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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

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32Q99 Complex manifolds
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##### References:
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