×

HyperKähler and quaternionic Kähler geometry. (English) Zbl 0711.53051

A (pseudo-) hyperKähler \((4n+4)\)-manifold \({\mathcal U}(M)\) is constructed for each quaternionic Kähler 4n-manifold in a way that is compatible with generalizations of the Marsden-Weinstein symplectic quotient. The manifold \({\mathcal U}(M)\) admits: (1) a certain type of SU(2)-action permuting the complex structures; (2) a function which is a Kähler potential for each complex structure; (3) quaternionic Kähler metrics in the same quaternionic class. Locally (1), (2) and (3) are equivalent and characterize \({\mathcal U}(M)\). Examples include, for a compact semisimple Lie group G, the nilpotent orbits in \({\mathfrak g}^{{\mathbb{C}}}\), which are seen to be fibred over quaternionic Kähler manifolds.
Reviewer: A.Swann

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Alekseevskiî, D.V.: Riemannian manifolds with exceptional holonomy groups Funkts. Anal. Prilozh.2(2), 1-10 (1968) [Engl. transl.: Funct. Anal. Appl.2, 97-105 (1968)] · Zbl 0174.45203
[2] Alekseevskiî, D.V.: Compact quaternion spaces. Funkts. Anal. Prilozh.2(2), 11-20 (1968) [Engl. transl.: Funct. Anal. Appl.2, 106-114 (1968)] · Zbl 0175.19001
[3] Alekseevskiî, D.V.: Quaternion Riemannian spaces with transitive reductive or solvable group of motions. Funkts. Anal. Prilozh.4 (4), 68-69 (1970) [Engl. transl.: Funct. Anal. Appl.4, 321-322 (1970)] · Zbl 0221.32008
[4] Alekseevskiî, D.V.: Classification of quaternionic spaces with a transitive solvable group of motions. Izv. Akad. Nauk SSSR Ser. Mat.9 (2), 315-362 (1975) [Engl. transl.: Math. USSR, Izv.9, 297-339 (1975)] · Zbl 0309.53045
[5] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond., Ser. A362, 425-461 (1978) · Zbl 0389.53011
[6] Atiyah, M.F., Drinfeld, D.G., Hitchin, N.J., Manin, Y.I.: Construction of instantons. Phys. Lett. A65, 185-187 (1978) · Zbl 0424.14004
[7] Atiyah, M.F., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles. New Jersey: Princeton University Press 1988 · Zbl 0671.53001
[8] Bagger, J., Witten, E.: Matter couplings inn=2 supergravity. Nucl. Phys. B222, 1-10 (1983)
[9] Besse, A.L.: Einstein manifolds. Erg. Math. Grenzgeb. Berlin Heidelberg New York: Springer 1987
[10] Brinkman, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann.94, 119-145 (1925) · JFM 51.0568.03
[11] Bryant, R.L., Salamon, S.M.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J.58, 829-850 (1989) · Zbl 0681.53021
[12] Calabi, E.: Métriques kähl?iennes et fibrés holomorphes. Ann. Sci. Éc. Norm. Supér., IV. Sér.12, 269-294 (1979)
[13] Calabi, E.: Isometric families of Kähler structures. In: The Chern Symposium, 1979, Hsiang, W.-Y., et al. (eds.), pp. 23-39. Springer 1980
[14] Cordero, L.A., Fernández, M., Gray, A.: Variétés symplectiques sans structures kählériennes. C.R. Acad. Sci., Paris, Sér. I301, 217-218 (1985) · Zbl 0581.53021
[15] Cordero, L.A., Fernández, M., de Léon, M.: Examples of compact non-Kähler almost Kähler manifolds. Proc. Am. Math. Soc.95, 280-286 (1985) · Zbl 0575.53015
[16] Cordero, L.A., Fernández, M., Gray, A.: Symplectic manifolds with no Kähler structure. Topology25, 375-380 (1986) · Zbl 0596.53030
[17] Fernández, M., Gray, A.: The Iwasawa manifold. In: Differential geometry, Peñíscola 1985, Proceedings, Naveira, A., Ferrandez, A., Mascaro, F. (eds.) (Lect. Notes Math., vol. 1209). Berlin Heidelberg New York: Springer 1985
[18] Galicki, K.: Quaternionic Kähler and hyperKähler non-linear ?-models. Nucl. Phys. B271, 402-416 (1986)
[19] Galicki, K.: A generalization of the momentum mapping construction for quaternionic Kähler manifolds. Commun. Math. Phys.108, 117-138 (1987) · Zbl 0608.53058
