Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Embeddings of symplectic balls into the complex projective plane

Sílvia Anjos, Jarek Kędra, Martin Pinsonnault

Abstract

We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of $\mathrm{PGL}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$ embedded symplectic balls.

Embeddings of symplectic balls into the complex projective plane

Abstract

We investigate spaces of symplectic embeddings of balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of on the configuration space of ordered points in with the action of the symplectomorphism group on the space of embedded symplectic balls.
Paper Structure (45 sections, 43 theorems, 173 equations, 8 figures)

This paper contains 45 sections, 43 theorems, 173 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background.
  3. Main results.
  4. Organization of the paper.
  5. Acknowledgements.
  6. Symplectic balls in rational 4-manifolds
  7. General setting
  8. Configurations of balls and points
  9. Stability for embedding spaces
  10. Stability chambers for symplectic balls in CP2
  11. Stability chambers for $n=1$ or $n=2$ balls in ${\mathbf C}{\mathbf P}^2$
  12. Stability chambers for $n=3$ balls in ${\mathbf C}{\mathbf P}^2$
  13. Stability chambers for $n=4$ balls in ${\mathbf C}{\mathbf P}^2$
  14. Stability chambers for $n=5$ balls in ${\mathbf C}{\mathbf P}^2$
  15. Stratifications of Conf, J, A
  16. ...and 30 more sections

Key Result

Theorem 1.1

Consider ${\mathbf C}{\mathbf P}^2$ endowed with its standard Fubini-Study symplectic form and let $c_1,c_2,c_3 \in (0,1)$ be such that $c_i+c_j < 1$.

Figures (8)

  • Figure 1: Toric configurations for $n=1$ and $n=2$ balls.
  • Figure 2: Configurations for $n=3$ balls.
  • Figure 3: A configuration $S_0$ for $k=4$
  • Figure 4: Configurations of type $\mathcal{T}_{ijk}$ for $n=4$ balls.
  • Figure 5: Configurations of type $\mathcal{T}_{1234}$ for $n=4$ balls.
  • ...and 3 more figures

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • ...and 74 more