Compact Kähler surfaces with harmonic anti-self-dual Weyl tensor

Jelonek, Włodzimierz

doi:10.1016/s0926-2245(02)00076-1

Cited by 9 publications

(7 citation statements)

References 12 publications

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“…[13]), i.e., symmetric 2-tensors S satisfying sym ∇S = 0. Indeed φ is hamiltonian if and only if A = φ + σω is closed and S = J(φ − σω) is a Killing tensor.…”

Section: Strongly Conformally Einstein Kähler Metricsmentioning

confidence: 99%

Hamiltonian 2-Forms in Kähler Geometry, I General Theory

Apostolov¹,

Calderbank²,

Gauduchon³

2006

J. Differential Geom.

129

506

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Abstract. We introduce the notion of a hamiltonian 2-form on a Kähler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kähler geometry. In particular, on any Kähler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kähler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms also arise on conformally Einstein Kähler manifolds and provide an Ansatz for extremal Kähler metrics unifying and extending many previous constructions.In a previous paper [3], while investigating Kähler 4-manifolds whose antiselfdual Weyl tensor is co-closed, we happened upon a remarkable linear differential equation for (1, 1)-forms φ on a Kähler manifold. This equation states (in any dimension)(1)for all vector fields X, where (g, J, ω) is the Kähler structure with Levi-Civita connection ∇. A hamiltonian 2-form is a (nontrivial) solution φ of (1).Hamiltonian 2-forms underpin many explicit constructions in Kähler geometry. They arise in particular on Bochner-flat Kähler manifolds and on Kähler manifolds (of dimension greater than four) which are conformally Einstein, both of which have been classified recently, respectively by Bryant [5], and Derdziński and Maschler [8]. In this paper we obtain an explicit local classification of all Kähler metrics with a hamiltonian 2-form, which provides a unifying framework for these works, and at the same time extends Bryant's local classification to the much larger class of Kähler manifolds with co-closed Bochner tensor, called weakly Bochner-flat.The key feature of hamiltonian 2-forms φ on Kähler 2m-manifolds M -and the reason for the name-is that if σ 1 , . . . σ m are the elementary symmetric functions of the m eigenvalues of φ (viewed as a hermitian operator via the Kähler form ω), then the hamiltonian vector fields K r = J grad g σ r are Killing. Further, the Poisson brackets {σ r , σ s } are all zero, so that the vector fields K 1 , . . . K m commute.If K 1 , . . . K m are linearly independent, then the Kähler metric is toric. However, not every toric Kähler metric arises in this way: the hamiltonian property also implies that the eigenvalues of φ have orthogonal gradients. We say that a toric manifold is orthotoric if there is a momentum map (σ 1 , . . . σ m ) for the torus action (with respect to some basis of the Lie algebra) such that the gradients of the roots of the polynomial m r=0 (−1) r σ r t m−r are orthogonal-here σ 0 = 1. Of course K 1 , . . . K m need not be independent; then on an open set where the span is ℓ-dimensional, there is a local hamiltonian ℓ-torus action by isometries, so the Kähler metric on M may be described (locally) by the Pedersen-Poon construction [20], as a fibration, with 2ℓ-dimensional toric fibres, over a 2(m−ℓ)-dimensional complex manifold S equipped with a family of Kähler quotient metrics parameterized by the momentum map of the local ℓ-torus action.The hamiltonian property of φ has fu...

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“…[13]), i.e., symmetric 2-tensors S satisfying sym ∇S = 0. Indeed φ is hamiltonian if and only if A = φ + σω is closed and S = J(φ − σω) is a Killing tensor.…”

Section: Strongly Conformally Einstein Kähler Metricsmentioning

confidence: 99%

Hamiltonian 2-Forms in Kähler Geometry, I General Theory

Apostolov¹,

Calderbank²,

Gauduchon³

2006

J. Differential Geom.

129

506

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“…Gray [1978] called such manifolds ᏭC ⊥ manifolds. Many interesting manifolds are of this type, including (compact) Einstein-Weyl manifolds [Jelonek 1999], weakly self-dual Kähler surfaces [Jelonek 2002a;Apostolov et al 2003] and D'Atri spaces.…”

Section: Introductionmentioning

confidence: 99%

“…In [Jelonek 2002a] we showed that every Kähler surface has a harmonic antiself-dual part W − of the Weyl tensor W (i.e. such that δW − = 0) if and only if it is an ᏭC ⊥ -manifold.…”

Section: Introductionmentioning

confidence: 99%

“…In [Jelonek 2002a] we also showed that any simply connected 4-dimensional ᏭC ⊥ -manifold (M, g) whose Ricci tensor has exactly two eigenvalues of multiplicity 2 admits two mutually opposite hermitian structures commuting with the Ricci tensor. Surfaces admitting two oppositely oriented complex structures will be called bihermitian surfaces.…”

Section: Introductionmentioning

confidence: 99%

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Bihermitian Gray surfaces

Jelonek¹

2005

Pacific J. Math.

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“…On the other hand, we shall show that on a Kähler manifold with harmonic Bochner curvature tensor, the Gray's result does not take place. More precisely, we prove the following: Note that there exists an irreducible compact Kähler manifold [4] of dimension 4 with harmonic Bochner curvature tensor whose Ricci tensor is not parallel. Therefore, in Lemma 1.3 and Theorem 1.1, the constant scalar curvature is an essential condition.…”

mentioning

confidence: 98%