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KIAS Lectures on Symplectic Aspects of Degenerations

Jonathan David Evans

Abstract

This is a series of three lectures I gave at the Korea Institute of Advanced Study in June 2019 at a workshop about "Algebraic and Symplectic Aspects of Degenerations of Complex Surfaces". I focus on the symplectic aspects, in particular on the case of cyclic quotient surface singularities. These notes have been available on a public Git repository since 2019, and I noticed that people occasionally cited them in the years since. For that reason, I decided to post them on arXiv for a more permanent record; I have made some small corrections and annotations but otherwise they are unchanged. These notes are a purely expository account of stuff I was thinking about 2016-2019, and are largely self-aggrandising.

KIAS Lectures on Symplectic Aspects of Degenerations

Abstract

This is a series of three lectures I gave at the Korea Institute of Advanced Study in June 2019 at a workshop about "Algebraic and Symplectic Aspects of Degenerations of Complex Surfaces". I focus on the symplectic aspects, in particular on the case of cyclic quotient surface singularities. These notes have been available on a public Git repository since 2019, and I noticed that people occasionally cited them in the years since. For that reason, I decided to post them on arXiv for a more permanent record; I have made some small corrections and annotations but otherwise they are unchanged. These notes are a purely expository account of stuff I was thinking about 2016-2019, and are largely self-aggrandising.
Paper Structure (29 sections, 14 theorems, 27 equations)

This paper contains 29 sections, 14 theorems, 27 equations.

Table of Contents

  1. Lecture 1
  2. Hunting for singularities
  3. Degenerations and symplectic parallel transport
  4. Signs left by singularities
  5. Cyclic quotient singularities
  6. The link
  7. Milnor fibre
  8. Vanishing cycle
  9. Lecture 2
  10. Toric and almost toric pictures
  11. Toric varieties
  12. Lagrangian torus fibrations
  13. Action coordinates
  14. Mutation
  15. Smoothings of other cyclic quotient singularities
  16. ...and 14 more sections

Key Result

Lemma 1.1

The smooth fibres are all symplectomorphic. More precisely, given a path $\gamma\colon[0,1]\to\Delta$ avoiding $0\in\Delta$, there is a diffeomorphism $\phi_t\colon X_{\gamma(0)}\to X_{\gamma(t)}$ for all $t\in[0,1]$ such that $\phi_t^*\omega_{\gamma(t)}=\omega_{\gamma(0)}$.

Theorems & Definitions (65)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Definition 1.3: Vanishing cycle
  • Remark 1.4
  • Definition 1.5: Link
  • Remark 1.6
  • Lemma 1.7
  • proof
  • ...and 55 more