Stability of vector bundles and extremal metrics (Q1103204)

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.