zbMATH — the first resource for mathematics

Algebraic dimension of twistor spaces. (English) Zbl 0665.32014
Associated to any self-dual Riemannian four manifold M is a twistor space. This is the set of all complex structures (compatible with the metric) on all tangent spaces to the manifold. It fibres over the manifold with fibres isomorphic to one dimensional complex projective space and is itself a complex manifold.
The author proves: Theorem. If the twistor space of a compact self-dual manifold is Moishezon, the self-dual conformal class contains a metric with positive scalar curvature.
Reviewer: M.K.Murray

32J99 Compact analytic spaces
32L25 Twistor theory, double fibrations (complex-analytic aspects)
Full Text: DOI EuDML
[1] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. R. Soc. Lond., Ser.A362, 425-461 (1978) · Zbl 0389.53011
[2] Besse, A.: Einstein manifolds. Berlin Heidelberg New York: Springer 1987 · Zbl 0613.53001
[3] Donaldson, S.K., Friedman, R.: Private communication
[4] Godement, R.: Topologie algébrique et théorie des Faisceaux. Paris: Hermann 1958 · Zbl 0080.16201
[5] Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen. Publ. Math., Inst. Hautes Etud. Sci.5, 223-292 (1960) · Zbl 0100.08001
[6] Hartshorne, R.: Algebraic geometry. Graduate Texts in Math., Vol 52). Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[7] Hitchin, N.J.: Linear field equations on self-dual spaces. Proc. R. Soc. Lond., Ser.A370, 173-191 (1980) · Zbl 0436.53058
[8] Hitchin, N.J.: Kählerian twistor spaces. Proc. Lond. Math. Soc. (3)43, 133-150 (1981) · Zbl 0474.14024
[9] Moishézon, B.G.: Onn-dimensional compact varieties withn algebraically independent meromorphic functions. Am. Math. Soc. Transl.63, 51-177 (1967)
[10] Poon, Y.S.: Compact self-dual manifolds with positive scalar curvature. J. Differ. Geom.24, 97-132 (1986) · Zbl 0583.53054
[11] Poon, Y.S.: Small resolution of double solids as twistor spaces. J. Differ. Geom. (in press)
[12] Schoen, R.: Conformal deformation of a Riemannian metric to construct scalar curvature. J. Differ. Geom.20, 479-495 (1984) · Zbl 0576.53028
[13] Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. (Lecture Notes Math., Vol. 439). Berlin Heidelberg New York: Springer 1975 · Zbl 0299.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.