×

zbMATH — the first resource for mathematics

Geometry of some twistor spaces of algebraic dimension one. (English) Zbl 1326.32034
Summary: It is shown that there exists a twistor space on the \(n\)-fold connected sum of complex projective planes \(n\mathbb{CP}^2\), whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over \(n\mathbb{CP}^2\) for any \(n \geq 5\), while the latter kind of example is constructed over \(5\mathbb{CP}^2\). Both of these seem to be the first such example on \(n\mathbb{CP}^2\). The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.
MSC:
32L25 Twistor theory, double fibrations (complex-analytic aspects)
53C28 Twistor methods in differential geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[2] M. Atiyah, N. Hitchin, I. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London, Ser. A 362 (1978) 425-461. · Zbl 0389.53011
[3] F. Campana, The class C is not stable by small deformations, Math. Ann. 229 (1991) 19-30. · Zbl 0722.32014
[4] I. Enoki, Surfaces of class VII0 with curves, Tohoku Math. J. 33 (1981) 453-492. · Zbl 0476.14013
[5] S. K. Donaldson, R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces Non-linearlity 2 (1989) 197-239. · Zbl 0671.53029
[6] R. Friedman, D. Morrison, The birational geometry of degenerations: An overview, The birational geometry of degenerations (R. Friedman, D. Morrison, eds.) Progress Math. 29 (1983) 1-32. · Zbl 0508.14024
[7] A. Fujiki. On the structure of compact complex manifolds in C, Adv. Stud. Pure Math. 1 (1983) 231-302. · Zbl 0513.32027
[8] A. Fujiki. Compact self-dual manifolds with torus actions, J. Differential Geom. 55 (2000) 229-324. · Zbl 1032.57036
[9] A. Fujiki, Algebraic reduction of twistor spaces of Hopf surfaces, Osaka J. Math. 37 (2000) 847-858. · Zbl 1002.32014
[10] P. Griffiths and J. Harris, “Principle of Algebraic Geometry”, Wiley-Interscience. · Zbl 0408.14001
[11] J. Hausen, Zur Klassifikation of glatter kompakter C*-Flachen, Math. Ann. 301 (1995), 763-769.
[12] N. Hitchin, Linear field equations on self-dual spaces, Proc. Roy. Soc. London Ser. A 370 (1980) 173-191. · Zbl 0436.53058
[13] N. Hitchin, K¨ahlerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981) 133-150. · Zbl 0474.14024
[14] N. Honda and M. Itoh, A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes, J. Math. Soc. Japan 52 (2000) 139-160. · Zbl 0979.53082
[15] N. Honda, Double solid twistor spaces II: general case, J. reine angew. Math. 698 (2015) 181-220. · Zbl 1316.32015
[16] E. Horikawa, Deformations of holomorphic maps III, Math. Ann. 222 (1976) 275-282. · Zbl 0334.32021
[17] M. Inoue, New surfaces with no meromorphic functions, Proc. Int. Cong. Math., Vancouver 1 (1974), 423-426.
[18] D. Joyce, Explicit construction of self-dual 4-manifolds, Duke Math. J. 77 (1995) 519-552. · Zbl 0855.57028
[19] B. Kreußler and H. Kurke, Twistor spaces over the connected sum of 3 projective planes. Compositio Math. 82:25-55, 1992. · Zbl 0766.53049
[20] C. LeBrun, Y. Poon Twistors, K¨ahler manifolds, and bimeromorphic geometry II, J. Amer. Math. Soc. 5 (1992), 317-325.
[21] K. Nishiguchi Degenerations of K3 surfaces, J. Math. Kyoto Univ. 82 (1988) 267-300. · Zbl 0707.14035
[22] P. Orlik, P. Wagreich, Algebraic surfaces with k*-actions, Acta Math. 138 (1977) 43-81. · Zbl 0352.14016
[23] H. Pedersen, Y. S. Poon, Self-duality and differentiable structures on the connected sum of complex projective planes, Proc. Amer. Math. Soc. 121 (1994) 859-864. · Zbl 0808.32028
[24] Y. S. Poon, On the algebraic structure of twistor spaces, J. Differential Geom. 36 (1992), 451-491. · Zbl 0742.53024
[25] K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lect. Note Math. 439 (1975) · Zbl 0299.14007
[26] O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surfaces, Ann. Math. 76 (1962), 560-615. · Zbl 0124.37001
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.