×

Conformal Killing forms in Kähler geometry. (English) Zbl 1508.53026

Let \((M, \omega)\) be a connected Kähler manifold of complex dimension \(m \ge 3\). This paper studies the following overdetermined system of first-order equations on a pair of differential forms \((\varphi , \tau )\): \[ \nabla_X \varphi = X^{1,0} \wedge \tau + \frac{i}{2} \omega \wedge (X \lrcorner \tau ), \] for any vector field \(X\) on \(M\), where \(\varphi\) is a primitive \((1,m-1)\)-form and \(\tau\) is a \((0,m-1)\)-form. In fact, for a solution \((\varphi , \tau )\) of the above equation, \(\tau\) is determined by \(\varphi\) as \(\tau = - \frac{2}{m+1} \partial^* \varphi\). The solution \(\varphi\) is a primitive conformal Killing \((1,m-1)\)-form.
The authors first prove a general result which gives a local structure of solutions \((\varphi,\tau )\). Namely, assuming that \(\bar{\partial} \tau\) is not identically zero, they prove that \(\varphi\) is parallel, or else either \(\omega\) is Ricci-flat and equipped with a cone vector field or it is locally of Calabi type with local Kähler-Einstein base. They use this result to prove that primitive conformal Killing \((1,m-1)\)-forms on a compact Kähler manifolds are parallel, and establishes a connection between conformal Killing forms with degree \(\neq 1, 2m-1\) and Hamiltonian \(2\)-forms as introduced in [V. Apostolov et al., J. Differ. Geom. 73, No. 3, 359–412 (2006; Zbl 1101.53041)]. This result completes the classification of conformal Killing forms on compact Kähler manifolds.

MSC:

53B35 Local differential geometry of Hermitian and Kählerian structures
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C12 Foliations (differential geometric aspects)

Citations:

