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**Hamiltonian 2-forms in Kahler geometry, II Global Classification.** / Apostolov, V.; Gauduchon, P.; Tønnesen-Friedman, C.W.; Calderbank, D.M.J.

Research output: Contribution to journal › Article › peer-review

Apostolov, V, Gauduchon, P, Tønnesen-Friedman, CW & Calderbank, DMJ 2004, 'Hamiltonian 2-forms in Kahler geometry, II Global Classification', *Journal of Differential Geometry*, vol. 68, no. 1, pp. 277-345. <http://www.intlpress.com/jdg/2004/v68.htm>

Apostolov, V., Gauduchon, P., Tønnesen-Friedman, C. W., & Calderbank, D. M. J. (2004). Hamiltonian 2-forms in Kahler geometry, II Global Classification. *Journal of Differential Geometry*, *68*(1), 277-345. http://www.intlpress.com/jdg/2004/v68.htm

Apostolov V, Gauduchon P, Tønnesen-Friedman CW, Calderbank DMJ. Hamiltonian 2-forms in Kahler geometry, II Global Classification. Journal of Differential Geometry. 2004 Sep;68(1):277-345.

@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}hler–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 metrics—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",

}

TY - JOUR

T1 - Hamiltonian 2-forms in Kahler geometry, II Global Classification

AU - Apostolov, V.

AU - Gauduchon, P.

AU - Tønnesen-Friedman, C.W.

AU - Calderbank, D.M.J.

PY - 2004/9

Y1 - 2004/9

N2 - We present a classification of compact Kähler 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 Kähler 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 Kähler manifolds with a rigid hamiltonian torus action are bundles of toric Kähler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kähler 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 Kähler–Einstein 4-orbifolds. Combining these two themes, we prove that compact Kähler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kähler products, and we describe explicitly how the Kähler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kähler metrics—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 Kähler manifolds, since we need it and find the existing literature incomplete.

AB - We present a classification of compact Kähler 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 Kähler 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 Kähler manifolds with a rigid hamiltonian torus action are bundles of toric Kähler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kähler 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 Kähler–Einstein 4-orbifolds. Combining these two themes, we prove that compact Kähler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kähler products, and we describe explicitly how the Kähler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kähler metrics—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 Kähler manifolds, since we need it and find the existing literature incomplete.

M3 - Article

VL - 68

SP - 277

EP - 345

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -