Workshop on Singularities and Symplectic Geometry

Venue

Room 201, Research building No. 15, Kyoto University

Speakers

• Hakho Choi (KIAS)
• Masahiro Futaki (Chiba University)
• Naohiko Kasuya (Kyoto Sanyo Univerisity)
• Myeonggi Kwon (University of Giessen)
• Ryo Yamagishi (Kavli IPMU)

Program

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January 8

13:30 - 14:30 Myeonggi Kwon I

15:00 - 16:00 Myeonggi Kwon II

16:30 - 17:30 Ryo Yamagishi I

January 9

9:30 - 10:30 Ryo Yamagishi II

11:00 - 12:00 Hakho Choi I

13:30 - 14:30 Hakho Choi II

15:00 - 16:00 Naohiko Kasuya I

16:30 - 17:30 Naohiko Kasuya II

January 10

13:30 - 14:30 Masahiro Futaki I

15:00 - 16:00 Masahiro Futaki II

Title and Abstract

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Myeonggi Kwon (University of Giessen)

Symplectic topology via Floer theory and singularities

For an isolated hypersurface singularity, its link admits a canonical contact structure, and its Milnor fiber serves as a natural symplectic filling of the link. This provides an interesting playground to explore relationship between symplectic topology and singularity theory. In this talk, we introduce various results on interactions between symplectic topology and singularity theory, in particular, in terms of Floer theory. Symplectic homology and wrapped Floer homology in Minor fibers and their applications will be discussed.

Ryo Yamagishi (Kavli IPMU)

Crepant resolutions of canonical singularities

In order to study singularity of a (complex) algebraic variety, it is natural to ask for nice resolutions of it. When the singularity is canonical in the sense of birational geometry, crepant resolutions should be candidates of such nice resolutions and indeed the geometry of crepant resolutions nicely reflect the properties of the given singularity in many cases.

In the talks I will introduce two particular classes of canonical singularities: quotient singularities and (holomorphic) symplectic singularities. Crepant resolutions of these singularities are interesting not only in algebraic geometry but also in representation theory. I will give a brief review of these topics.

Hakho Choi (KIAS)

On symplectic fillings of small Seifert 3-manifolds

One of the fundamental problems in symplectic 4-manifold topology is in classifying symplectic fillings of certain 3-manifolds equipped with a natural contact structure. If we get the classification result, then it is natural to ask that Is there any surgery description of those fillings.

In this talk, we discuss classification of minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we demonstrate that every minimal symplectic filling of small Seifert 3-manifolds satisfying the conditions can be obtained by a sequence of rational blowdowns from the minimal resolution. This is joint work with Jongil Park.

Naohiko Kasuya (Kyoto Sangyo University)

Topology of the Milnor fiber of a cusp singularity

Ebeling and Wall extended Arnold's strange duality to cusp singularities. The duality can be explained from the viewpoint of Inoue-Hirzebruch surfaces. Namely, there is a certain duality between the dual resolution graph of a cusp singularity and that of its dual cusp. It can be also explained by the link topology. The links of these singularities are hyperbolic mapping tori which are orientaion reversing diffeomorphic. Hence, we obtain a smooth 4-manifold by gluing the two Milnor fibers of dual cusps along thier boundaries. We show that the 4-manifold is a K3 surface by constructing a genus-one Lefschetz fibration from the Milnor fiber to the 2-disk. This is a joint work with Hiroki Kodama, Yoshihiko Mitsumatsu, and Atsuhide Mori.

Masahiro Futaki (Chiba University)

Introduction to the Fukaya-Seidel category and homological mirror symmetry for singularities

In this talk I will give an introductory survey on the Fukaya-Seidel category. It was first introduced by Hori-Vafa as the A-model category for Landau-Ginzburg models and then formulated by Seidel. Several known results will be reviewed.I will then demonstrate how the framework of Ganatra-Pardon-Shende can be applied to the calculations of the Fukaya-Seidel category of the potential f+y^n.

Organizer

Takahiro Oba (RIMS, Kyoto University) oba [at mark] kurims.kyoto-u.ac.jp

Support

Grant-in-Aid for JSPS Research Fellows: 18J01373