Hamiltonian 2-forms in Kähler geometry, III Extremal metrics and stability

This paper concerns the existence and explicit construction of extremal Kahler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of Hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory. We obtain a characterization, on a large family of projective bundles, of the 'admissible' Kahler classes (i.e., those compatible with the bundle structure in a way we make precise) which contain an extremal K hler metric. In many cases every Kahler class is admissible. In particular, our results complete the classification of extremal Kahler metrics on geometrically ruled surfaces, answering several long-standing questions. We also find that our characterization agrees with a notion of K-stability for admissible Kahler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.