Stein fillings vs. Milnor fibers

TL;DR

This work shows that Milnor fillings of a fixed link $(Y,\xi)$ have bounded topology, in contrast to potential unbounded Stein fillings, and proves finiteness of Milnor fibers for sandwiched singularities by translating smoothings into deformations of decorated plane curves. It develops a framework combining Laufer–Durfee–Wahl relations, geometric-genus bounds, and monodromy/Lefschetz-fibration techniques to produce and distinguish Stein fillings, including many unexpected ones, for various singularity classes (cusps, triangles, minimally elliptic). The paper also analyzes how Milnor fibers sit inside the broader landscape of Stein fillings, and demonstrates, via specific constructions (e.g., monodromy factorization and lantern/daisy substitutions), that Milnor fibers can be strictly fewer than the available Stein fillings in many interesting cases. The results deepen the understanding of how analytic structure, resolution data, and open-book monodromy govern the topology of fillings, and give concrete finite vs. infinite phenomena across rational, minimally elliptic, and sandwiched singularities. The methods connect complex-analytic deformations with symplectic and contact-topological techniques, providing explicit obstructions and procedures to realize or preclude Milnor-fiber realizations of various Stein fillings.

Abstract

Given a link of a normal surface singularity with its canonical contact structure, we compare the collection of its Stein fillings to its Milnor fillings (that is, Milnor fibers of possible smoothings). We prove that, unlike Stein fillings, Milnor fillings of a given link have bounded topology; for links of sandwiched singularities, we further establish that there are only finitely many Milnor fillings. We discuss some other obstructions for a Stein filling to be represented by a Milnor fiber, and for various types of singularities, including simple classes like cusps and triangle singularities, we produce Stein fillings that do not come from Milnor fibers or resolutions.

Figures (7)

• Figure 1: The Gay-Mark open book for the resolution of a non-rational singularity which admits infinitely many unexpected fillings. The monodromy is the product of positive Dehn twists along the curves shown in blue. The shaded subsurface of genus $g=4$ with $b=4$ boundary components is the one where the arbitrarily long positive factorizations are supported.
• Figure 2: The curves $c_i, d, e$ on $\Sigma_g^{2g-4}$ with boundary components $\delta_j$.
• Figure 3: The curves $x_j$ drawn on the two subsurfaces of $\Sigma_g^{2g-4}$ containing all the boundary components.
• Figure 4: Minimally elliptic singularities with unexpected fillings.
• Figure 5: A family of smoothable triangle singularities and the monodromy curves for the open book on the link.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• Remark 2.2
• Proposition 2.3: DurfLaufermuSteenbrink, see also Wahl-smooth
• Proposition 2.4
• proof : Proof of Theorem \ref{['bounds']}
• Remark 2.5