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Existence of twistor spaces
of algebraic dimension two
over the connected sum
of four complex projective planes


Authors: F. Campana and B. Kreußler
Journal: Proc. Amer. Math. Soc. 127 (1999), 2633-2642
MSC (1991): Primary 32L25, 32J17, 32J20
DOI: https://doi.org/10.1090/S0002-9939-99-05406-4
Published electronically: May 19, 1999
MathSciNet review: 1676299
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes $4 \mathbb{C}\mathbb{P}^2$. These are the first examples of twistor spaces of algebraic dimension two over a simply connected Riemannian four-manifold with positive scalar curvature. For this purpose we develop a method to distinguish between twistor spaces of algebraic dimension one and two by looking at the order of a certain point in the Picard group of an elliptic curve.


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  • [AHS] M.F. Atiyah, N.J. Hitchin, I.M. Singer: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978) 425-461. MR 80d:53023
  • [C1] F. Campana: The class $\mathcal{C}$ is not stable by small deformations, Math. Ann. 290 (1991) 19-30. MR 93d:32047
  • [C2] F. Campana: On Twistor Spaces of the Class $\mathcal{C}$, J. Diff. Geom. 33 (1991) 541-549. MR 92g:32059
  • [C3] F. Campana: The class $\mathcal{C}$ is not stable by small deformations II, Contemp. Math. 162 (1994) 65-76. MR 95i:32026
  • [Don] S. Donaldson: An application of gauge theory to the topology of 4-manifolds, J. Diff. Geom. 18 (1983) 279-315. MR 85c:57015
  • [DonF] S. Donaldson, R. Friedman: Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity 2 (1989) 197-239. MR 90e:32027
  • [ES] M.G. Eastwood, M.A. Singer: The Fröhlicher Spectral Sequence on a Twistor Space, J. Diff. Geom. 38 (1993) 653-669. MR 94k:32050
  • [F] M. Freedman: The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982) 357-454. MR 84b:57006
  • [FK] T. Friedrich, H. Kurke: Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. Nach. 106 (1982) 271-299. MR 84b:53043
  • [G] P. Gauduchon: Structures de Weyl et théorèmes d'annulation sur une variété conforme autoduale, Ann. Scuola Norm. Sup. Pisa 18 (1991) 563-629. MR 93d:32046
  • [H] J. Harris: Algebraic Geometry, Graduate Texts in Mathematics 133, Springer-Verlag, 1992. MR 93j:14001
  • [H1] N.J. Hitchin: Linear field equations on self-dual spaces, Proc. Roy. Soc. London Ser. A 370 (1980) 173-191. MR 81i:81057
  • [H2] N.J. Hitchin: Kählerian twistor spaces, Proc. Lond. Math. Soc. , III Ser. 43 (1981) 133-150. MR 84b:32014
  • [Kr] B. Kreußler: Small resolutions of double solids, branched over a 13-nodal quartic surface, Annals of Global Analysis and Geometry 7 (1989) 227-267. MR 91b:14011
  • [KK] B. Kreußler, H. Kurke: Twistor spaces over the connected sum of 3 projective planes, Compositio Math. 82 (1992) 25-55. MR 93d:32049
  • [Ku] H. Kurke: Classification of twistor spaces with a pencil of surfaces of degree 1, Part I, Math. Nachr. 158 (1992) 67-85. MR 94k:32051
  • [LeB1] C. LeBrun: Explicit self-dual metrics on $\mathbb{C}\mathbb{P}^2\#\dots \#\mathbb{C}\mathbb{P}^2$, J. Diff. Geom. 34 (1991) 223-253. MR 92g:53040
  • [LeB2] C. LeBrun: Twistors, Kähler manifolds, and bimeromorphic geometry. I, J. AMS 5 (1992) 289-316. MR 92m:32052
  • [LeBP] C. LeBrun, Y.S. Poon: Twistors, Kähler manifolds, and bimeromorphic geometry. II, J. AMS 5 (1992) 317-325. MR 92m:32053
  • [PP1] H. Pedersen, Y.S. Poon: Self-duality and differentiable structures on the connected sum of complex projective planes, Proc. Amer. Math. Soc. 121 (1994) 859-864. MR 94i:32049
  • [PP2] H. Pedersen, Y.S. Poon: A relative deformation of Moishezon twistor spaces, J. Alg. Geom. 3 (1994) 685-701. MR 95j:32045
  • [PP3] H. Pedersen, Y.S. Poon: Equivariant connected sums of compact self-dual manifolds, Math. Ann. 301 (1995) 717-749. MR 95m:53069
  • [Pe] R. Penrose: Nonlinear gravitons and curved twistor theory, General Relativity and Gravitation 7 (1976) 31-52. MR 55:11905
  • [Pon1] M. Pontecorvo: Hermitian Surfaces and a Twistor Space of Algebraic Dimension 2, Proc. Amer. Math. Soc. 113 (1991) 177-186. MR 91k:32028
  • [Pon2] M. Pontecorvo: Algebraic dimension of twistor spaces and scalar curvature of anti-self-dual metrics, Math. Ann. 291 (1991) 113-122. MR 92j:32113
  • [Po1] Y.S. Poon: Compact self-dual manifolds with positive scalar curvature, J. Diff. Geom. 24 (1986) 97-132. MR 88b:32022
  • [Po2] Y.S. Poon: Algebraic dimension of twistor spaces, Math. Ann. 282 (1988) 621-627. MR 90f:32029
  • [Po3] Y.S. Poon: On the algebraic structure of twistor spaces, J. Diff. Geom. 36 (1992) 451-491. MR 94a:32045
  • [Sch] R. Schoen: Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984) 479-495. MR 86i:58137
  • [V] M. Ville: Twistor examples of algebraic dimension zero treefolds, Invent. math. 103 (1991) 537-545. MR 92c:32034

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Additional Information

F. Campana
Affiliation: Department of Mathematics, Université de Nancy, F–54506 Vandœuvre les Nancy, France
Email: Frederic.Campana@iecn.u-nancy.fr

B. Kreußler
Affiliation: Department of Mathematics, Universität Kaiserslautern, D–67653 Kaiserslautern, Germany
Email: kreusler@mathematik.uni-kl.de

DOI: https://doi.org/10.1090/S0002-9939-99-05406-4
Received by editor(s): June 20, 1996
Published electronically: May 19, 1999
Communicated by: Ron Donagi
Article copyright: © Copyright 1999 American Mathematical Society

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