Selfdual Einstein hermitian four-manifolds

Apostolov, Vestislav; Gauduchon, Paul

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 203-243.

We provide a local classification of selfdual Einstein riemannian four-manifolds admitting a positively oriented hermitian structure and characterize those which carry a hyperhermitian, non-hyperkähler structure compatible with the negative orientation. We show that selfdual Einstein 4-manifolds obtained as quaternionic quotients of $ℍP^2$ and $ℍH^2$ are hermitian.

Classification : 53B35, 53C55

[1] V. Apostolov - J. Davidov - O. Muškarov, Compact Self-dual Hermitian Surfaces, Trans. Amer. Math. Soc. 348 (1996), 3051-3063. | MR | Zbl

[2] V. Apostolov - P. Gauduchon, The Riemannian Goldberg-Sachs Theorem, Int. J. Math. 8 (1997), 421-439. | MR | Zbl

[3] J. Armstrong, “Almost Kähler Geometry”, Ph.D. Thesis, Oxford, 1998.

[4] M. F. Atiyah - N. J. Hitchin - I. M. Singer, Self-duality in four dimensional geometry, Proc. Roy. Soc. London, A 362 (1979), 425-461. | MR | Zbl

[5] L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Publications de l'Institut É. Cartan (Nancy) 4 (1982), 1-60. | Zbl

[6] A. L. Besse, “Einstein manifolds”, Ergeb. Math. Grenzgeb.3, Folge 10, Springer-Verlag, Berlin, Heidelberg, New York, 1987. | MR | Zbl

[7] J.-P. Bourguignon, Les variétés de dimension $4$ à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent. Math. 63 (1981), 263-286. | MR | Zbl

[8] C. Boyer - J. Finley, Killing vectors in self-dual Euclidean Einstein spaces, J. Math. Phys. (1982), 1126-1130. | MR | Zbl

[9] C. Boyer, Conformal duality and compact complex surfaces, Math. Ann. 274 (1986), 517-526. | MR | Zbl

[10] C. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), 157-164. | MR | Zbl

[11] C. Boyer, Self-dual and anti-self-dual Hermitian metrics on compact complex surfaces, In “Mathematics and General Relativity”, Proceedings, Santa Cruz 1986, J. Isenberg (ed.), Contemp. Math. 71 (1988), 105-114. | MR | Zbl

[12] R. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001) 623-715. | MR | Zbl

[13] D. M. J. Calderbank, The Faraday 2-form in Einstein-Weyl geometry, Math. Scand. 89 (2001) 97-116. | MR | Zbl

[14] D. M. J. Calderbank, The geometry of the Toda equation, J. Geom. Phys. 36 (2000) 152-162. | MR | Zbl

[15] D. M. J. Calderbank, “Selfdual Einstein metrics and conformal submersions”, Edinburgh Preprint MS-00-001 (2000), available at arXiv: math.DG/0001041.

[16] D. M. J. Calderbank - K. P. Tod, Einstein metrics, hyper-complex structures and the Toda field equation, Differential Geom. Appl. 14 (2001) 199-208. | MR | Zbl

[17] D. M. J. Calderbank - H. Pedersen, Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, Ann. Inst. Fourier 50 (2000), 921-963. | Numdam | MR | Zbl

[18] D. M. J. Calderbank - H. Pedersen, “Selfdual Einstein metrics with torus symmetry”, Edinburgh Preprint MS-00-022 (2000), available at arXiv:math.DG/0105263. | MR | Zbl

[19] B.-Y. Chen, Some topological obstructions to Bochner-Kähler metrics and their applications, J. Differential Geom. 13 (1978), 574-588. | MR | Zbl

[20] A. Dancer - I. Strachan, Kähler-Einstein metrics with SU(2) action, Math. Proc. Cambridge Philos. Soc. 115 (1994), 513-525. | MR | Zbl

[21] A. Derdziński, Exemples de métriques de Kähler et d'Einstein autoduales sur le plan complexe, In: “Géometrie riemannienne en dimension 4”, Séminaire Arthur Besse, L. Bérard-Bergery, M. Berger C. Houzel (eds.), CEDIC/Fernand Nathan, 1981. | Zbl

[22] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405-433. | Numdam | MR | Zbl

[23] M. G. Eastwood - K. P. Tod, Local constraints on Einstein-Weyl geometries, J. reine angew. Math. 491 (1997), 183-198. | MR | Zbl

[24] K. Galicki, A generalization of the momentum mapping construction for quaternionic-Kähler manifolds, Comm. Math. Phys. 108 (1987), 108, 117-138. | MR | Zbl

[25] K. Galicki, New metrics with $S{p}_{n}S{p}_1$ holonomy, Nucl. Phys. B 289 (1987), 573.

