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Unexpected Stein fillings, rational surface singularities and plane curve arrangements

Olga Plamenevskaya and Laura Starkston

Geometry & Topology 27 (2023) 1083–1202

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 5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).

symplectic fillings, singularities, surface singularities, curve arrangements
Mathematical Subject Classification
Primary: 14J17, 32S30, 32S50, 57K33, 57K43
Secondary: 14H50, 32S25
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
Olga Plamenevskaya
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
Laura Starkston
Mathematics Department
University of California, Davis
Davis, CA
United States

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