Zbl 1101.53041
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] A. Andrada and I. G. Dotti, Conformal Killing-Yano 2-forms, Differential Geom. Appl. 58 (2018), 103-119. · Zbl 1391.53032
[2] V. Apostolov, T. Drăghici, and A. Moroianu, A splitting theorem for Kähler manifolds whose Ricci tensors have constant eigenvalues, Internat. J. Math. 12 (2001), no. 7, 769-789. · Zbl 1111.53303
[3] V. Apostolov, D. Calderbank, and P. Gauduchon, Hamiltonian 2-forms in Kähler geometry, I. General theory, J. Differential Geom. 73 (2006), no. 3, 359-412. · Zbl 1101.53041
[4] V. Apostolov, D. Calderbank, P. Gauduchon, and C.W. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry, II. Global classification, J. Differential Geom. 68 (2004), no. 2, 277-345. · Zbl 1079.32012
[5] V. Apostolov, D. Calderbank, P. Gauduchon, and C. W. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry, III. Extremal metrics and stability, Invent. Math. 173 (2008), no. 3, 547-601. · Zbl 1145.53055
[6] V. Apostolov, D. Calderbank, and E. Legendre, Weighted K-stability of polarized varieties and extremality of Sasaki manifolds, Adv. Math. 391 (2021), Paper No. 107969, 63 pp. · Zbl 1478.53081
[7] V. Apostolov and G. Maschler, Conformally Kähler, Einstein-Maxwell geometry, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 5, 1319-1360. · Zbl 1477.53077
[8] V. Apostolov, G. Maschler, and C. W. Tønnesen-Friedman, Weighted extremal Kähler metrics and the Einstein-Maxwell geometry of projective bundles, to appear in Comm. Anal. Geom., preprint, arXiv:1808.02813 [math.DG].
[9] C. Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), no. 3, 509-521. · Zbl 0778.53037
[10] F. Belgun, A. Moroianu, and U. Semmelmann, Killing forms on symmetric spaces, Differential Geom. Appl. 24 (2006), no. 3, 215-222. · Zbl 1096.53029
[11] I. M. Benn and P. Charlton, Dirac symmetry operators from conformal Killing-Yano tensors, Classical Quantum Gravity 14 (1997), no. 5, 1037-1042. · Zbl 0879.58079
[12] T. Branson, Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), no. 2, 334-383. · Zbl 0904.58054
[13] E. Calabi, “Extremal Kähler metrics” in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, NJ, 1982, 259-290. · Zbl 0471.00020
[14] S. G. Chiossi and P.-A. Nagy, Complex homothetic foliations on Kähler manifolds, Bull. Lond. Math. Soc. 44 (2012), no. 1, 113-124. · Zbl 1241.53021
[15] T. Collins and G. Székelyhidi, K-semistability for irregular Sasakian manifolds, J. Differential Geom. 109 (2018), no. 1, 81-109. · Zbl 1403.53039
[16] A. Derdzinski and G. Maschler, Local classification of conformally-Einstein Kähler metrics in higher dimensions, Proc. London Math. Soc. (3) 87 (2003), no. 3, 779-819. · Zbl 1049.53017
[17] N. Ginoux and U. Semmelmann, Imaginary Kählerian Killing spinors, I, Ann. Global Anal. Geom. 40 (2011), no. 4, 467-495. · Zbl 1239.53065
[18] A. Hwang and M. Singer, A momentum construction for circle-invariant Kähler metrics, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2285-2325. · Zbl 0987.53032
[19] K.-D. Kirchberg, Killing spinors on Kähler manifolds, Ann. Global Anal. Geom. 11 (1993), no. 2, 141-164. · Zbl 0810.53033
[20] W. Kühnel, “Conformal transformations between Einstein spaces” in Conformal Geometry (Bonn, 1985/1986), Aspects Math. E12, Friedr. Vieweg, Braunschweig, 1988, 105-146. · Zbl 0667.53039
[21] D. Martelli, J. Sparks, and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Math. Phys. 280 (2008), no. 3, 611-673. · Zbl 1161.53029
[22] V. Matveev and S. Rosemann, Conification construction for Kähler manifolds and its application in c-projective geometry, Adv. Math. 274 (2015), 1-38. · Zbl 1370.53052
[23] A. Moroianu, Conformally related Riemannian metrics with non-generic holonomy, J. Reine Angew. Math. 755 (2019), 279-292. · Zbl 1429.53071
[24] A. Moroianu and U. Semmelmann, Twistor forms on Kähler manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 823-845. · Zbl 1121.53050
[25] A. Moroianu and U. Semmelmann, Killing forms on quaternion-Kähler manifolds, Ann. Global Anal. Geom. 28 (2005), no. 4, 319-335. · Zbl 1092.53037
[26] A. Moroianu and P. Gauduchon, “Killing 2-forms in dimension 4” in Special metrics and Group Actions in Geometry, Springer INdAM Ser. 23, Springer, Cham, 2017, 161-205. · Zbl 1414.53026
[27] P.-A. Nagy, “Connections with totally skew-symmetric torsion and nearly-Kähler geometry” in Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theor. Phys. 16, Eur. Math. Soc., Zürich, 2010, 347-398. · Zbl 1235.53052
[28] P.-A. Nagy and L. Ornea, Conformal foliations, Kähler twists and the Weinstein construction, preprint, arXiv:1909.11499 [math.DG].
[29] M. Pontecorvo, On twistor spaces of anti-self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 331 (1992), no. 2, 653-661. · Zbl 0754.53053
[30] U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Math. Z. 245 (2003), no. 3, 503-527. · Zbl 1061.53033
[31] U. Semmelmann, Killing forms on \[{\text{G}_2} - and {\text{Spin}_7} -manifolds \], J. Geom. Phys. 56 (2006), no. 9, 1752-1766. · Zbl 1105.53040
[32] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275. · Zbl 0136.17701
[33] C. Tønnesen-Friedman, Extremal Kähler metrics and Hamiltonian functions, II., Glasg. Math. J. 44 (2002), no. 2, 241-253. · Zbl 1031.58006
[34] I. Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl. (4) 132 (1982), 1-18. · Zbl 0512.53058
[35] M. Walker and R. Penrose, On quadratic first integrals of the geodesic equations for type \[\{22\} spacetimes \], Comm. Math. Phys. 18 (1970), 265-274. · Zbl 0197.26404
[36] K. Yano, Some remarks on tensor fields and curvature, Ann. Math. (2) 55 (1952), 328-347 · Zbl 0046.40002
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.