Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Affiliation

• Hokkaido University, Faculty of Science, Department of Mathematics, Associate Professor

Degree

• Ph.D. (Mathematical Sciences)（2014/03 The University of Tokyo）

Profile and Settings

• Name (Japanese)

  Kasuya

• Name (Kana)

  Naohiko

• Name

  201601003743631849

Affiliation

• Hokkaido University, Faculty of Science, Department of Mathematics, Associate Professor

Achievement

Research Interests

• Embeddings   Complex structures   Contact structures   Differential topology   

Research Areas

• Natural sciences / Geometry / Differential Topology

Research Experience

• 2017/04 - 2021/03 Kyoto Sangyo University Faculty of Science Department of Mathematics Associate professor
• 2015/04 - 2017/03 Aoyama Gakuin University School of Social Informatics Assistant professor
• 2014/04 - 2015/03 The University of Tokyo Graduate School of Mathematical Sciences
• 2012/04 - 2014/03 JSPS Research Fellowships for Young Scientists DC2

Education

• 2011/04 - 2014/03  The University of Tokyo  Graduate School of Mathematical Sciences Doctoral Program
• 2009/04 - 2011/03  The University of Tokyo  Graduate School of Mathematical Sciences
• 2007/04 - 2009/03  The University of Tokyo  Faculty of Science  Department of Mathematics
• 2005/04 - 2007/03  The University of Tokyo  College of Arts and Sciences

Published Papers

• Classification of orientable torus bundles over closed orientable surfaces

  Naohiko Kasuya, Issei Noda

  arXiv: 2406.14138 2024/06
• On the strongly pseudoconcave boundary of a compact complex surface

  Naohiko Kasuya, Daniele Zuddas

  Proceedings of the American Mathematical Society 152 (2) 709 - 723 2024/02 [Refereed][Not invited]
  
• A concave holomorphic filling of an overtwisted contact 3-sphere

  Naohiko Kasuya, Daniele Zuddas

  Algebraic & Geometric Topology 23 (5) 2141 - 2156 2023/07/25 [Refereed][Not invited]
  
• Lefschetz fibrations on the Milnor fibers of cusp and simple elliptic singularities

  Naohiko Kasuya, Hiroki Kodama, Yoshihiko Mitsumatsu, Atsuhide Mori

  arXiv: 2111.00749 2021/11
• On the deformation of the exceptional unimodal singularities

  Naohiko Kasuya, Atsuhide Mori

  Journal of Singularities 23 1 - 14 2021/02/04 [Refereed][Not invited]
  
• CR regular embeddings of $S^{4n-1}$ in $\mathbb{C}^{2n+1}$

  Naohiko Kasuya

  Proceedings of the American Mathematical Society 148 (7) 3021 - 3024 2020/03/17 [Refereed][Not invited]
  
• Erratum: Generic immersions and totally real embeddings

  Naohiko Kasuya, Masamichi Takase

  International Journal of Mathematics 30 (12) 1992003  0129-167X 2019/11 [Refereed][Not invited]
   

  This note corrects an error in Theorem 5.2(c) of our paper "Generic immersions and totally real embeddings".
• Non-Kähler complex structures on $\mathbb{R}^4$, II

  Antonio Di Scala, Naohiko Kasuya, Daniele Zuddas

  Journal of Symplectic Geometry 16 (3) 631 - 644 2018/11/26 [Refereed][Not invited]
  
• Generic immersions and totally real embeddings

  Naohiko Kasuya, Masamichi Takase

  International Journal of Mathematics 29 (11) 1850073  0129-167X 2018/11 [Refereed][Not invited]
   

  We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n - 1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k + 1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.
• Knots and links of complex tangents

  Naohiko Kasuya, Masamichi Takase

  Transactions of the American Mathematical Society 370 (3) 2023 - 2038 0002-9947 2018 [Refereed][Not invited]
   

