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Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Yusuke Kawamoto

Abstract

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

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Abstract

We study the relation between isolated hypersurface singularities (e.g. ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic invariants coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety (or the symplectic manifold) degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an -singularity, (2) if the symplectic manifold contains an -configuration of Lagrangian spheres, then there are consequences on the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.
Paper Structure (19 sections, 16 theorems, 146 equations, 3 figures)

This paper contains 19 sections, 16 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Isolated hypersurface singularities and quantum cohomology
  4. $A_m$ singularities and Hofer geometry
  5. Dehn twist and spectral invariants
  6. Structure of the paper
  7. Acknowledgements
  8. Preliminaries
  9. Spectral invariant theory
  10. Entov--Polterovich quasimorphisms and (super)heaviness
  11. Semi-simplicity of the quantum cohomology ring
  12. Degeneration
  13. Isolated hypersurface singularities
  14. Proofs
  15. Spectral rigidity of Lagrangian spheres
  16. ...and 4 more sections

Key Result

Theorem A

Let $X$ be a complex $n$ dimensional smooth Fano variety. Assume $QH(X,\omega)$ is semi-simple, where $\omega$ is the anti-canonical form of $X$. If $X$ degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be

Figures (3)

  • Figure 1: Dynkin diagrams of type $A_{n},D_{n},E_6,E_7,E_8$.
  • Figure 2: $D_4$-configuration: The sphere $S$ corresponds to the sphere in the middle.
  • Figure 3: The sphere $S$ corresponds to, the sphere at the end of the straight line in the $D_n$ diagram, and the third sphere in the $E_6,E_7,E_8$ diagrams, respectively.

Theorems & Definitions (36)

  • Theorem A: Algebraic geometry version
  • Theorem B: Symplectic topology version
  • Corollary 1.2.2
  • Theorem C
  • Theorem 1.3.2: Kapovich--Polterovich question, Entov--Polterovich--Py question
  • Theorem D
  • Proposition 2.1.2: [BC08][LZ18][FOOO19]
  • Theorem 2.2.1: [EP03]
  • Definition 2.2.4: [EP09],[EP06]
  • Proposition 2.2.6: [EP09]
  • ...and 26 more