Sandwiched singularities and nearly Lefschetz fibrations

TL;DR

This paper explores the symplectic topology of sandwiched singularities, developing a theory of symplectic fillings analogous to algebraic deformations, and reveals unexpected Stein fillings through new fibration techniques.

Contribution

It introduces a symplectic analog of deformation theory for sandwiched singularities using nearly Lefschetz fibrations and spinal open books, extending previous work to a broader class.

Findings

All minimal symplectic fillings are generated by immersed disk arrangements.

Unexpected Stein fillings are identified that differ from Milnor fibers.

The approach encodes fillings via multisections of Lefschetz fibrations.

Abstract

We study Milnor fibers and symplectic fillings of links of sandwiched singularities, with the goal of contrasting their algebro-geometric deformation theory and symplectic topology. In the algebro-geometric setting, smoothings of sandwiched singularities are described by de Jong--van Straten's theory: all Milnor fibers are generated from deformations of a singular plane curve germ associated to the surface singularity. We develop an analog of this theory in the symplectic setting, showing that all minimal symplectic fillings of the links are generated by certain immersed disk arrangements resembling de Jong--van Straten's picture deformations. This paper continues our previous work for a special subclass of singularities; the general case has additional difficulties and new features. The key new ingredient in the present paper is given by spinal open books and nearly Lefschetz fibrations: we use recent work of Min--Roy--Wang to understand symplectic fillings and encode them via multisections of certain Lefschetz fibrations. As an application, we discuss arrangements that generate unexpected Stein fillings that are different from all Milnor fibers, showing that the links of a large class of sandwiched singularities admit unexpected fillings.