  It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the 3-dimensional complex space. We show in fact that a 1-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the 3-dimensional complex space if and only if it represents the trivial integral homology class in the 3-manifold. The proof involves a new application of singularity theory of differentiable maps.
• Non-Kahler complex structures on R-4

  Antonio J. Di Scala, Naohiko Kasuya, Daniele Zuddas

  GEOMETRY & TOPOLOGY 21 (4) 2461 - 2473 1465-3060 2017/05/19 [Refereed][Not invited]
   

  We construct the first examples of non-Kahler complex structures on R-4. These complex surfaces have some analogies with the complex structures constructed in the early fifties by Calabi and Eckmann on the products of two odd-dimensional spheres. However, our construction is quite different from that of Calabi and Eckmann.
• An obstruction for codimension two contact embeddings in the odd dimensional Euclidean spaces

  Naohiko Kasuya

  JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 68 (2) 737 - 743 0025-5645 2016/04 [Refereed][Not invited]
   

  We prove that the first Chern class of a codimension two closed contact submanifold of the odd dimensional Euclidean space is trivial. For any closed co-oriented contact 3-manifold with trivial Chern class, we prove that there is a contact structure on the 5-dimensional Euclidean space which admits a contact embedding of it.
• On embeddings of almost complex manifolds in almost complex Euclidean spaces

  Antonio J. Di Scala, Naohiko Kasuya, Daniele Zuddas

  JOURNAL OF GEOMETRY AND PHYSICS 101 19 - 26 0393-0440 2016/03 [Refereed][Not invited]
   

  We prove that any compact almost complex manifold (M-2m, J) of real dimension 2m admits a pseudo-holomorphic embedding in (R4m+2, (J) over tilde) for a suitable positive almost complex structure (J) over tilde. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class s(m) (M, J), for the existence of an embedding or an immersion in (R-4m, (J) over tilde). We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in (R-6, (J) over tilde). (C) 2015 Elsevier B.V. All rights reserved.
• On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

  Naohiko Kasuya

  Demonstratio Mathematica 48 (2) 0420-1213 2015/06/01 [Refereed][Not invited]
   

  This is a survey article about the study of the links of some complex hypersurface singularities in ℂ
• On contact embeddings of contact manifolds in the odd dimensional Euclidean spaces

  Naohiko Kasuya

  INTERNATIONAL JOURNAL OF MATHEMATICS 26 (7) 1550045  0129-167X 2015/06 [Refereed][Not invited]
   

  We prove that a closed co-oriented contact (2m + 1)-manifold (M2m+1, xi) can be a contact submanifold of the standard contact structure on R4m+1, if it satisfies one of the following conditions: (1) m is odd (m >= 3) and H-1 (M2m+1; Z) = 0, (2) m is even (m = 4) and M2m+1 is 2-connected, (3) m = 2 and M-5 is simply-connected.
• The Canonical Contact Structure on the Link of a Cusp Singularity

  Naohiko Kasuya

  TOKYO JOURNAL OF MATHEMATICS 37 (1) 1 - 20 0387-3870 2014/06 [Refereed][Not invited]
   

  Caubel, Nemethi, and Popescu-Pampu in [2] proved that an oriented 3-manifold admits at most one positive contact structure which can be realized as the complex tangency along the link of a complex surface singularity. They call it the Milnor fillable contact structure. Lekili and Ozbagci in [10] showed that a Milnor finable contact structure is universally tight. In particular, by Honda's classification [5], the link of a cusp singularity is contactomorphic to the positive contact structure associated to the Anosov flow on a Sol-manifold (see [1]). We describe the contact structure on the link of a cusp singularity in two different ways without using Honda's classification theorem. One description is based on the toric method introduced in Mori [15]. The other description is based on Hirzebruch's construction of the Hilbert modular cusps. Consequently, we give certain answers to the problems in Mori [14] concerning the relation between the cusp singularities and the simple elliptic singularities, and the higher dimensional extension of the local Lutz-Mori twist.
• ON LAGRANGIAN EMBEDDINGS OF PARALLELIZABLE MANIFOLDS

