Twistors, Kähler manifolds, and bimeromorphic geometry. I

LeBrun, Claude

Journal of the American Mathematical Society, Tome 05 (1992) no. 2, pp. 289-316

Résumé

By considering deformations of the Moishezon twistor spaces of $\mathbb {C}{\mathbb {P}_2}\# \cdot \cdot \cdot \# \mathbb {C}{\mathbb {P}_2}$ constructed in [20], we show that the blow up of ${\mathbb {C}^2}$ at $n$ points in general position admits an asymptotically flat scalar-flat Kähler metric in each Kähler class, at least provided that the given points are nearly collinear.

Détail

DOI : 10.1090/S0894-0347-1992-1137098-5

Bibliographie

[1] Atiyah, M. F. Green’s functions for self-dual four-manifolds 1981 129 158

[2] Atiyah, M. F., Hitchin, N. J., Singer, I. M. Self-duality in four-dimensional Riemannian geometry Proc. Roy. Soc. London Ser. A 1978 425 461

[3] Baston, Robert J., Eastwood, Michael G. The Penrose transform 1989

[4] Calabi, Eugenio Extremal Kähler metrics 1982 259 290

[5] Donaldson, S., Friedman, R. Connected sums of self-dual manifolds and deformations of singular spaces Nonlinearity 1989 197 239

[6] Flaherty, Edward J., Jr. The nonlinear graviton in interaction with a photon Gen. Relativity Gravitation 1978 961 978

[7] Floer, Andreas Self-dual conformal structures on 𝑙𝐶𝑃² J. Differential Geom. 1991 551 573

[8] Fujiki, Akira On a compact complex manifold in \cal𝐶 without holomorphic 2-forms Publ. Res. Inst. Math. Sci. 1983 193 202

[9] Grauert, Hans, Remmert, Reinhold Coherent analytic sheaves 1984

[10] Hitchin, N. J. Polygons and gravitons Math. Proc. Cambridge Philos. Soc. 1979 465 476

[11] Hitchin, N. J. Linear field equations on self-dual spaces Proc. Roy. Soc. London Ser. A 1980 173 191

[12] Hitchin, N. J. Kählerian twistor spaces Proc. London Math. Soc. (3) 1981 133 150

[13] Jones, P. E., Tod, K. P. Minitwistor spaces and Einstein-Weyl spaces Classical Quantum Gravity 1985 565 577

[14] Kodaira, K. A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds Ann. of Math. (2) 1962 146 162

[15] Kodaira, K., Nirenberg, L., Spencer, D. C. On the existence of deformations of complex analytic structures Ann. of Math. (2) 1958 450 459

[16] Kodaira, K., Spencer, D. C. On deformations of complex analytic structures. I, II Ann. of Math. (2) 1958 328 466

[17] Lebrun, Claude Poon’s self-dual metrics and Kähler geometry J. Differential Geom. 1988 341 343

[18] Lebrun, Claude Explicit self-dual metrics on 𝐶𝑃₂#⋯#𝐶𝑃₂ J. Differential Geom. 1991 223 253

[19] Lebrun, Claude Scalar-flat Kähler metrics on blown-up ruled surfaces J. Reine Angew. Math. 1991 161 177

[20] Lebrun, Claude, Poon, Yat Sun Twistors, Kähler manifolds, and bimeromorphic geometry. II J. Amer. Math. Soc. 1992 317 325

[21] Pontecorvo, Massimiliano On twistor spaces of anti-self-dual Hermitian surfaces Trans. Amer. Math. Soc. 1992 653 661

[22] Poon, Y. Sun Compact self-dual manifolds with positive scalar curvature J. Differential Geom. 1986 97 132

[23] Ward, R. S. On self-dual gauge fields Phys. Lett. A 1977 81 82

[24] Yau, Shing Tung Problem section 1982 669 706