arXiv subjects

Laura Starkston

Publications and source records attributed to Laura Starkston.

At least 19 recordsLinked to original sources

arXiv2025

An Unexpected Rational Blowdown

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-Szab\'o--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.

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arXiv2025

Sandwiched singularities and nearly Lefschetz fibrations

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.

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arXiv2025

Contact 3-manifolds that admit a non-free toric action

We classify contact toric 3-manifolds up to contactomorphism, through explicit descriptions, building off of work by Lerman [Lerman03]. As an application, we classify all contact structures on 3-manifolds that can be realised as a concave boundary of linear plumbing over spheres. The later result is inspired by the work [MNRSTW25].

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arXiv2025

Properties of contact toric structures and concave boundaries of linear plumbings

We consider plumbings of symplectic disk bundles over spheres admitting concave contact boundary, with the goal of understanding the geometric properties of the boundary contact structure in terms of the data of the plumbing. We focus on the linear plumbing case in this article. We study the properties of the contact structure using two different sets of tools. First, we prove that all such contact manifolds have a global contact toric structure, and use tools from toric geometry to identify when the contact structure is tight versus overtwisted. Second, we study algebraic torsion measurements from embedded contact homology (ECH) for these concavely induced contact manifolds, which has largely been unexplored. We develop a toolkit establishing existence and constraints of pseudoholomorphic curves adapted to the Morse-Bott Reeb dynamics of these plumbing examples, to provide the ECH algebraic torsion and contact invariant calculations for the concave boundaries of linear plumbings.

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arXiv2024

The contact cut graph and a Weinstein $\mathcal{L}$-invariant

We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein $\mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's $\mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein $\mathcal{L}$-invariant. We present topological and geometric constraints of Weinstein domains with $\mathcal{L}=0$. We also give two families of examples of multisections with divides that have arbitrarily large $\mathcal{L}$-invariant.

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arXiv2023

Multisections with divides and Weinstein 4-manifolds

We show how to encode a Weinstein 4-manifold using a multisection diagram with divides, which is a sequence of cut systems on a surface, together with a separating collection of curves. We give two algorithms to construct a multisection diagram with divides of a given Weinstein manifold, one starting with a Kirby-Weinstein handle decomposition and one starting with a positive, allowable Lefschetz fibration (PALF). Through the connections with PALFs, we define a monodromy of a multisection and show how to symplectically carry out monodromy substitution on multisections with divides.

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arXiv2022

Weinstein handlebodies for complements of smoothed toric divisors

We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centered condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface $F$, where the 2-handles are attached along the co-oriented conormal lifts of curves on $F$. We discuss how to use these diagrams to calculate invariants and provide numerous examples applying this procedure. For example, we provide Weinstein handlebody diagrams for the complements of the smooth and nodal cubics in $\mathbb{CP}^2$.

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arXiv2021

The symplectic isotopy problem for rational cuspidal curves

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to 5, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from 4-dimensional topology.

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arXiv2021

Rational cuspidal curves and symplectic fillings

A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no symplectic fillings, and further examples where the fillings can be fully classified.

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arXiv2021

Symplectic 4-manifolds admit Weinstein trisections

We prove that every symplectic 4-manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in $\mathbb{CP}^2$ can be symplectically isotoped into bridge position.

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arXiv2021

An introduction to Weinstein handlebodies for complements of smoothed toric divisors

In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we include one complicated example at the end of the article to showcase the algorithm and the types of Weinstein Kirby diagrams it produces.

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arXiv2020

Unexpected Stein fillings, rational surface singularities, and plane curve arrangements

We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong-van Straten in [dJvS98]; they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong-van Straten's construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most -5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).

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arXiv2019

Arboreal Singularities in Weinstein Skeleta

We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler's program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse-Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its symplectic neighborhood. We then study the singularities of the skeleta in this class and show that after a certain type of generic perturbation either (1) these singularities fall into the class of (signed Lagrangian versions of) Nadler's arboreal singularities which are combinatorially classified into finitely many types in a given dimension or (2) there are singularities of tangency in associated front projections. We then turn to the singularities of tangency to try to reduce them also to collections of arboreal singularities. We give a general localization procedure to isolate the Liouville flow to a neighborhood of these non-arboreal singularities, and then show how to replace the simplest singularities of tangency (those of type $Σ^{1,0}$) by arboreal singularities.

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arXiv2019

A new approach to the symplectic isotopy problem

The symplectic isotopy conjecture states that every smooth symplectic surface in $CP^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov's pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian's proof of the conjecture up to degree 17 [ST05], but further progress has stalled. In this article we provide a new direction of attack on this problem. Using a solution to a nodal symplectic isotopy problem we guide model symplectic isotopies of smooth surfaces. This results in an equivalence between the smooth symplectic isotopy problem and an existence problem of certain embedded Lagrangian disks. This redirects study of this problem from the realm of pseudoholomorphic curves of high genus to the realm of Lagrangians and Floer theory. Because the main theorem is an equivalence going both directions, it could theoretically be used to either prove or disprove the symplectic isotopy conjecture.

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arXiv2016

Topological realizations of line arrangements

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of $2$-spheres embedded, in the symplectic, smooth, or topological (locally flat) categories, in the complex projective plane. We find obstructions to realizability in the topological category, which apply to configurations specified by all projective planes over a finite field. Such obstructions are used to show that certain contact graph manifolds are not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, $2$-spheres.

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arXiv2016

Comparing star surgery to rational blow-down

We compare the star surgery operations introduced in [KS] to the generalized rational blow-down. We show that star surgery shares the properties that make rational blow-down useful for constructions of small exotic symplectic 4-manifolds. Then we show that star surgery operations provide a strictly more general class of operations by proving that there is an infinite family of star surgeries which are inequivalent to any sequence of generalized symplectic rational blow-downs. This answers a question posed to the author by Ozbagci. It also demonstrates that the monodromy substitutions coming from star surgery operations yield relations in planar mapping class monoids which cannot be positively generated by the relations determined in [EMVHM11] which come from the generalized rational blow-downs.

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arXiv2015

Surgery along star shaped plumbings and exotic smooth structures on 4-manifolds

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times.

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arXiv2013

Symplectic fillings of Seifert fibered spaces

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbing of spheres. In other cases, we produce new manifolds with convex symplectic boundary, thus yielding new cut-and-paste operations on symplectic manifolds containing certain configurations of symplectic spheres.

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