An-type surface singularity and nondisplaceable Lagrangian tori

Sun, Yuhan

doi:10.1142/s0129167x20500202

Cited by 4 publications

(1 citation statement)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Vianna [31] also showed a continuum of nondisplaceable tori in (CP 1 ) 2n . Sun [30] found nondisplaceable Lagrangian tori near a chain of Lagrangian 2-spheres in a closed symplectic 4-manifold and del Pezzo surfaces.…”

Section: Remark 14mentioning

confidence: 99%

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Cho

Kim

2021

Kyoto J. Math.

View full text Add to dashboard Cite

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya-Oh-Ohta-Ono's bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n ≥ 3) with a monotone Kirillov-Kostant-Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S 3 -fiber in Fl( 3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

show abstract

Section: Remark 14mentioning

confidence: 99%

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Cho

Kim

2021

Kyoto J. Math.

View full text Add to dashboard Cite

show abstract

Hofer geometry via toric degeneration

Kawamoto

2023

Math. Ann.

View full text Add to dashboard Cite

The main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for $$n=2$$ n = 2 , the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to $$C^0$$ C 0 symplectic topology.

show abstract

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Kawamoto

2024

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

We study the relation between isolated hypersurface singularities (e.g., ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m {A_{m}} -singularity, (2) if the symplectic manifold contains an A m {A_{m}} -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.

show abstract

An-type surface singularity and nondisplaceable Lagrangian tori

Cited by 4 publications

References 41 publications

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Hofer geometry via toric degeneration

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Contact Info

Product

Resources

About