On the Milnor fibres of cyclic quotient singularities

Némethi, András; Popescu-Pampu, Patrick

doi:10.1112/plms/pdq007

Cited by 28 publications

(33 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For cyclic quotient surface singularities, Ohta-Ono [34] proves that a minimal symplectic filling of an A n -singularity is diffeomorphic to its Milnor fiber. Then Némethi-Popescu-Pampu [32] shows that every minimal symplectic filling of a cyclic quotient surface singularity is diffeomorphic to a Milnor fiber of the singularity, and provides an explicit one-to-one correspondence between Milnor fibers and minimal symplectic fillings.…”

mentioning

confidence: 96%

See 1 more Smart Citation

Milnor fibers and symplectic fillings of quotient surface singularities

Park

Shin

et al. 2018

Advances in Mathematics

View full text Add to dashboard Cite

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 ...

show abstract

mentioning

confidence: 96%

“…The conjecture was solved affirmatively by Némethi-Popescu-Pampu [32]. In order to identify Milnor fibers with Stein fillings, Némethi-Popescu-Pampu [32] uses the explicit equations of reduced versal base space of cyclic quotient surface singularities developed by Riemenschneider [42] and Arndt [1].…”

mentioning

confidence: 99%

Milnor fibers and symplectic fillings of quotient surface singularities

Park

Shin

et al. 2018

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…The proof of the following theorem was completed by Stevens [180], using deep results of Kollár and Shepherd-Barron Christophersen and Stevens gave moreover equations describing the restriction of the miniversal deformation to each such component (generalizing Pinkham's equations described in Section 4.4 and Riemenschneider's equations for the deformation over the Artin component given in [169]). Némethi and myself used those equations in [137] in order to prove the following theorem, which answers affirmatively a conjecture of Lisca [113], first formulated in [112]: In the paper [137] we got a second proof of the last statement by using the work [89] of de Jong and van Straten on sandwiched surface singularities, which form a class of rational surface singularities containing the cyclic quotients. We extended partially our results to all sandwiched singularities in [138].…”

Section: Iii-65mentioning

confidence: 71%

“…This is a consequence of the proof of a conjecture of Lisca [112,Page 16] by Némethi and myself [137]. This conjecture related the smoothings of cyclic quotient singularities with the Stein fillings of their contact boundaries.…”

Section: Iii-51mentioning

confidence: 74%

Complex singularities and contact topology

Popescu-Pampu¹

2017

Winter Braids Lecture Notes

Self Cite

View full text Add to dashboard Cite

This text is a greatly expanded version of the mini-course I gave during the school Winter Braids VI organized in Lille between 22-25 February 2016. It is an introduction to the study of interactions between singularity theory of complex analytic varieties and contact topology. I concentrate on the relation between the smoothings of singularities and the Stein fillings of their contact boundaries. I tried to explain basic intuitions and facts in both fields, for the sake of the readers who are not accustomed with one of them.

show abstract

“…There are several important contributions to this line of research done by various authors and we refer to [234,Section 6.2] for an account of this. We finish this subsection with the following theorem from [200] that provides a generalization of Eliashberg's theorem 11.6:…”

Section: Open-books and Contact Structuresmentioning

confidence: 99%

On Milnor’s fibration theorem and its offspring after 50 years

Seade

2018

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

To Jack, whose profoundness and clarity of vision seep into our appreciation of the beauty and depth of mathematics.Abstract. Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading. IntroductionMilnor's fibration theorem in [189] is a milestone in singularity theory that has opened the way to a myriad of insights and new understandings. This is a beautiful piece of mathematics, where many different branches, aspects and ideas, come together. The theorem concerns the geometry and topology of analytic maps near their critical points.Consider the simplest case, a holomorphic map (C n+1 , 0) f → (C, 0) taking the origin into the origin, with an isolated critical point at 0. As an example one can have in mind the Pham-Brieskorn polynomials:(0.1) z → z a 0 0 + · · · + z an n , a i ≥ 2 for all i = 0, 1, · · · , n .Since f is analytic, there exists r > 0 sufficiently small so that 0 ∈ C is the only critical value of the restriction f | Br , where B r is the open ball of radius r and center at 0. Set V := f −1 (0) and V * := V \ {0} .

show abstract

On the Milnor fibres of cyclic quotient singularities

Cited by 28 publications

References 36 publications

Milnor fibers and symplectic fillings of quotient surface singularities

Milnor fibers and symplectic fillings of quotient surface singularities

Complex singularities and contact topology

On Milnor’s fibration theorem and its offspring after 50 years

Contact Info

Product

Resources

About