Milnor fibers and symplectic fillings of quotient surface singularities

Abstract

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 ...

Complex singularities and contact topology

“…We extended partially our results to all sandwiched singularities in [138]. Nevertheless, the question to know if the analog of the last statement of The last statement of Theorem 6.16 was extended by Park, Park, Shin and Urzúa [151] to all quotient surface singularities. They concentrated on the cases not covered by Theorem 6.16.…”

Symplectic Fillings of Links of Quotient Surface Singularities – Corrigendum

“…D. Shin informed us of his joint work [3] and that one case is missing in our list in [2]. We claimed that the link of the quotient surface singularity has four minimal symplectic fillings; however, there is a fifth one.…”

“…Details of the algorithm found in the program in the case of Type (3, 1) singularities follow. The algorithm for the case of Type (3,2) singularities is similar and the details are omitted.…”

“…As mentioned above, corresponding to each link of a quotient surface singularity of Type (3,1) there is a standard compactifying divisor K consisting of a cuspidal rational curve D, a string of rational curves C 1 , . .…”

“…. , −c k ), i, j is the data of a preadmissible configuration of a Case II filling of the link of a quotient surface singularity of Type (3,1) if k = 1 then if d < 9 and c 1 =0 then return if no such s exists then return 0 end if let Q denote the data of the configuration K = A ∪ B ∪ D ∪ C 1 ∪ · · · ∪ C k−1 obtained by blowing down C s define i and j by the condition that D intersects the string C 1 , . .…”