Non-Moishezon twistor spaces of $4{\mathbf {CP}}^2$ with non-trivial automorphism group
HTML articles powered by AMS MathViewer
- by Nobuhiro Honda PDF
- Trans. Amer. Math. Soc. 358 (2006), 1897-1920 Request permission
Abstract:
We show that a twistor space of a self-dual metric on $4{\mathbf {CP}}^2$ with $U(1)$-isometry is not Moishezon iff there is a $\mathbf {C}^*$-orbit biholomorphic to a smooth elliptic curve, where the $\mathbf C^*$-action is the complexification of the $U(1)$-action on the twistor space. It follows that the $U(1)$-isometry has a two-sphere whose isotropy group is $\mathbf Z_2$. We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.References
- M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 131423, DOI 10.1112/plms/s3-7.1.414
- M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
- F. Campana, The class ${\scr C}$ is not stable by small deformations, Math. Ann. 290 (1991), no. 1, 19–30. MR 1107661, DOI 10.1007/BF01459236
- F. Campana and B. Kreußler, A conic bundle description of Moishezon twistor spaces without effective divisors of degree one, Math. Z. 229 (1998), no. 1, 137–162. MR 1649326, DOI 10.1007/PL00004646
- F. Campana and B. Kreussler, Existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2633–2642. MR 1676299, DOI 10.1090/S0002-9939-99-05406-4
- S. Donaldson and R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity 2 (1989), no. 2, 197–239. MR 994091
- A. Douady, Déformations Régulières, Séminaire Henri Cartan 13e année, 1960/61, n$^\circ$ 3.
- Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR 755331, DOI 10.1007/978-3-642-69582-7
- Ronald Fintushel, Circle actions on simply connected $4$-manifolds, Trans. Amer. Math. Soc. 230 (1977), 147–171. MR 458456, DOI 10.1090/S0002-9947-1977-0458456-6
- N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), no. 1, 133–150. MR 623721, DOI 10.1112/plms/s3-43.1.133
- Nobuhiro Honda, Donaldson-Friedman construction and deformations of a triple of compact complex spaces, Osaka J. Math. 36 (1999), no. 3, 641–672. MR 1740826
- Nobuhiro Honda, On some twistor spaces over $4\textbf {C}\textrm {P}^2$, Compositio Math. 122 (2000), no. 3, 323–336. MR 1781334, DOI 10.1023/A:1002077103075
- Nobuhiro Honda, Equivariant deformations of meromorphic actions on compact complex manifolds, Math. Ann. 319 (2001), no. 3, 469–481. MR 1819878, DOI 10.1007/PL00004443
- Nobuhiro Honda, Donaldson-Friedman construction and deformations of a triple of compact complex spaces. II, Math. Nachr. 256 (2003), 48–57. MR 1989377, DOI 10.1002/mana.200310069
- Nobuhiro Honda and Mitsuhiro Itoh, A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes, J. Math. Soc. Japan 52 (2000), no. 1, 139–160. MR 1727196, DOI 10.2969/jmsj/05210139
- B. Kreussler, Moishezon twistor spaces without effective divisors of degree one, J. Algebraic Geom. 6 (1997), no. 2, 379–390. MR 1489120
- B. Kreußler, On the algebraic dimension for twistor spaces over the connected sum of four complex projective planes, Geom. Dedicata 71 (1998), no. 3, 263–285. MR 1631683, DOI 10.1023/A:1005038726026
- B. Kreussler, Twistor spaces with a pencil of fundamental divisors, Doc. Math. 4 (1999), 127–166. MR 1683286
- Bernd Kreussler and Herbert Kurke, Twistor spaces over the connected sum of 3 projective planes, Compositio Math. 82 (1992), no. 1, 25–55. MR 1154160
- Claude LeBrun, Explicit self-dual metrics on $\textbf {C}\textrm {P}_2\#\cdots \#\textbf {C}\textrm {P}_2$, J. Differential Geom. 34 (1991), no. 1, 223–253. MR 1114461
- Claude LeBrun, Self-dual manifolds and hyperbolic geometry, Einstein metrics and Yang-Mills connections (Sanda, 1990) Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 99–131. MR 1215284
- Claude LeBrun and Michael Singer, A Kummer-type construction of self-dual $4$-manifolds, Math. Ann. 300 (1994), no. 1, 165–180. MR 1289837, DOI 10.1007/BF01450482
- Henrik Pedersen and Yat Sun Poon, Self-duality and differentiable structures on the connected sum of complex projective planes, Proc. Amer. Math. Soc. 121 (1994), no. 3, 859–864. MR 1195729, DOI 10.1090/S0002-9939-1994-1195729-1
- Henrik Pedersen and Yat Sun Poon, Equivariant connected sums of compact self-dual manifolds, Math. Ann. 301 (1995), no. 4, 717–749. MR 1326765, DOI 10.1007/BF01446656
- Massimiliano Pontecorvo, Hermitian surfaces and a twistor space of algebraic dimension $2$, Proc. Amer. Math. Soc. 113 (1991), no. 1, 177–186. MR 1074754, DOI 10.1090/S0002-9939-1991-1074754-2
- Y. Sun Poon, Compact self-dual manifolds with positive scalar curvature, J. Differential Geom. 24 (1986), no. 1, 97–132. MR 857378
- Y. Sun Poon, On the algebraic structure of twistor spaces, J. Differential Geom. 36 (1992), no. 2, 451–491. MR 1180390
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
Additional Information
- Nobuhiro Honda
- Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi Hiroshima, 739-8526, Japan
- Address at time of publication: Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, 152-8551, Japan
- Email: honda@math.titech.ac.jp
- Received by editor(s): January 22, 2003
- Published electronically: December 20, 2005
- Additional Notes: This work was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 1897-1920
- MSC (2000): Primary 32L25, 32G05, 32G07, 53A30, 53C25
- DOI: https://doi.org/10.1090/S0002-9947-05-04141-3
- MathSciNet review: 2197434