An Unexpected Rational Blowdown

TL;DR

The paper establishes a new symplectic rational blowdown by constructing Stein fillings that are rational homology disks for the links $(Y_{k,n}, \xi_{k,n})$ of a two-parameter family of sandwiched singularities $G_{k,n}$, where the singularities do not admit $\,\mathbb{Q}$HD smoothings. It develops a unified framework using de Jong--van Straten picture deformations, DJVS immersed disk arrangements, and braided wiring diagrams with tangencies, along with boundary-preserving diagrammatic moves, to realize these fillings from Scott deformations of decorated germs. The authors provide explicit constructions for all $k\ge 0$, $n\ge 1$, including a concrete example embedded in an elliptic fibration that yields an exotic 4-manifold, demonstrating the practical impact of the new blowdown technique. This work broadens the landscape of $\mathbb{Q}$HD fillings beyond weighted homogeneous cases and supplies versatile diagrammatic tools that may apply to broader classes of singularities and symplectic 4-manifolds.

Abstract

The rational blowdown operation in 4-manifold topology replaces a neighborhood of a configuration of spheres by a rational homology ball. Such configurations typically arise from resolutions of surface singularities that admit rational homology disk smoothings. Conjecturally, all such singularities must be weighted homogeneous and belong to certain specific families: Stipsicz-Szabó--Wahl constructed QHD smoothings for these families and used Donaldson's theorem to obtain very restrictive necessary conditions on the resolution graphs for singularities with this property. In particular, these results, as well as subsequent work of Bhupal-Stipsicz, show that for certain resolution graphs, the canonical contact structure on the link of the singularity cannot admit a QHD symplectic filling. By contrast, we exhibit Stein rational homology disk fillings for the contact links of an infinite family of rational singularities that are {\em not} weighted homogeneous, producing a new symplectic rational blowdown. Inspiration for our construction comes from de Jong-van Straten's description of Milnor fibers of sandwiched singularities; we use the symplectic analog of de Jong-van Straten theory developed by the second and third authors. The unexpected Stein fillings are built using spinal open books and nearly Lefschetz fibrations.

Figures (19)

• Figure 1: The family $G_{k,n}$ for $k\geq -1,n\geq 0$, where $G_{k,n}$ has $k+n+7$ vertices, with $k$ vertices between the two nodes. The graph $G_{-1,1}$ has a single node of valency $4$, with 6 vertices total.
• Figure 2: The decorated germ for the graph $G_{0,1}$. Each component $C_i$ of the germ has a cusp singularity modeled on $y^2=x^3$. The intersection multiplicities are $C_0 \cdot C_i=7$ for $i=1, \dots, 6$, and $C_i\cdot C_j = 8$ for $i, j=1, \dots, 6$. The weights are as indicated, $w(C_0)=8$, $w(C_i)=7$ for $i=1, \dots, 6$.
• Figure 3: The Scott deformation of the decorated germ of Figure \ref{['fig:djvs']}. The blue curve corresponds to $C_0$ in the decorated germ. Yellow, orange and red curves show deformations of three of the six curves $C_1, \dots, C_6$; each of the other three curves, not pictured, goes through the same intersection points and have one free marked point.
• Figure 4: For two nearby vertical tangencies on curves with nested real parts as on the left, the wiring diagram has braiding of the strands near the tangencies, as shown. The branches of the same curve are shown by strands of the same color.
• Figure 5: The Scott deformation of the germ of Example \ref{['ex:Scott-warmup']}: the real part of the deformed curve (left) and the braided wiring diagram (right). The figure is taken from PS2.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Example 3.1
• Proposition 3.2
• proof
• Remark 4.1
• proof : Proof of Move (i):
• proof : Proof of Move (ii):