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Takahiro Oba

Publications and source records attributed to Takahiro Oba.

16 recordsLinked to original sources
arXiv2026

Every cusp singularity link admits infinitely many strong symplectic fillings

In this paper, we show that if the link of an isolated complex surface singularity is either a $Sol^3$-manifold or an $\widetilde{SL}(2;\mathbb{R})$-manifold with its canonical contact structure, then it admits infinitely many strong symplectic fillings that are pairwise non-diffeomorphic and not related by a sequence of blow-ups or blow-downs. As a consequence, the link of any cusp singularity, exceptional unimodal singularity, or hyperbolic Brieskorn singularity admits infinitely many pairwise non-diffeomorphic minimal strong symplectic fillings.

math.GT↗
arXiv2024

A note on Stein fillability of circle bundles over symplectic manifolds

We show that, given a closed integral symplectic manifold $(Σ, ω)$ of dimension $2n \geq 4$, for every integer $k>\int_Σω^{n}$, the Boothby-Wang bundle over $(Σ, kω)$ carries no Stein fillable contact structure. This negatively answers a question raised by Eliashberg. A similar result holds for Boothby-Wang orbibundles. As an application, we prove the non-smoothability of some isolated singularities.

math.GT↗
arXiv2024

On dynamically convex contact manifolds and filtered symplectic homology

In this paper we are interested in characterizing the standard contact sphere in terms of dynamically convex contact manifolds which admit a Liouville filling with vanishing symplectic homology. We first observe that if the filling is flexible, then those contact manifolds are contactomorphic to the standard contact sphere. We then investigate quantitative geometry of those contact manifolds focusing on similarities with the standard contact sphere in filtered symplectic homology.

math.SG↗
arXiv2024

Symplectic fillings of unit cotangent bundles of spheres and applications

We prove the uniqueness, up to diffeomorphism, of symplectically aspherical fillings of the unit cotangent bundle of odd-dimensional spheres. As applications, we first show the non-existence of exact symplectic cobordisms between some 5-dimensional Brieskorn manifolds. We also determine the diffeomorphism types of closed symplectic 6-manifolds with certain codimension 2 symplectic submanifolds.

math.SG↗
arXiv2023

Rational ruled surfaces as symplectic hyperplane sections

We study embeddability of rational ruled surfaces as symplectic hyperplane sections into closed integral symplectic manifolds. From this we obtain results on Stein fillability of Boothby--Wang bundles over rational ruled surfaces.

math.SG↗
arXiv2023

Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations

We provide a closed, simply connected, symplectic $6$-manifold having infinitely many codimension $2$ symplectic submanifolds. These are mutually homologous but homotopy inequivalent, and furthermore, they cannot admit complex structures. The key ingredient for the construction is hyperelliptic Lefschetz fibrations on $4$-manifolds. As a corollary, we present a similar result on symplectic submanifolds of codimension $2$ in higher dimensions. In the appendix, we give a proof of the well-known fact that all symplectic submanifolds of codimension $2$ in $(\mathbb{CP}^3, \omega_{\mathrm{FS}})$ of a fixed degree $\leq 3$ are mutually diffeomorphic.

math.SG↗
arXiv2021

A four-dimensional mapping class group relation

In the symplectic mapping class group of a $4$-dimensional Weinstein domain, we give a relation between two products of (right-handed) Dehn twists via holomorphic curve techniques. A key ingredient of the construction is a solution to the symplectic isotopy problem on symplectic submanifolds in del Pezzo surfaces. In the appendix, we provide an alternative proof of a relation between a fibered Dehn twist and a product of Dehn twists.

math.GT↗
arXiv2019

Lefschetz-Bott fibrations on line bundles over symplectic manifolds

We describe Lefschetz-Bott fibrations on complex line bundles over symplectic manifolds explicitly. As an application, we construct more than one strong symplectic filling of the link of the $A_{k}$-type singularity. In the appendix, we show that the total space of a Lefschetz-Bott fibration over the unit disk serves as a strong symplectic filling of a contact manifold compatible with an open book induced by the fibration.

math.GT↗
arXiv2017

Surfaces in $D^4$ with the same boundary and fundamental group

We construct a family of pairs of non-isotopic symplectic surfaces in the standard symplectic $4$-disk such that they are bounded by the same transverse knot in the standard contact $3$-sphere and fundamental groups of their complements are isomorphic. In the appendix, we prove explicitly that one can obtain a symplectic surface in the standard symplectic $4$-disk from a braided surface in a bidisk.

math.GT↗
arXiv2016

Planar Lefschetz fibrations and Stein structures with distinct Ozsvath-Szabo invariants on corks

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks.

math.GT↗
arXiv2016

Compact Stein surfaces as branched covers with same branch sets

Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$ with boundary admits a Stein structure if and only if $X$ is a simple branched cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$, we construct a braided surface $S_{N}$ in $D^4$ and simple branched covers $X_{N, 1}, X_{N, 2}, \dots , X_{N, N}$ of $D^{4}$ branched along $S_{N}$ such that the covers have the same degrees, and they are mutually diffeomorphic, but the Stein structures associated to the covers are mutually not homotopic. Furthermore, by reinterpreting this result in terms of contact topology, for each integer $N \geq 2$, we also construct a transverse link $L_{N}$ in the standard contact $3$-sphere $(S^3, ξ_{std})$ and simple branched covers $M_{N,1}, M_{N,2}, \ldots, M_{N, N}$ of $S^3$ branched along $L_{N}$ such that the covers have the same degrees, and they are mutually diffeomorphic, but the contact structures associated to the covers are mutually not isotopic.

math.GT↗
arXiv2014

Stein fillings of homology $3$-spheres and mapping class groups

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillings of a rational homology $3$-spheres into strongly symplectic fillings, we also show that a symplectically fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique symplectic filling up to diffeomorphism and blow-up.

math.SG↗