zbMATH — the first resource for mathematics

Stability of vector bundles and extremal metrics. (English) Zbl 0645.53037
The problem of finding Calabi extremal metrics on a compact Kähler manifold M depends on the existence of holomorphic vector fields on M and on the structure of its algebra. In the present paper negative examples are constructed. The authors take a complex surface \(S_ 0=C\times {\mathbb{P}}^ 1,\) where C is a compact Riemann surface of genus \(g\geq 2\), and the Kähler metric \(g_ 0\) which is the product of the metric of constant curvature -1 on C and that of constant curvature \(+1\) on \({\mathbb{P}}^ 1.\) (This metric has scalar curvature \(R\equiv 0).\)
Writing \(S_ 0\) in terms of vector bundles over C, namely \(S_ 0={\mathbb{P}}(E_ 0)\), \(E_ 0=C\times {\mathbb{C}}^ 2,\) the authors deform \(E_ 0\) appropriately in order to construct new ruled surfaces S over C such that 1) S does not admit an extremal Kähler metric g whose Kähler class \(=[\omega_ 0]\) in \(H^ 2(S,{\mathbb{R}})= H^ 2(S_ 0,{\mathbb{R}})\) (here \(\omega_ 0\) denotes the Kähler form of \(g_ 0\) on \(S_ 0)\). 2) there are no non-trivial holomorphic vector fields on S. The found obstruction involves the borderline semi-stability properties of Hermitian vector bundles with Hermite-Einstein connections.
Reviewer: S.Dimiev

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q99 Complex manifolds
Full Text: DOI EuDML
[1] Bourguignon. J.-P. et al.: Première classe de Chern et courbure de Ricci: preuve de la conjecture de Calabi. Astérisque58 (1978)
[2] Burns, D., Bartolomeis, P. de: Stable harmonic maps to IP n . (To appear)
[3] Calabi, E.: Extremal Kähler metrics. In: Yau, S.T. (ed) Seminar on Differential Geometry (Ann. Math. Stud.102, pp.259-290). Princeton University Press 1982 · Zbl 0487.53057
[4] Calabi, E.: Extremal Kähler metrics, II. In: Chavel, I., Farkas, H.M. (eds.) Differential Geomerty and Complex Analysis. Berlin Heidelberg New York: Springer 1985, pp. 95-114
[5] Derdzinski, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compos. Math.49, 405-433 (1983) · Zbl 0527.53030
[6] Futaki, A.: An obstruction to the existence of Einstein-Kähler metrics. Invent. Math.73, 437-443 (1983) · Zbl 0514.53054 · doi:10.1007/BF01388438
[7] Kobayashi, S.: Curvature and stability of vector bundles. Proc. Jpn. Acad. Ser. A, Math. Sci.58, 158-162 (1982) · Zbl 0546.53041 · doi:10.3792/pjaa.58.158
[8] Levine, M.: A remark on extremal metrics. J. Differ. Geom.21, 73-77 (1985) · Zbl 0601.53056
[9] Lübke, M.: Stability of Einstein-Hermitian vector bundles. Manuscr. Math.42, 245-257 (1983) · Zbl 0558.53037 · doi:10.1007/BF01169586
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.