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

Abstract

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.

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[Abr98] Abreu, M., Kähler geometry of toric varieties and extremal metrics , Int. J. Math. 9 (1998), 641651.
[ACG06] Apostolov, V., Calderbank, D. M. J. and Gauduchon, P., Hamiltonian 2-forms in Kähler geometry. I. General theory , J. Differential Geom. 73 (2006), 359412.
[ACGT04] 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), 277345.
[ACGT08] 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), 547601.
[Arm08a] Armstrong, S., Projective holonomy I: Principles and properties , Ann. Global Anal. Geom. 33 (2008), 4769.
[Arm08b] Armstrong, S., Projective holonomy II: Cones and complete classifications , Ann. Global Anal. Geom. 33 (2008), 137160.
[BMR15] 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.
[BKM09] 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), 341356; MR 2545256.
[Cal57] 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), 206207.
[Cal79] Calabi, E., Métriques kählériennes et fibrés holomorphes , Ann. Sci. École Norm. Supér. (4) 12 (1979), 269294.
[Cal82] Calabi, E., Extremal Kähler metrics , inSeminar on Differential Geometry (Princeton University Press, Princeton, NJ, 1982).
[CEMN] Calderbank, D. M. J., Eastwood, M., Matveev, V. S. and Neusser, K., C-projective geometry, in preparation.
[Eis23] Eisenhart, L. P., Symmetric tensors of the second order whose first covariant derivatives are zero , Trans. Amer. Math. Soc. 25 (1923), 297306.
[FR11] Fedorova, A. and Rosemann, S., The Tanno theorem for Kählerian metrics with arbitrary signature , Differential Geom. Appl. 29 (2011), 7179.
[FKMR12] 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), 153188.
[Gra67] Gray, A., Pseudo-Riemannian almost product manifolds and submersions , J. Math. Mech. 16 (1967), 715737.
[IT61] 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), 193200.
[KT11] Kiyohara, K. and Topalov, P. J., On Liouville integrability of h-projectively equivalent Kähler metrics , Proc. Amer. Math. Soc. 139 (2011), 231242.
[Kos55] Kostant, B., Holonomy and the Lie algebra of infinitesimal motions of a Riemann manifold , Trans. Amer. Math. Soc. 80 (1955), 528542.
[LT97] Lerman, E. and Tolman, S., Hamiltonian torus actions on symplectic orbifolds and toric varieties , Trans. Amer. Math. Soc. 349 (1997), 42014230.
[HS02] Hwang, A. D. and Singer, M. A., A momentum construction for circle-invariant Kähler metrics , Trans. Amer. Math. Soc. 354 (2002), 22852325.
[MR12] Matveev, V. S. and Rosemann, S., Proof of the Yano–Obata conjecture for h-projective transformations , J. Differential Geom. 92 (2012), 221261.
[MR15] Matveev, V. S. and Rosemann, S., Conification construction for Kaehler manifolds and its application in c-projective geometry , Adv. Math. 274 (2015), 138.
[Sch14] Schöbel, K., The variety of integrable Killing tensors on the 3-sphere , Symmetry Integrability Geom. Methods Appl. 10 (2014), 080.
[MD78] Mikeš, J. and Domashev, V. V., On the theory of holomorphically projective mappings of Kaehlerian spaces , Math. Zametki 23 (1978), 297303.
[Mik98] Mikeš, J., Holomorphically projective mappings and their generalizations , J. Math. Sci. 89 (1998), 13341353.
[O’Ne66] O’Neill, B., The fundamental equations of a submersion , Michigan Math. J. 13 (1966), 459469.
[OT54] Otsuki, T. and Tashiro, Y., On curves in Kählerian spaces , Math. J. Okayama Univ. 4 (1954), 5778.
[Tan78] Tanno, S., Some differential equations on Riemannian manifolds , J. Math. Soc. Japan 30 (1978), 509531.
[Yos78] Yoshimatsu, Y., H-projective connections and H-projective transformations , Osaka J. Math. 15 (1978), 435459.
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Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms

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