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On the embedding complexity of Liouville manifolds

Sheel Ganatra, Kyler Siegel

Abstract

We define a family of symplectic invariants which obstruct exact symplectic embeddings between Liouville manifolds, using the general formalism of linearized contact homology and its L-infinity structure. As our primary application, we investigate embeddings between normal crossing divisor complements in complex projective space, giving a complete characterization in many cases. Our main embedding results are deduced explicitly from pseudoholomorphic curves, without appealing to Hamiltonian or virtual perturbations.

On the embedding complexity of Liouville manifolds

Abstract

We define a family of symplectic invariants which obstruct exact symplectic embeddings between Liouville manifolds, using the general formalism of linearized contact homology and its L-infinity structure. As our primary application, we investigate embeddings between normal crossing divisor complements in complex projective space, giving a complete characterization in many cases. Our main embedding results are deduced explicitly from pseudoholomorphic curves, without appealing to Hamiltonian or virtual perturbations.

Paper Structure

This paper contains 35 sections, 34 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Main results
  4. Obstructions
  5. Constructions
  6. Setting the stage
  7. Geometric preliminaries
  8. Contact manifolds and symplectizations
  9. Flavors of open symplectic manifolds
  10. Embeddings
  11. The Conley--Zehnder index
  12. SFT moduli spaces
  13. Admissible almost complex structures
  14. SFT moduli spaces
  15. SFT compactness and neck stretching
  16. ...and 20 more sections

Key Result

Theorem 1.5

Fix $n \in \mathbb{Z}_{\geq 1}$ and tuples $\vec{d} = (d_1,\dots,d_k) \in \mathbb{Z}_{\geq 1}^k$ and $\vec{d'} = (d_1',\dots,d_{k'}') \in \mathbb{Z}_{\geq 1}^{k'}$ with $\sum\limits_{i=1}^kd_i,\sum\limits_{i=1}^{k'}d_i' \geq n+1$. Assume that we have $\sum\limits_{i=1}^{k'} d_i' < 2\sum\limits_{i=1}

Theorems & Definitions (83)

  • Theorem 1.5
  • Corollary 1.6
  • Example 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10: on virtual perturbations
  • Theorem 1.11
  • Example 1.12
  • Example 1.13
  • Definition 1.14
  • ...and 73 more