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Unexpected Stein
fillings, rational surface singularities and plane curve
arrangements
Olga Plamenevskaya and Laura Starkston |
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Geometry & Topology 27 (2023) 1083–1202
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Abstract |
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We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong and van Straten (1998); they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong and van Straten’s construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber). |
Keywords
symplectic fillings, singularities, surface singularities,
curve arrangements
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Mathematical Subject Classification
Primary: 14J17, 32S30, 32S50, 57K33, 57K43
Secondary: 14H50, 32S25
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References |
Publication
Received: 21 July 2020
Revised: 14 June 2021
Accepted: 22 July 2021
Published: 8 June 2023
Proposed: András I Stipsicz
Seconded: Dan Abramovich, Paul Seidel
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Authors |
| Olga Plamenevskaya | |
| Department of Mathematics Stony Brook University Stony Brook, NY United States |
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| Laura Starkston | |
| Mathematics Department University of California, Davis Davis, CA United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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