Sandwiched singularities and nearly Lefschetz fibrations

TL;DR

The paper investigates sandwiched surface singularities by contrasting algebro-geometric deformations (Milnor fibers via decorated plane curve germs) with symplectic topology (Stein fillings of links). It develops a symplectic counterpart to De Jong--van Straten’s picture-deformation theory using DJVS immersed disk arrangements, encoded by planar spinal open books and nearly Lefschetz fibrations, and shows that all minimal fillings arise in this framework. The authors introduce braided wiring diagrams to capture arrangements and provide a blueprint to reconstruct fillings from diagram data, combining LVHMW-type results with Min--Roy--Wang’s findings to prove that every minimal filling can be realized as a nearly Lefschetz fibration compatible with the open book. They further demonstrate that many links admit unexpected Stein fillings not homeomorphic to Milnor fibers, distinguished via incidence matrices and non-realizable DJVS arrangements, thereby highlighting a rich interaction between algebraic and symplectic structures with potential implications for 4-manifold topology.

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.

Figures (14)

• Figure 1: A resolution graph of a sandwiched singularity together with an augmentation with $-1$ curves, decorated by transverse disks. Below is the sequence of blow-downs and the image of the disks under these blow-downs. In this case, the resulting curve configuration consists of two simple cusps intersecting each other with multiplicity $7$. By tracking the multiplicities of the exceptional spheres at each stage, one can see that the weight $w_i$ of each component is $6$.
• Figure 2: Top: A resolution graph $G$ of a sandwiched singularity together with an augmentation. Bottom: A larger sandwiched resolution graph $K$ with an augmentation whose (non-decorated) plane curve germ is the same as $C_G$, but the weights are higher. As in Figure \ref{['fig:examplesandwich']}, the weights associated to the curve components of $C_G$ with the chosen augmentation $G'$ are both $8$. By adding the additional legs of $-2$ spheres, the weights of the two curve components $C_K$ (with the indicated augmentation $K'$) are $11$ and $12$.
• Figure 3: Dehn surgery diagram to recover normal Euler numbers in resolution graph from plumbing structure fibrations.
• Figure 4: The boundary interchange is the monodromy of the complement of the tubular neighborhood of the curve $\{y^2 = x\}$.
• Figure 5: Top left: an augmentation of the plumbing graph for a normal crossing resolution of a sandwiched singularity. Top right: its corresponding Kirby diagram. Bottom left: the cap with additional disk bundles of Euler number $0$ over spheres $\tilde{\tilde{\Sigma}}_i$ together with meridian curves of $\tilde{\tilde{\Sigma}}_i$ corresponding to the markings. Bottom right: the result of canceling the $0$-surgery with the $(-1)$ surgery, tracking the images of the markings.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8
• Remark 2.9