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

• 成果类型: Article
• 署名作者: Apostolov, Vestislav; Calderbank, DavidM. J.; Gauduchon, Paul; Tonnesen-Friedman, Christina W.
• 署名单位: University of Quebec; University of Quebec Montreal; Bulgarian Academy of Sciences; University of Bath; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Union College
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0126-x
• 发表日期: 2008
• 页码: 547-601
• 关键词: constant scalar curvature; projective embeddings; einstein metrics; compact; EXISTENCE; MANIFOLDS; bundles; uniqueness; criterion; SPACE

摘要

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.