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 -