[20] Galicki, K., Lawson, H.B.: Quaternionic reduction and quaternionic orbifolds. Math. Ann.282, 1-21 (1988) · Zbl 0628.53060
[21] Gibbons, G.W., Pope, C.N.: The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys.66, 267-290 (1979)
[22] Gray, A.: A note on manifolds whose holonomy group is a subgroup ofSp(n){\(\cdot\)}Sp(1). Mich. Math. J.16, 125-128 (1969) · Zbl 0177.50001
[23] Gudmundsson, S.: On the geometry of harmonic morphisms. Preprint · Zbl 0715.53029
[24] Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press 1984 · Zbl 0576.58012
[25] Hitchin, N.J.: Metrics on moduli spaces. In: Lefschetz Centennial Conference. Part I: Proceedings on algebraic geometry, Mexico City, 1984. Sundararaman, D. (ed.) (Contemp. Math., vol. 58, pp. 157-178). AMS, Providence, R.I. (1986)
[26] Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc.55, 59-126 (1987) · Zbl 0634.53045
[27] Hitchin, N.J., Karlhede, A., Lindström, U., Ro?ek, M.: HyperKähler metrics and supersymmetry. Commun. Math. Phys.108, 535-589 (1987) · Zbl 0612.53043
[28] Ishihara, S.: Quaternion Kählerian manifolds. J. Differ. Geom.9, 483-500 (1974) · Zbl 0297.53014
[29] Jensen, G.: Einstein metrics in principal fibre bundles. J. Differ. Geom.8, 599-614 (1973) · Zbl 0284.53038
[30] Kobayashi, S.: Remarks on complex contact manifolds. Proc. Am. Math. Soc.10, 164-167 (1959) · Zbl 0090.38502
[31] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. 2 volumes. New York: Wiley 1963 · Zbl 0119.37502
[32] Kronheimer, P.B.: The construction of ALE spaces as hyperKähler quotients. J. Differ. Geom.29, 665-683 (1989) · Zbl 0671.53045
[33] Kronheimer, P.B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom.29, 685-697 (1989) · Zbl 0671.53046
[34] Kronheimer, P.B.: Instantons and the geometry of the nilpotent variety. J. Differ. Geom.32, 473-490 (1990) · Zbl 0725.58007
[35] Lang, S.: Algebra. Second edition. Reading, Mass.: Addison-Wesley 1984
[36] LeBrun, C.: Spaces of complex null geodesics in complex-Riemannian geometry. Trans. Am. Math. Soc.278, 209-231 (1983) · Zbl 0562.53018
[37] McDuff, D.: Examples of simply-connected symplectic non-Kählerian manifolds. J. Differ. Geom.20, 267-277 (1984) · Zbl 0567.53031
[38] Maciocia, A.: Metrics on the moduli spaces of instantons over Euclidean 4-space. Commun. Math. Phys.135, 467-482 (1991) · Zbl 0734.53025
[39] Marchiafava, S., Romani, G.: Sui fibrati construttura quaternionale generalizzata. Ann. Mat. Pura Appl., IV. Ser.107, 131-157 (1976) · Zbl 0356.55003
[40] Marchiafava, S.: Su alouna sottovarietà che ha interesse considerare i una varietà Kahleriana quaternionale. Preprint
[41] Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121-130 (1974) · Zbl 0327.58005
[42] Obata, M.: Affine connections on manifolds with almost complex, quaternion or Hermititian structure. Jap. J. Math.26, 43-73 (1956)
[43] O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J.13, 459-469 (1966) · Zbl 0145.18602
[44] Page, D.N., Pope, C.N.: Einstein metrics on quaternionic line bundles. Classical Quantum Gravity3, 249-259 (1986) · Zbl 0633.53068
[45] Salamon, S.M.: Quaternionic Kähler manifolds. Invent. Math.67, 143-171 (1982) · Zbl 0486.53048
[46] Salamon, S.M.: Differential geometry of quaternionic manifolds. Ann. Sci. Éc. Norm. Supér., IV. Sér.19, 31-55 (1986) · Zbl 0616.53023
[47] Swann, A.F.: Aspects symplectiques de la géométrie quaternionique. C.R. Acad. Sci., Paris, Sér. I308, 225-228 (1989) · Zbl 0661.53023
[48] Swann, A.F.: Some quaternionically equivalent Einstein metrics. Twistor Newsl.30, 18-20 (1990)
[49] Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc.55, 467-468 (1976) · Zbl 0324.53031
[50] Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech.14, 1033-1047 (1965) · Zbl 0141.38202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.