ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities

Suvaina, Ioana

doi:10.1007/s10455-011-9273-1

Cited by 32 publications

(33 citation statements)

References 19 publications

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“…Thanks to the work of [HRS16] and Theorem 6 in Section 5.2, in complex dimension two we can also state a converse classificatory statement (which can be compared to [Suv12]).…”

Section: Examplesmentioning

confidence: 98%

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

Borbon

Spotti

2019

International Mathematics Research Notices

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We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of C n /Γ with non-positive discrepancies. In particular, this includes the case of the minimal resolution of two dimensional quotient singularities for any finite subgroup Γ ⊂ U (2) acting freely on the three-sphere, hence generalizing Kronheimer's construction of smooth ALE gravitational instantons. Finally, we show how our results extend to the more general asymptotically conical setting.for every smooth compactly supported test function φ.

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“…Thanks to the work of [HRS16] and Theorem 6 in Section 5.2, in complex dimension two we can also state a converse classificatory statement (which can be compared to [Suv12]).…”

Section: Examplesmentioning

confidence: 98%

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

Borbon

Spotti

2019

International Mathematics Research Notices

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“…Note that this group is covered by the groupΓ = 1 rs (1, rs − 1), quotiented by a Z r -action. The spaces X in the non-Artin component admit Ricci-flat metrics which are isometric quotients of an A rs−1 hyperkähler metric [Şuv12,Wri12]. We also note that the embedding dimension is r + 3, and the base of the non-Artin component has dimension s [KSB88, BC94].…”

Section: Artin Component Examplesmentioning

confidence: 99%

“…However, this case was explicitly constructed by Kronheimer using the hyperkähler quotient construction [Kro89], so we do not devote any extra attention to this case. Note also that the Q-Gorenstein smoothings of the type T cyclic singularities admit Ricci-flat Kähler metrics which are just quotients of the A k -type hyperkähler metrics by finite groups of isometries [Şuv12,Wri12]. These metrics play a crucial role in our analysis of non-Artin components.…”

Section: Introductionmentioning

confidence: 99%

Existence and Compactness Theory for Ale Scalar-Flat Kähler Surfaces

Han¹,

Viaclovsky²

2020

Forum of Mathematics, Sigma

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Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.(1.7)Define global base spaces1.8) Case (a) follows easily from [HV16, Theorem 1.4]. Cases (b) and (c) are obtained by applying a generalization of a result of Biquard-Rollin to the ALE case [BR15]. For the precise statement, see Theorem 6.2 below.Recall that for integers p, q satisfying (p, q) = 1, the cyclic action 1 p (1, q) is that generated by (z 1 , z 2 ) → (ζ p z 1 , ζ q p z 2 ) where ζ p is a primitive pth root of unity. Corollary 1.12. Let Γ = 1 p (1, q) be any cyclic group with (p, q) = 1, and let J M k be any component of J M . Then for any J ∈ J M k (J is away from the central fiber if k > 0), there exists a scalar-flat Kähler metric ω J in some Kähler class.This is obtained by using the Calderbank-Singer construction from [CS04], together with Theorem 1.11. 1.2. Global existence results. We now turn our attention to existence of global moduli spaces of ALE SFK metrics for certain groups Γ. The following theorem is an application of Case (a) in Theorem 1.11 together with Corollary 1.7.

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“…It is worth emphasizing that, despite persistent rumors to the contrary, M really might not be simply connected, even in the Ricci-flat case. For pertinent examples and classification results, see [21,27].…”

Section: Capping Off the Endsmentioning

confidence: 99%

Mass, Kähler manifolds, and symplectic geometry

LeBrun

2019

Ann Glob Anal Geom

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In the author's previous joint work with , a mass formula for asymptotically locally Euclidean (ALE) Kähler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension 4 presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chruściel fall-off conditions that sufficed in higher dimensions. The present article, however, shows that techniques of 4-dimensional symplectic geometry can be used to obtain all the major results of [7], assuming only Chruściel-type falloff. In particular, the present article presents a new a proof of our Penrose-type inequality for the mass of an asymptotically Euclidean Kähler manifold that only requires Chruściel metric fall-off.

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ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities

Cited by 32 publications

References 19 publications

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

Existence and Compactness Theory for Ale Scalar-Flat Kähler Surfaces

Mass, Kähler manifolds, and symplectic geometry

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