@article{2b96800301b84f54825f9989a6fa0aed, title = "Hamiltonian 2-forms in Kahler geometry, II Global Classification", abstract = "We present a classification of compact K{\"a}hler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a K{\"a}hler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact K{\"a}hler manifolds with a rigid hamiltonian torus action are bundles of toric K{\"a}hler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric K{\"a}hler manifolds are diffeomorphic to complex projective space, but we extend our analysis to orthotoric orbifolds, where the geometry is much richer. We thus obtain new examples of K{\"a}hlerâ€“Einstein 4-orbifolds. Combining these two themes, we prove that compact K{\"a}hler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over K{\"a}hler products, and we describe explicitly how the K{\"a}hler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal K{\"a}hler metricsâ€”in particular a subclass of such metrics which we call weakly Bochner-flat. We also provide a self-contained treatment of the theory of compact toric K{\"a}hler manifolds, since we need it and find the existing literature incomplete.", author = "V. Apostolov and P. Gauduchon and C.W. T{\o}nnesen-Friedman and D.M.J. Calderbank", year = "2004", month = sep, language = "English", volume = "68", pages = "277--345", journal = "Journal of Differential Geometry", issn = "0022-040X", publisher = "International Press of Boston, Inc.", number = "1", }