Journal of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6834(e) ISSN 0894-0347(p)

Twistors, Kähler manifolds, and bimeromorphic geometry. II

Author(s): Claude LeBrun; Yat Sun Poon
Journal: J. Amer. Math. Soc. 5 (1992), 317-325.
MSC: Primary 32J27; Secondary 32G05, 32J17, 32L25, 53C55
Part I: J. Amer. Math. Soc. 5 (1992), 289-316
MathSciNet review: 1137099
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Using examples [13] of compact complex -manifolds that arise as twistor spaces, we show that the class of compact complex manifolds bimeromorphic to Kähler manifolds is not stable under small deformations of complex structure.

References:

[1]: M. Atiyah, N. Hitchin, and I. Singer, Self-duality in four dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461. MR 506229 (80d:53023)
[2]: F. Campana, On twistor spaces of the class $\EuScript{C}$ , J. Differential Geometry 33 (1991), 541-549. MR 1094468 (92g:32059)
[3]: -, The Class $\EuScript{C}$ is not stable under deformations, preprint.
[4]: S. K. Donaldson and R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity 2 (1989), 197-239. MR 994091 (90e:32027)
[5]: A. Floer, Self-dual conformal structures on $\ell \mathbb{C}{\mathbb{P}^2}$ , J. Differential Geom. 33 (1991), 551-573. MR 1094469 (92e:53049)
[6]: A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), 225-258. MR 0481142 (58:1285)
[7]: -On a compact complex manifold in $\EuScript{C}$ without holomorphic -forms, Publ. Res. Inst. Math. Sci. 19 (1983).
[8]: H. Grauert and R. Remmert, Coherent analytic sheaves, Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 755331 (86a:32001)
[9]: N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), 133-150. MR 623721 (84b:32014)
[10]: K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. 75 (1962), 146-162. MR 0133841 (24:A3665b)
[11]: K. Kodaira and D. Spencer, On deformations of complex analytic structure, I, II, III, Ann. of Math. 67 (1958), 328-466; 71 (1960), 43-76. MR 0112154 (22:3009)
[12]: C. LeBrun, Explicit self-dual metrics on $\mathbb{C}{\mathbb{P}_2}\char93 \cdot\cdot\cdot\char93 \mathbb{C}{\mathbb{P}_2}$ , J. Differential Geom. 34 (1991), 223-253. MR 1114461 (92g:53040)
[13]: -, Twistors, Kähler manifolds, and bimeromorphic geometry. I, J. Amer. Math. Soc. 5 (1992), 289-316 (this issue). MR 1137098 (92m:32052)
[14]: B. Moishezon, On -dimensional compact varieties with algebraically independent meromorphic functions, Amer. Math. Soc. Transl., vol. 63, Amer. Math. Soc., Providence, RI, 1967, pp. 51-177.
[15]: J. Morrow and K. Kodaira, Complex manifolds, Holt, New York, 1971. MR 0302937 (46:2080)
[16]: R. Penrose, Nonlinear gravitons and curved twistor theory, Gen. Rel. Grav. 7 (1976), 31-52. MR 0439004 (55:11905)
[17]: Y. S. Poon, On the algebraic dimension of twistor spaces, Math. Ann. 282 (1988), 621-627. MR 970223 (90f:32029)
[18]: -, Algebraic structure of twistor spaces, J. Differential Geom. (to appear). MR 1180390 (94a:32045)
[19]: K. Ueno, Open problems, Classification of Algebraic and Analytic Manifolds, Birkhäuser, Basel and Boston, MA, 1983. MR 728620 (85b:14001)
[20]: R. Ward, On self-dual gauge-fields, Phys. Lett. A 61 (1977), 81-82. MR 0443823 (56:2186)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|