The Bochner-flat geometry of weighted
                    projective spaces

David, Liana; Gauduchon, Paul

doi:10.1090/crmp/040/06

Cited by 16 publications

(29 citation statements)

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“…To summarize, g given by (13) is globally defined on a toric orbifold whose rational Delzant polytope is [α, β] ⊂ t * , with normals u α , u β ∈ t, if and only if Θ smooth on [α, β], with Θ(α) = 0 = Θ(β),…”

Section: Compatible Kähler Metrics: Local Theorymentioning

confidence: 99%

Hamiltonian 2-forms in Kähler geometry, II Global Classification

Apostolov¹,

Calderbank²,

Gauduchon³

et al. 2004

J. Differential Geom.

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280

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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.This paper is concerned with the construction of explicit Kähler metrics on compact manifolds, and has several interrelated motivations. The first is the notion of a hamiltonian 2-form, introduced in part I of this series [4].Definition 1. Let φ be any (real) J-invariant 2-form on the Kähler manifold (M, g, J, ω) of dimension 2m. We say φ is hamiltonian iffor any vector field X, where tr φ = φ, ω is the trace with respect to ω. When M is a Riemann surface (m = 1), this equation is vacuous and we require instead that tr φ is a Killing potential, i.e., a hamiltonian for a Killing vector field J grad g tr φ.A second motivation is the notion of a weakly Bochner-flat (WBF) Kähler metric, by which we mean a Kähler metric whose Bochner tensor (which is part of the curvature tensor) is co-closed. By the differential Bianchi identity, this is equivalent (for m ≥ 2) to the condition that ρ + Scal 2(m+1) ω is a hamiltonian 2-form, where ρ is the Ricci form. WBF Kähler metrics are extremal in the sense of Calabi, i.e., the symplectic gradient of the scalar curvature is a Killing vector field, and provide Date: November 3, 2018. We would like to thank C. a class of extremal Kähler metrics which include the Bochner-flat Kähler metrics studied by Bryant [10] and products of Kähler-Einstein metrics. The geometry of WBF Kähler metrics is tightly constrained, because the more specific the normalized Ricci form is, the closer the metric is to being Kähler-Einstein, while the more generic it is, the stronger the consequences of the hamiltonian property.A hamiltonian 2-form φ induces...

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Section: Compatible Kähler Metrics: Local Theorymentioning

confidence: 99%

Hamiltonian 2-forms in Kähler geometry, II Global Classification

Apostolov¹,

Calderbank²,

Gauduchon³

et al. 2004

J. Differential Geom.

Self Cite

280

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show abstract

“…Similarly, since s gw = s g T w − 2n, and since the mean transverse scalar curvature of g T w is 4n n j=0 w j [16], we have that the projection s 0 gw of s gw onto the constants is given by s 0 gw = 2n(2 n j=0 w j − 1) .…”

Section: Openness Of the Canonical Sasaki Setmentioning

confidence: 90%

“…The space of metrics M(ξ w ,J) associated with the polarized Sasakian manifold (S 2n+1 , ξ w ,J) has a representative g w whose transverse Kähler metric g T w is Bochner flat [8] on CP n w , and thus, extremal. Computing in an affine orbifold chart, it can be determined [16] that the scalar curvature of g T w is given by s g T w = 4(n + 1) n j=0 w j (2( n k=0 w k ) − (n + 2)w j )|z j | 2 n j=0 w j |z j | 2 , at z ∈ S 2n+1 . Since the volume µ g T w (CP n w ) = π n /(n!…”

Section: Openness Of the Canonical Sasaki Setmentioning

confidence: 99%

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Kähler–Einstein Metrics

Boyer¹,

Galicki²

2007

Sasakian Geometry

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Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2 -norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that just the standard polarization can be represented by a Sasaki-Einstein metric.

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“…A Kähler manifold is Bochner-flat if its Bochner tensor vanishes. Bochner-flat Kähler manifolds represent an important class of Kähler manifolds and have been intensively studied: the local geometry of Bochner-flat Kähler manifolds and its interactions with Sasaki geometry has been studied, using the Webster's correspondence, in [DG06]; complete Bochner-flat Kähler structures on simply connected manifolds have been classified in [Bry01]; generalisations of Bochner-flat Kähler manifolds (like weakly Bochner-flat Kähler manifolds and Kähler manifolds with a hamiltonian 2-form) have also been developed (see, for example, [ACG04,ACG06,Gau01]).…”

Section: Introductionmentioning

confidence: 99%

The Bochner-flat cone of a CR manifold

David¹

2008

Compositio Math.

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We construct a Kähler structure (which we call a generalised Kähler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a one-parameter family of compatible Sasaki structures. We determine those generalised Kähler cones which are Bochner-flat and we study their local geometry. We prove that any Bochner-flat Kähler manifold of complex dimension bigger than two is locally isomorphic to a generalised Kähler cone.

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The Bochner-flat geometry of weighted projective spaces

Cited by 16 publications

References 0 publications

Hamiltonian 2-forms in Kähler geometry, II Global Classification

Hamiltonian 2-forms in Kähler geometry, II Global Classification

Kähler–Einstein Metrics

The Bochner-flat cone of a CR manifold

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