  Naohiko Kasuya, Toru Yoshiyasu

  INTERNATIONAL JOURNAL OF MATHEMATICS 24 (9) 1350073  0129-167X 2013/08 [Refereed][Not invited]
   

  We prove that for any closed parallelizable n-manifold M-n, if the dimension n not equal 7, or if n = 7 and the Kervaire semi-characteristic chi(1/2) (M-7) is zero, then M-n can be embedded in the Euclidean space R-2n with a certain symplectic structure as a Lagrangian submanifold. By the results of Gromov and Fukaya, our result gives rise to symplectic structures of R-2n(n >= 3) which are not conformally equivalent to open domains in standard ones.

Research Projects

• 力学的微分トポロジーによる葉層・接触・シンプレクティック構造の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2021/04 -2026/03 

  Author : 三松 佳彦, 直江 央寛, 高倉 樹, 太田 啓史, 三好 重明, 粕谷 直彦
• 強擬凹複素曲面とその境界に現れる接触構造

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2021/04 -2025/03 

  Author : 粕谷 直彦

   

  研究課題(17K14193)において、強擬凹曲面への正則ハンドルの接着法を確立することにより「任意の3次元閉多様体はケーラーな強擬凹曲面の境界として実現可能である」という結果を得た。本研究課題ではその続きとして「強擬凹曲面上のケーラー形式と境界との相性がよい場合、境界上の接触構造はtightか？」という問題に取り組んでいる。現在のところ、以下の2つのアプローチを試みている。一つ目は、強擬凹曲面について何らかの意味で小平の埋め込み定理の類似が成り立てば、そこから境界のfillabilityが示され、特に接触構造がtightであることが示されるだろうというものである。しかし、強擬凹曲面上ではコンパクト複素曲面におけるホッジ理論に相当するものがないため、この方針はまだ糸口が見えない。何とか状況を打開するために、小平の複素曲面論や多変数関数論におけるL^2理論をよく理解しようと論文を勉強している段階である。二つ目の方針は、ハンドル接着によって具体的に反例を構成するというものである。つまり、境界との相性がよいケーラー形式を持ちかつ境界がovertwistedである強擬凹曲面を作るという方針である。現段階では、こちらの方が有望ではないかと考えており、contact (+1)-surgeryに相当するハンドル接着のみを使うことが鍵となるだろうと予想している。
• ケーラーでない開複素多様体の幾何と4次元トポロジー

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2017/04 -2023/03 

  Author : 粕谷 直彦

   

  本研究課題の目標の一つである「任意の3次元閉接触多様体は強擬凹複素曲面の境界として実現可能か？」という問いを肯定的に解決し、さらに接触多様体を充填する複素曲面はケーラーにも非ケーラーにもとれることを証明した。この結果は2002年にEtnyre-Hondaによって示された「任意の3次元閉接触多様体はconcave symplectic fillingを許容する」という定理のholomorphic versionと解釈することができる。また、任意の2つの3次元閉接触多様体を複素コボルディズムでつなげること、さらにそのコボルディズムはケーラーにとれることを証明した。ただし、この場合のケーラー構造は境界の接触構造との相性が悪く、得られる複素コボルディズムは必ずしもシンプレクティックコボルディズムとは限らない。 今回我々がとった手法は、研究実施計画に記した通り、Eliashbergのハンドル接着によるシュタイン多様体の構成法を参考にしたものである。Eliashbergの方法では、強擬凸境界上のルジャンドル結び目に沿って、ラグランジュ円板をcoreとする正則ハンドルを接着するが、我々の方法では、強擬凹境界上の横断的結び目に沿って、正則円板をcoreとする正則ハンドルを接着する。この違いが我々のアイディアの本質である。これらの内容は、Daniele Zuddas氏との共著論文としてまとめ、現在学術雑誌へ投稿中である。