Rational blowdown graphs for symplectic fillings of lens spaces

In a previous work, we proved that each minimal symplectic filling of any oriented lens space, viewed as the singularity link of some cyclic quotient singularity and equipped with its canonical contact structure, can be obtained from the minimal resolution of the singularity by a sequence of symplectic rational blowdowns along linear plumbing graphs. Here we give a dramatically simpler visual presentation of our rational blowdown algorithm in terms of the triangulations of a convex polygon. As a consequence, we are able to organize the symplectic deformation equivalence classes of all minimal symplectic fillings of any given lens space equipped with its canonical contact structure, as a graded, directed, rooted, and connected graph, where the root is the minimal resolution of the corresponding cyclic quotient singularity and each directed edge is a symplectic rational blowdown along an explicit linear plumbing graph. Moreover, we provide an upper bound for the rational blowdown depth of each minimal symplectic filling.

💡 Research Summary

This paper provides a groundbreaking combinatorial framework for organizing and understanding the entire set of minimal symplectic fillings of a lens space L(p,q) equipped with its canonical contact structure ξ_can. The authors construct a graded, directed, rooted, and connected graph—the “rational blowdown graph”—that captures how all these fillings are related through sequences of symplectic rational blowdown surgeries.

The work builds upon Lisca’s classification of such fillings, which are parameterized by a set Z_k(p/(p-q)) of integer k-tuples, where k is related to the Hirzebruch-Jung continued fraction expansion of p/(p-q). A key insight, originally due to Stevens, is that this parameter set can be bijectively identified with the set of triangulations T(P_{k+1}) of a convex (k+1)-gon. The authors’ major contribution is to reinterpret the complex geometric process of obtaining a filling from the “minimal resolution” (a canonical filling) via “symplectic rational blowdowns along linear plumbing graphs” in terms of simple combinatorial moves on these triangulations.

They establish that performing a “lantern substitution” in the monodromy factorization of a planar Lefschetz fibration (which corresponds to a rational blowdown) is equivalent to a sequence of “diagonal flips” on the corresponding triangulation. Specifically, a flip along a “distinguished diagonal” (an edge emanating from a fixed vertex V_⋆) increases a natural “height” invariant.

Using this correspondence, the authors define the rational blowdown graph G_{k}^{p,q} (Theorem 1):

• Vertices correspond to a specific subset T_{p,q}(P_{k+1}) of triangulations, each representing a symplectic deformation equivalence class of a minimal filling.
• The root vertex is the “initial triangulation,” which uses all distinguished diagonals and represents the minimal resolution filling.
• A directed edge from one vertex to another represents a specific sequence of diagonal flips. Crucially, each such sequence corresponds to a single symplectic rational blowdown along an explicit linear plumbing graph.
• The graph is graded by the second Betti number (b_2) of the filling. The root has the highest b_2, and traversing a directed edge decreases the grade by the number of flips composing that edge.

This graph offers a complete visual roadmap, showing all possible ways to construct any filling from the minimal resolution via rational blowdowns. Different paths in the graph from the root to a vertex represent distinct sequences of blowdowns yielding the same filling.

Furthermore, the paper introduces the concept of the “rational blowdown depth” of a filling W_{p,q}(n): the minimum number of successive rational blowdowns required to obtain it from the minimal resolution. Defining the combinatorial depth dpt(n) of its parameter tuple n as the number of 1’s in its interior entries, the authors prove that the rational blowdown depth of W_{p,q}(n) is at most dpt(n) (Proposition 5). This implies, for instance, that fillings with dpt(n)=1 are obtained by a single blowdown. They conjecture this upper bound is exact (Conjecture 6) and provide supporting examples of fillings with depth 2 and 3.

Finally, the authors note that, due to a known correspondence between Lisca’s fillings and Milnor fibres of cyclic quotient singularities established by Némethi and Popescu-Pampu, all results hold verbatim when “minimal symplectic fillings” are replaced by “Milnor fibres” and the “minimal resolution” is replaced by the “Milnor fibre of the Artin smoothing component” (Corollary 7).

In summary, this work translates a deep geometric classification problem into an elegant combinatorial language, providing both a powerful structural overview and new quantitative insights into the space of symplectic fillings of lens spaces.