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Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces

Julian Chaidez, Ben Wormleighton

Abstract

We develop new methods of both constructing and obstructing symplectic embeddings into non-toric rational surfaces using the theory of Newton-Okoukov bodies. Applications include sharp embedding results for concave toric domains into non-toric rational surfaces, and new cases of non-existence for infinite staircases in the non-toric setting.

Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces

Abstract

We develop new methods of both constructing and obstructing symplectic embeddings into non-toric rational surfaces using the theory of Newton-Okoukov bodies. Applications include sharp embedding results for concave toric domains into non-toric rational surfaces, and new cases of non-existence for infinite staircases in the non-toric setting.
Paper Structure (37 sections, 36 theorems, 156 equations, 6 figures)

This paper contains 37 sections, 36 theorems, 156 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Newton--Okounkov bodies
  3. Symplectic embeddings via Newton--Okounkov bodies
  4. Algebraic capacities via Newton--Okounkov bodies
  5. Applications
  6. Computation of algebraic capacities
  7. Concave to non-toric embeddings
  8. Obstructing staircases
  9. Asymptotics via Zariski decomposition
  10. Future directions
  11. Algebraic preliminaries
  12. Birational geometry
  13. Newton--Okounkov bodies
  14. Zariski decomposition
  15. Zariski chambers
  16. ...and 22 more sections

Key Result

Proposition 2.11

Let $Y$ be a toric variety with torus-invariant ample divisor $A$ and moment polytope $\Omega$. Then for any torus-invariant flag $Y_\bullet$ we have

Figures (6)

  • Figure 1: Transferring embeddings from ${X_{\frac{1}{m}\Delta_m}}^\circ$ to $(Y,\omega_A)$
  • Figure 2: Concave and convex moment domains
  • Figure 3: Weight decompositions
  • Figure 4: Newton--Okounkov body for del Pezzo $5$
  • Figure 5: Weight sequence decomposition for $\Delta$
  • ...and 1 more figures

Theorems & Definitions (106)

  • proof
  • Remark 1.1
  • Remark 1.2: Abelian surfaces
  • Remark 1.3: Other connections
  • Remark 1.4: Limitations
  • Example 2.1
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 96 more