On conformally Kähler, Einstein manifolds

Chen, Xiuxiong; LeBrun, Claude; Weber, Brian

doi:10.1090/s0894-0347-08-00594-8

Cited by 110 publications

(167 citation statements)

References 60 publications

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“…The more complicated terms such as 'multiple principal totally null field of 2-planes having locally constant real index' will be explained in Sect. 3.…”

Section: Theorem 14 Let (M G) Be a 4-dimensional Manifold Equipped mentioning

confidence: 99%

“…If the metric has Lorentzian signature (+, +, +, −) then we chose the basis so that g(e 1 , e 1 ) = g(e 2 , e 2 ) = g(e 3 , e 3 ) = 1 = −g(e 4 , e 4 ), and as an example of N we take…”

Section: Totally Null 2-planes In Four Dimensionsmentioning

confidence: 99%

“…In the case of split signature (+, +, −, −), we have g(e 1 , e 1 ) = g(e 2 , e 2 ) = 1, g(e 3 , e 3 ) = g(e 4 , e 4 ) = −1, and we distinguish two different classes of 2-dimensional totally null N s. As an example of the first class, we take…”

Section: Totally Null 2-planes In Four Dimensionsmentioning

confidence: 99%

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Sharp version of the Goldberg–Sachs theorem

Gover

Hill

Nurowski

2010

Annali di Matematica

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We reexamine from first principles the classical Goldberg-Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)⇒(algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes, we associate a natural connection, in terms of which these conditions have a very simple form.

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“…The more complicated terms such as 'multiple principal totally null field of 2-planes having locally constant real index' will be explained in Sect. 3.…”

Section: Theorem 14 Let (M G) Be a 4-dimensional Manifold Equipped mentioning

confidence: 99%

“…If the metric has Lorentzian signature (+, +, +, −) then we chose the basis so that g(e 1 , e 1 ) = g(e 2 , e 2 ) = g(e 3 , e 3 ) = 1 = −g(e 4 , e 4 ), and as an example of N we take…”

Section: Totally Null 2-planes In Four Dimensionsmentioning

confidence: 99%

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Sharp version of the Goldberg–Sachs theorem

Gover

Hill

Nurowski

2010

Annali di Matematica

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“…The proof of Theorem B. Using Theorem A it is a simple task to classify asymptotically locally Euclidean (ALE) scalar-flat Kähler toric 4-orbifolds, following the procedure suggested by Chen, LeBrun and Weber in [5]. Definition 12.3 (Joyce [15]).…”

Section: Psfrag Replacementsmentioning

confidence: 99%

Compact anti-self-dual orbifolds with torus actions

Wright

2010

Sel. Math. New Ser.

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Abstract. We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat Kähler orbifold. We use this observation to classify ALE scalar-flat Kähler 4-orbifolds whose isometry group contain a 2-torus.

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“…The case where g/τ 2 is Einstein, was the subject of the study of [6,7,8], where local and global classifications were given, and, in all even dimensions larger than four, τ turns out to be, in fact, a special Kähler-Ricci potential. In dimension four this need not be the case, and different compact examples where recently shown to exist in [5].…”

Section: Introductionmentioning

confidence: 99%

Special Kähler–Ricci potentials and Ricci solitons

Maschler

2008

Ann Glob Anal Geom

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Abstract. On a manifold of dimension at least six, let (g, τ ) be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function τ . Off the zero set of τ , if the metric g = g/τ 2 is a gradient Ricci soliton which has soliton function 1/τ , we show that g is Kähler with respect to another complex structure, and locally of a type first described by Koiso. Moreover, τ is a special Kähler-Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci-Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs (g, τ ) satisfying a Ricci-Hessian equation is invariant, in a suitable sense, under the map (g, τ ) → ( g, 1/τ ).

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On conformally Kähler, Einstein manifolds

Abstract: We prove that any compact complex surface with c 1 > 0 c_1>0 admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blow-up C P 2 # 2 C P 2 ¯ \mathbb {CP}_2\# 2\overline {\… Show more

Cited by 110 publications

References 60 publications

Sharp version of the Goldberg–Sachs theorem

Sharp version of the Goldberg–Sachs theorem

Compact anti-self-dual orbifolds with torus actions

Special Kähler–Ricci potentials and Ricci solitons

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