𝑆¹-quotients of quaternion-Kähler manifolds

Battaglia, Fiammetta

doi:10.1090/S0002-9939-96-03208-X

$S^1$-quotients of quaternion-Kähler manifolds
HTML articles powered by AMS MathViewer

by Fiammetta Battaglia PDF

Proc. Amer. Math. Soc. 124 (1996), 2185-2192 Request permission

Abstract:

The notion of symplectic reduction has been generalized to manifolds endowed with other structures, in particular to quaternion-Kähler manifolds, namely Riemannian manifolds with holonomy in $Sp(n)Sp(1)$. In this work we prove that the only complete quaternion-Kähler manifold with positive scalar curvature obtainable as a quaternion-Kähler quotient by a circle action is the complex Grassmannian $Gr_2(\mathbb {C}^n)$.

References

D. V. Alekseevskiĭ, Riemannian spaces with unusual holonomy groups, Funkcional. Anal. i Priložen 2 (1968), no. 2, 1–10 (Russian). MR 0231313
D. V. Alekseevskiĭ, Classification of quaternionic spaces with transitive solvable group of motions, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 315–362, 472 (Russian). MR 0402649
F. Battaglia: Tesi di Dottorato, Università di Firenze, 1993.
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
B. de Wit and A. Van Proeyen, Special geometry, cubic polynomials and homogeneous quaternionic spaces, Comm. Math. Phys. 149 (1992), no. 2, 307–333. MR 1186031, DOI 10.1007/BF02097627
K. Galicki, A generalization of the momentum mapping construction for quaternionic Kähler manifolds, Comm. Math. Phys. 108 (1987), no. 1, 117–138. MR 872143, DOI 10.1007/BF01210705
Krzysztof Galicki, Multi-centre metrics with negative cosmological constant, Classical Quantum Gravity 8 (1991), no. 8, 1529–1543. MR 1122250, DOI 10.1088/0264-9381/8/8/017
K. Galicki and H. B. Lawson Jr., Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988), no. 1, 1–21. MR 960830, DOI 10.1007/BF01457009
Toru Gocho and Hiraku Nakajima, Einstein-Hermitian connections on hyper-Kähler quotients, J. Math. Soc. Japan 44 (1992), no. 1, 43–51. MR 1139657, DOI 10.2969/jmsj/04410043
N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. MR 877637, DOI 10.1007/BF01214418
Dominic Joyce, The hypercomplex quotient and the quaternionic quotient, Math. Ann. 290 (1991), no. 2, 323–340. MR 1109637, DOI 10.1007/BF01459248
Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
C.R. LeBrun: On the topology of quaternionic manifolds, Twistor Newsletter, 32 (1991), 6–7.
Claude LeBrun, On complete quaternionic-Kähler manifolds, Duke Math. J. 63 (1991), no. 3, 723–743. MR 1121153, DOI 10.1215/S0012-7094-91-06331-3
C.R. LeBrun, S.M. Salamon : Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118, (1994), 109-132.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4
Takashi Nitta, Yang-Mills connections on quaternionic Kähler quotients, Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 8, 245–247. MR 1077606
Y. S. Poon and S. M. Salamon, Quaternionic Kähler $8$-manifolds with positive scalar curvature, J. Differential Geom. 33 (1991), no. 2, 363–378. MR 1094461
Simon Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), no. 1, 143–171. MR 664330, DOI 10.1007/BF01393378
S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31–55. MR 860810, DOI 10.24033/asens.1503
Simon Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics Series, vol. 201, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1004008
Andrew Swann, Hyper-Kähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), no. 3, 421–450. MR 1096180, DOI 10.1007/BF01446581
Joseph A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 (1965), 1033–1047. MR 0185554