Twistors, Kähler manifolds, and bimeromorphic geometry. II

LeBrun, Claude; Poon, Yat Sun

doi:10.1090/S0894-0347-1992-1137099-7

Twistors, Kähler manifolds, and bimeromorphic geometry. II
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by Claude LeBrun and Yat Sun Poon

J. Amer. Math. Soc. 5 (1992), 317-325

DOI: https://doi.org/10.1090/S0894-0347-1992-1137099-7

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Part I: J. Amer. Math. Soc. (1992), 289-316

Abstract:

Using examples [13] of compact complex $3$-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

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, On twistor spaces of the class $\scr C$, J. Differential Geom. 33 (1991), no. 2, 541–549. MR 1094468, DOI 10.4310/jdg/1214446329

The Class $\mathcal {C}$ is not stable under deformations

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, DOI 10.1088/0951-7715/2/2/002
Andreas Floer, Self-dual conformal structures on $l\textbf {C}\textrm {P}^2$, J. Differential Geom. 33 (1991), no. 2, 551–573. MR 1094469
Akira Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. MR 481142, DOI 10.1007/BF01403162

On a compact complex manifold in $\mathcal {C}$ without holomorphic

-forms

19

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
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
K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2) 75 (1962), 146–162. MR 133841, DOI 10.2307/1970424
K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328–466. MR 112154, DOI 10.2307/1970009
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, Twistors, Kähler manifolds, and bimeromorphic geometry. I, J. Amer. Math. Soc. 5 (1992), no. 2, 289–316. MR 1137098, DOI 10.1090/S0894-0347-1992-1137098-5

On

-dimensional compact varieties with

algebraically independent meromorphic functions

James Morrow and Kunihiko Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR 0302937
Roger Penrose, Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), no. 1, 31–52. MR 439004, DOI 10.1007/bf00762011
Y. Sun Poon, Algebraic dimension of twistor spaces, Math. Ann. 282 (1988), no. 4, 621–627. MR 970223, DOI 10.1007/BF01462887
Y. Sun Poon, On the algebraic structure of twistor spaces, J. Differential Geom. 36 (1992), no. 2, 451–491. MR 1180390
Open problems, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 591–630. MR 728620
R. S. Ward, On self-dual gauge fields, Phys. Lett. A 61 (1977), no. 2, 81–82. MR 443823, DOI 10.1016/0375-9601(77)90842-8