Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-20T04:55:44.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms

Part of: Classical differential geometry Global differential geometry Local differential geometry Compact analytic spaces

Published online by Cambridge University Press:  26 April 2016

David M. J. Calderbank ,
Vladimir S. Matveev  and
Stefan Rosemann
David M. J. Calderbank
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK email D.M.J.Calderbank@bath.ac.uk
Vladimir S. Matveev
Affiliation:
Institute of Mathematics, FSU Jena, 07737 Jena, Germany email vladimir.matveev@uni-jena.de
Stefan Rosemann
Affiliation:
Institute of Mathematics, FSU Jena, 07737 Jena, Germany email stefan.rosemann@uni-jena.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

Keywords

Kähler metricprojective geometrycurvature

MSC classification

Primary: 53B35: Hermitian and Kählerian structures
Secondary: 53C55: Hermitian and Kählerian manifolds 53B10: Projective connections 53A20: Projective differential geometry 32J27: Compact Kähler manifolds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1555 - 1575
Copyright
© The Authors 2016 

References

Abreu, M., Kähler geometry of toric varieties and extremal metrics , Int. J. Math. 9 (1998), 641–651.CrossRefGoogle Scholar
Apostolov, V., Calderbank, D. M. J. and Gauduchon, P., Hamiltonian 2-forms in Kähler geometry. I. General theory , J. Differential Geom. 73 (2006), 359–412.CrossRefGoogle Scholar
Apostolov, V., Calderbank, D. M. J., Gauduchon, P. and Tønnesen-Friedman, C., Hamiltonian 2-forms in Kähler geometry. II. Global classification , J. Differential Geom. 68 (2004), 277–345.CrossRefGoogle Scholar
Apostolov, V., Calderbank, D. M. J., Gauduchon, P. and Tønnesen-Friedman, C., Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability , Invent. Math. 173 (2008), 547–601.CrossRefGoogle Scholar
Armstrong, S., Projective holonomy I: Principles and properties , Ann. Global Anal. Geom. 33 (2008), 47–69.CrossRefGoogle Scholar
Armstrong, S., Projective holonomy II: Cones and complete classifications , Ann. Global Anal. Geom. 33 (2008), 137–160.CrossRefGoogle Scholar
Bolsinov, A. V., Matveev, V. S. and Rosemann, S., Local normal forms for c-projectively equivalent metrics and proof of the Yano–Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics, Preprint (2015), arXiv:1510.00275.Google Scholar
Bolsinov, A. V., Kiosak, V. and Matveev, V. S., A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics , J. Lond. Math. Soc. (2) 80 (2009), 341–356; MR 2545256.CrossRefGoogle Scholar
Calabi, E., The space of Kähler metrics , in Proceedings of the International Congress of Mathematicians 1954, vol. 2 (Noordhoff and North-Holland, Groningen and Amsterdam, 1957), 206–207.Google Scholar
Calabi, E., Métriques kählériennes et fibrés holomorphes , Ann. Sci. École Norm. Supér. (4) 12 (1979), 269–294.CrossRefGoogle Scholar
Calabi, E., Extremal Kähler metrics , inSeminar on Differential Geometry (Princeton University Press, Princeton, NJ, 1982).Google Scholar
Calderbank, D. M. J., Eastwood, M., Matveev, V. S. and Neusser, K., C-projective geometry, in preparation.Google Scholar
Eisenhart, L. P., Symmetric tensors of the second order whose first covariant derivatives are zero , Trans. Amer. Math. Soc. 25 (1923), 297–306.CrossRefGoogle Scholar
Fedorova, A. and Rosemann, S., The Tanno theorem for Kählerian metrics with arbitrary signature , Differential Geom. Appl. 29 (2011), 71–79.CrossRefGoogle Scholar
Fedorova, A., Kiosak, V., Matveev, V. and Rosemann, S., The only Kähler manifold with degree of mobility at least 3 is (CP (n), g Fubini–Study) , Proc. Lond. Math. Soc. 105 (2012), 153–188.CrossRefGoogle Scholar
Gray, A., Pseudo-Riemannian almost product manifolds and submersions , J. Math. Mech. 16 (1967), 715–737.Google Scholar
Ishihara, S. and Tachibana, S., A note on holomorphic projective transformations of a Kählerian space with parallel Ricci tensor , Tohoku Math. J. (2) 13 (1961), 193–200.CrossRefGoogle Scholar
Kiyohara, K. and Topalov, P. J., On Liouville integrability of h-projectively equivalent Kähler metrics , Proc. Amer. Math. Soc. 139 (2011), 231–242.CrossRefGoogle Scholar
Kostant, B., Holonomy and the Lie algebra of infinitesimal motions of a Riemann manifold , Trans. Amer. Math. Soc. 80 (1955), 528–542.CrossRefGoogle Scholar
Lerman, E. and Tolman, S., Hamiltonian torus actions on symplectic orbifolds and toric varieties , Trans. Amer. Math. Soc. 349 (1997), 4201–4230.CrossRefGoogle Scholar
Hwang, A. D. and Singer, M. A., A momentum construction for circle-invariant Kähler metrics , Trans. Amer. Math. Soc. 354 (2002), 2285–2325.CrossRefGoogle Scholar
Matveev, V. S. and Rosemann, S., Proof of the Yano–Obata conjecture for h-projective transformations , J. Differential Geom. 92 (2012), 221–261.CrossRefGoogle Scholar
Matveev, V. S. and Rosemann, S., Conification construction for Kaehler manifolds and its application in c-projective geometry , Adv. Math. 274 (2015), 1–38.CrossRefGoogle Scholar
Schöbel, K., The variety of integrable Killing tensors on the 3-sphere , Symmetry Integrability Geom. Methods Appl. 10 (2014), 080.Google Scholar
Mikeš, J. and Domashev, V. V., On the theory of holomorphically projective mappings of Kaehlerian spaces , Math. Zametki 23 (1978), 297–303.Google Scholar
Mikeš, J., Holomorphically projective mappings and their generalizations , J. Math. Sci. 89 (1998), 1334–1353.CrossRefGoogle Scholar
O’Neill, B., The fundamental equations of a submersion , Michigan Math. J. 13 (1966), 459–469.Google Scholar
Otsuki, T. and Tashiro, Y., On curves in Kählerian spaces , Math. J. Okayama Univ. 4 (1954), 57–78.Google Scholar
Tanno, S., Some differential equations on Riemannian manifolds , J. Math. Soc. Japan 30 (1978), 509–531.CrossRefGoogle Scholar
Yoshimatsu, Y., H-projective connections and H-projective transformations , Osaka J. Math. 15 (1978), 435–459.Google Scholar