[26] K. Galicki - H. B. Lawson, Quaternionic Reduction and Quaternionic Orbifolds, Math. Ann. 282 (1988), 1-21. | MR | Zbl

[27] P. Gauduchon, La $1$-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), 495-518. | MR | Zbl

[28] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $S^1\times S^3$, J. reine angew. Math. 469 (1995), 1-50. | MR | Zbl

[29] P. Gauduchon, Complex structures on compact conformal manifolds of negative type, In: “Complex Analysis and Geometry”, V. Ancona, E. Ballico and A. Silva (eds.), Marcel Dekker, New York-Basel-Hong Kong, 1996, 201-212. | MR | Zbl

[30] P. Gauduchon, Connexion canonique et structures de Weyl en géométrie conforme, Preprint (unpublished).

[31] P. Gauduchon - K. P. Tod, Hyper-Hermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291-304. | MR | Zbl

[32] G. Gibbons - S. Hawking, Classification of Gravitational Instanton Symmetries, Comm. Math. Phys. 66 (1979), 291-310. | MR

[33] M. Itoh, Self-duality of Kähler surfaces, Compositio Math. 51 (1984), 265-273. | Numdam | MR | Zbl

[34] P. E. Jones - K. P. Tod, Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav. 2 (1985), 565-577. | MR | Zbl

[35] D. Joyce, Explicit construction of self-dual $4$-manifolds, Duke Math. J. 77 (1995), 519-552. | MR | Zbl

[36] C. Lebrun, Counter-example to the generalized positive action conjecture, Comm. Math. Phys. 118 (1988), 591-596. | MR | Zbl

[37] C. LeBrun, private communication.

[38] A. B. Madsen, Einstein-Weyl structures in the conformal classes of LeBrun metrics, Class. Quantum Grav. 14 (1997), 2635-2645. | MR | Zbl

[39] P. Nurowski, “Einstein equations and Cauchy-Riemann geometry”, Ph.D. Thesis, SISSA/ ISAS, Trieste, 1993.

[40] H. Pedersen, Einstein metrics, spinning top motions and monopoles, Math. Ann. 274 (1986), 35-59. | MR | Zbl

[41] H. Pedersen - A. Swann, Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. reine angew. Math. 441 (1993), 99-113. | MR | Zbl

[42] J. Plebañski - M. Przanowski, Hermite-Einstein four-dimensional manifolds with symmetry, Class. Quantum Grav. 15 (1998), 1721-1735. | MR | Zbl

[43] M. Przanowski - B. Broda, Locally Kähler gravitational instantons, Acta Phys. Pol. B 14 (1983), 637-661. | MR

[44] S. Salamon, Special structures on four-manifolds, Riv. Mat. Univ. Parma 17 (4) (1991), 109-123. | MR | Zbl

[45] S. Salamon, “Riemannian geometry and holonomy groups”, Pitman Research Notes in Mathematics Series, 201, New York, 1989. | MR | Zbl

[46] K. P. Tod, Cohomogeneity-one metrics with self-dual Weyl tensor, In “Twistor Theory”, S. Huggett (ed.), Marcel Dekker, New York (1995), 171-184. | MR | Zbl

[47] K. P. Tod, Scalar-flat Kähler and hyper-Kähler metrics from Painlevé-III, Class. Quantum Grav. 12 (1995), 1535-1547. | MR | Zbl

[48] K. P. Tod, The $SU\left(\infty \right)$-Toda field equation and special four-dimensional metrics, In: “Geometry and Physics”, J. E. Andrsen, J. Dupont, H. Pedersen and A. Swann (eds.), Marcel Dekker, New York (1997), 307-312. | MR | Zbl

[49] F. Tricerri - L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. | MR | Zbl

[50] I. Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc. 262 (1980), 533-542. | MR | Zbl

[51] I. Vaisman, Some curvature prperties of complex surfaces, Ann. Mat. Pura Appl. 32 (1982), 1-18. | MR | Zbl

[52] S. Webster, On the pseudo-conformal geometry of a Kähler manifolds, Math. Z. 157 (1977), 265-270. | MR | Zbl