebn. Math. Grenzgeb. 3. Flg., 15, Springer, Berlin, 1988 | MR

[22] Proudfoot N.J., “A survey of hypertoric geometry and topology”, Toric topology, Proc. Intern. Conf., Osaka, 2006, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 323–338 | DOI | MR | Zbl