M-Resolutions and Deformations of Quotient Singularities

Behnke, Kurt; Christophersen, Jan Arthur

doi:10.2307/2375004

Cited by 17 publications

(53 citation statements)

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“…More importantly, this shows a geometric way to connect directly Lisca's [26] and Kollár-Shepherd-Barron's [24] one-to-one correspondences. By Némethi-Popescu-Pampu [32], it also connects Christophersen-Stevens' [11,45] correspondence, together with the one induced by the M -resolutions of Behnke-Christophersen [3], and in particular we answer the question raised in Némethi-Popescu-Pampu [32, §11.2].…”

mentioning

confidence: 59%

“…The dual graph of the minimal resolution of (X, 0) is given by (3,1) and the dual graph of its minimal resolution is given by…”

Section: 21mentioning

confidence: 99%

“…In this section, we recall that there is a one-to-one correspondence between the components of Def(X) and certain partial resolutions of (X, 0). We refer to KSB [24], Stevens [45,46], Behnke-Christophersen [3] for details. It is known that a singularity of class T is a rational double point or a cyclic quotient surface singularity of type 1 dn 2 (1, dna − 1) with d ≥ 1, n ≥ 2, 1 ≤ a < n, and (n, a) = 1.…”

Section: Partial Resolutions and Versal Deformation Spacesmentioning

confidence: 99%

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Milnor fibers and symplectic fillings of quotient surface singularities

Park

Shin

et al. 2018

Advances in Mathematics

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We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that:-Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism.-Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called P -resolutions of a quotient surface singularity. We provide an explicit algorithm for identifying a given Milnor fiber as a minimal symplectic filling, that is, as complements Z − E ∞ . For this, we apply some techniques from the minimal model program for 3-folds such as divisorial contractions and flips.We first compactify a given smoothing of (X, 0). We briefly sketch the idea: Let M be the Milnor fiber of a smoothing π : X → ∆ of a quotient surface singularity X. Let Y → X be the P -resolution corresponding to π. According to Behnke-Christophersen [3], there is a special partial resolution, so called, M -resolution Y → X dominating Y (See Definition 6.14), and a Q-Gorenstein smoothing φ : Y → ∆ such that the smoothing φ blows down to π. We have a commutative diagram as described in Figure 1. It is easy to show that a general fiber Y t = φ −1 (t) is isomorphic to X t = π −1 (t). Therefore we haveWe then compactify X and Y to compact complex surfaces, following Lisca [26] and Pinkham [41], so that the two smoothings Y → ∆ and X → ∆ can be extended to the deformations Y → ∆ and X → ∆ of the natural compactifications X and Y of X and Y , where X and Y are obtained, roughly speaking, by pasting a regular neighborhood ν(E ∞ ) of the compactifying divisor E ∞ of X (See Section 7 for the definition of the natural compactification). We again have a commutative diagram, as described in Figure 1.The deformations Y → ∆ and X → ∆ are locally trivial along E ∞ . So the Milnor fiber Y t is given as the complement of the compactifying divisor E ∞ in a general fiber Y t (which is called a compactified Milnor fiber ; See Definition 7.6) of the deformation Y → ∆; See Proposition 7.5. So we haveWe need to recognize ( Y t , E ∞ ) as (Z, E ∞ ) in the lists of Lisca [26] and . For this, we show that Main Theorem 2 (Theorem 9.4). By applying specific divisorial contractions and flips in a controlled and explicit manner to the deformation Y → ∆, we obtain a new deformation W → ∆ such that all of its fibers are smooth.During this minimal model program (in short, MMP ) process, we ...

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confidence: 59%

“…The dual graph of the minimal resolution of (X, 0) is given by (3,1) and the dual graph of its minimal resolution is given by…”

Section: 21mentioning

confidence: 99%

Section: Partial Resolutions and Versal Deformation Spacesmentioning

confidence: 99%

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Milnor fibers and symplectic fillings of quotient surface singularities

Park

Shin

et al. 2018

Advances in Mathematics

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“…Furthermore, there exists a finite base change Der ′ (Z P k ) → Der k (Y 0 ). Next, we recall some facts from [BC94]. There exists an M-resolution Z M k → Z P k with only type T 0 singularities (type T singularities with s = 1), which has a local moduli space…”

Section: )mentioning

confidence: 99%

“…, r}. By [KSB88] and [BC94], for each irreducible component, there exists a unique Presolution Z P k → Y 0 , a unique M-resolution Z M k → Z P k , and finite base changes Der ′ (Z M k ) → Der ′ (Z P k ) → Der k (Y 0 ). Using the C * -action, we can extend Der ′ (Z M k ), Der ′ (Z P k ), Der k (Y 0 ) to global analytic spaces J M k , J P k , J k , which are bases spaces of deformations X k , Z k , Y k , respectively, and the total spaces admit C * -actions such that the following diagram is C *equivariant…”

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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P-Resolutions of Cyclic Quotients from the Toric Viewpoint

Altmann

1998

Singularities

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M-Resolutions and Deformations of Quotient Singularities

Cited by 17 publications

References 14 publications

Milnor fibers and symplectic fillings of quotient surface singularities

Milnor fibers and symplectic fillings of quotient surface singularities

Existence and Compactness Theory for Ale Scalar-Flat Kähler Surfaces

P-Resolutions of Cyclic Quotients from the Toric Viewpoint

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