Twistor forms on Kähler manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 823-845.

Twistor forms are a natural generalization of conformal vector fields on riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We study twistor forms on compact Kähler manifolds and give a complete description up to special forms in the middle dimension. In particular, we show that they are closely related to hamiltonian 2-forms. This provides the first examples of compact Kähler manifolds with non-parallel twistor forms in any even degree.

Classification : 53C55, 58J50
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     title = {Twistor forms on {K\"ahler} manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     volume = {Ser. 5, 2},
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Moroianu, Andrei; Semmelmann, Uwe. Twistor forms on Kähler manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 823-845. http://www.numdam.org/item/ASNSP_2003_5_2_4_823_0/

[1] M. F. Atiyah - N. J. Hitchin - I. M. Singer, Self-duality in four dimensional Riemannian geometry, Proc. R. Soc. Lond. A362 (1978), 425-461. | MR | Zbl

[2] V. Apostolov, private communication.

[3] V. Apostolov - D. Calderbank - P. Gauduchon, The geometry of weakly selfdual Kähler surfaces, Compositio Math. 135 (2003), 279-322. | MR | Zbl

[4] V. Apostolov - D. Calderbank - P. Gauduchon, Hamiltonian 2-forms in Kähler geometry I, math.DG/0202280 (2002). | Zbl

[5] C. Bär, Real Killing spinors and holonomy , Comm. Math. Phys. 154 (1993), 509-521. | MR | Zbl

[6] H. Baum - Th. Friedrich - R. Grunewald - I. Kath, “Twistor and Killing Spinors on Riemannian Manifolds”, Teubner-Verlag, Stuttgart-Leipzig, 1991. | MR | Zbl

[7] I. M. Benn - P. Charlton - J. Kress, Debye potentials for Maxwell and Dirac fields from a generalization of the Killing-Yano equation, J. Math. Phys. 38 (1997), 4504-4527. | MR | Zbl

[8] I. M. Benn - P. Charlton, Dirac symmetry operators from conformal Killing-Yano tensors, Classical Quantum Gravity 14 (1997), 1037-1042. | MR | Zbl

[9] A. Besse, “Einstein manifolds”, Springer-Verlag, New York 1987. | MR | Zbl

[10] T. Branson, Stein-Weiss operators and ellipticity, J. Functional Anal. 151 (1997), 334-383. | MR | Zbl

[11] S. Gallot - D. Meyer, Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), 259-284. | MR | Zbl

[12] J.-B. Jun - S. Ayabe - S. Yamaguchi, On the conformal Killing p-form in compact Kaehlerian manifolds, Tensor (N.S.) 42 (1985), 258-271. | MR | Zbl

[13] T. Kashiwada, On conformal Killing tensor, Natur. Sci. Rep. Ochanomizu Univ. 19 (1968), 67-74. | MR | Zbl

[14] R. Penrose - M. Walker, On quadratic first integrals of the geodesic equations for type {22} spacetimes, Comm. Math. Phys. 18 (1970) 265-274. | MR | Zbl

[15] U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Math. Z. 243 (2003), 503-527. | MR | Zbl

[16] S. Tachibana, On Killing tensors in Riemannian manifolds of positive curvature operator, Tohoku Math. J. (2) 28 (1976), 177-184. | MR | Zbl

[17] S. Tachibana - T. Kashiwada, On the integrability of Killing-Yano's equation, J. Math. Soc. Japan 21 (1969), 259-265. | MR | Zbl

[18] S. Tachibana, On conformal Killing tensor in a Riemannian space, Tohoku Math. J. (2) 21 (1969), 56-64. | MR | Zbl

[19] S. Yamaguchi, On a Killing p-form in a compact Kählerian manifold, Tensor (N.S.) 29 (1975), 274-276. | MR | Zbl

[20] K. Yano, Some remarks on tensor fields and curvature, Ann. of Math. (2) 55 (1952), 328-347. | MR | Zbl