Quotients of gravitational instantons

Wright, Evan P.

doi:10.1007/s10455-011-9272-2

Cited by 17 publications

(13 citation statements)

References 24 publications

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“…If we think of R 3 as the interior of the unit ball B 3 , then g F can be thought as an F -metric associated to the manifold with foliated boundary X = (B 3 × S 1 )/Z k . This example generalizes to multi-Taub-NUT metrics of type A k−1 admitting an action of Z k by isometries, see [43] and [46]. In this case, the circle fibration at infinity is replaced by a circle foliation when one passes to the quotient.…”

Section: Microlocalization For Certain Types Of Foliated Boundariesmentioning

confidence: 89%

Pseudodifferential operators on manifolds with foliated boundaries

Rochon

2012

Journal of Functional Analysis

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Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a 'resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration 'resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of Dirac-type operators, a result of independent interest generalizing the well-known formula of Bismut and Cheeger.

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Section: Microlocalization For Certain Types Of Foliated Boundariesmentioning

confidence: 89%

Pseudodifferential operators on manifolds with foliated boundaries

Rochon

2012

Journal of Functional Analysis

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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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“…Appendix A. Asymptotically conical Ricci-flat Kähler surfaces AC Ricci-flat Kähler manifolds of complex dimension n = 2 are completely classified [34,52,58]; they are precisely the Kronheimer ALE spaces [33] and certain quotients of Kronheimer spaces of type A by free holomorphic isometric actions of finite cyclic groups. The paper [44] essentially shows that all of these spaces can be constructed by the Tian-Yau method, although Theorem A is needed to get a definitive result.…”

Section: 33mentioning

confidence: 99%

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Conlon

Hein

2015

Geom. Funct. Anal.

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Let X be a compact Kähler orbifold without C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D ⊃ Sing(X) and −pKX = q[D] for some p, q ∈ N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kähler-Einstein, then, applying results from our previous paper [15], we show that each Kähler class on X \ D contains a unique asymptotically conical Ricci-flat Kähler metric, converging to its tangent cone at infinity at a rate of O(r −1−ε ) if X is smooth. This provides a definitive version of a theorem of Tian and Yau [54]. (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128 )) when X = BlpP 3 and D = Blp 1 ,p 2 P 2 is the strict transform of a smooth quadric through p in P 3 . Here D is no longer Kähler-Einstein, but the normal S 1 -bundle to D in X admits an irregular Sasaki-Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi-Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.Date: July 4, 2018. 1 arXiv:1301.5312v3 [math.DG] 8 Dec 2014 Comparing the two theorems and their proofs. Many of the refinements in Theorem A (no neat or almost ample condition, all Kähler classes, the parameter c, uniqueness and symmetry) are due to an improvement of general technique in [15], partly based on important earlier contributions by van Coevering [55], whereas asymptotics of the form (1.1) are already implicit in Tian-Yau [54]. Let us point out one useful consequence of our explicit estimate (1.2).Corollary B. If X is smooth, then the best possible convergence rate λ of the Ricci-flat metrics of Theorem A to their tangent cones at infinity is always strictly greater than 1.Proof. Since N q−p D = K −p D and Pic(D) is torsion-free because π 1 (D) = 0, there exists a line bundle L with L p = N D and L q−p = K −1 D . Thus, by [32, p. 32, Corollary], q − p n with equality if and only if D = P n−1 , so that α − 1 n and λ 1 with equality if and only if D = P n−1 , N D = O(1). But in the latter case, the cone, and hence (X \ D, g c ) itself, must be isometric to flat C n .It seems reasonable to expect that α − 1 n even if X is singular. Moreover, equality should still imply that the cone is C n /Γ (see Remark A.2 for some examples where Γ = {1}), so that λ 2n by 1 Cristiano Spotti pointed out to us that this phenomenon was first observed in [41] for the classical elliptic modular curve H/PSL(2, Z), which is isomorphic to C as a variety but whose π orb 1 and Pic orb are nontrivial.

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Quotients of gravitational instantons

Cited by 17 publications

References 24 publications

Pseudodifferential operators on manifolds with foliated boundaries

Pseudodifferential operators on manifolds with foliated boundaries

Existence and Compactness Theory for Ale Scalar-Flat Kähler Surfaces

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

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