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Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

Luis Crespo, Álvaro Pelayo, Francisco Santos

Abstract

We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the $2$-dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.

Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

Abstract

We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the -dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.
Paper Structure (20 sections, 45 theorems, 97 equations, 6 figures, 5 tables)

This paper contains 20 sections, 45 theorems, 97 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries on monotone polytopes
  4. Monotone, i.e. smooth reflexive, polytopes
  5. Three Ewald conditions and their relation to toric symplectic geometry
  6. Deeply smooth polytopes, face displacements, and the Ewald conditions
  7. Face displacements in smooth polytopes
  8. Deeply smooth polytopes satisfy the Ewald conditions
  9. Fiber bundles and neat polytopes
  10. Fiber bundles
  11. Neat polytopes and Ewald's Conjecture
  12. Simplex-segment bundles
  13. The number of Ewald points
  14. The maximum number of Ewald points
  15. Computation of the number of Ewald points
  16. ...and 5 more sections

Key Result

Corollary 1.4

If the momentum polytope of a monotone sympletic toric manifold satisfies the star Ewald condition then the central fiber is a stem.

Figures (6)

  • Figure 1: The five $2$-dimensional monotone polygons. We call them the monotone triangle, trapezoid, square, pentagon, and hexagon, respectively
  • Figure 2: The first two figures show unimodular bases $\mathcal{B}_1$ and $\mathcal{B}_2$ (yellow points) of two hyperplanes (blue), as obtained in the proof of Lemma \ref{['lemma:UT-free']} for the case of the $3$-cube, where $F$ is the facet pointing forward. The third figure shows the resulting unimodular basis of $\mathbb{Z}^3$.
  • Figure 3: The only two non-UT-free $3$-dimensional monotone polytopes.
  • Figure 4: Two monotone bundles. The first has a segment as base and a square as fiber. The second has a hexagon as base and a segment as fiber.
  • Figure 5: From left to right: $\textup{SSB}(3,0)$, $\textup{SSB}(3,1)$, $\textup{SSB}(3,2)$.
  • ...and 1 more figures

Theorems & Definitions (110)

  • Definition 1.1: Ewald set
  • Conjecture 1.2: Ewald's Conjecture 1988 Ewald
  • Remark 1.3
  • Corollary 1.4: McDuff McDuff-probes
  • Conjecture 1.5: General Ewald's Conjecture, Nill 2009 cctv
  • Theorem 2.1: Theorem \ref{['thm:deep-ewald']}
  • Definition 2.2: Neat polytope
  • Theorem 2.3: Corollary \ref{['cor:pmsym']}
  • Corollary 2.4
  • Conjecture 2.5: Oda, related to problems 1, 3, 4, 6 in Oda1997
  • ...and 100 more