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Concave symplectic toric fillings

Aleksandra Marinković

TL;DR

The paper proves that every contact toric 3-manifold admits infinitely many concave symplectic toric fillings that are mutually not equivariantly symplectomorphic and not related by blow ups. It achieves this by constructing toric structures on linear and cyclic plumbings over spheres, using moment-map data and $SL(2,\mathbb Z)$-equivalence to distinguish fillings, and by separating the analysis into non-free and free toric actions. For non-free actions, distinct cases determined by the angle between moment-cone rays yield infinite families of linear plumbings (and sometimes cyclic ones) with distinct self-intersection data; for free actions, cyclic plumbings are built from elongated linear ones, producing fillings of $(T^3,\xi_N)$ with distinct toric data. The results unify toric and contact-topological methods to significantly broaden the known landscape of concave toric fillings, with concrete consequences for lens spaces and the 3-torus.

Abstract

As shown by Etnyre and Honda in [EH], every contact 3-manifold admits infinitely many concave symplectic fillings that are mutually not symplectomorphic and not related by blow ups. In this note we refine this result in the toric setting by showing that every contact toric 3-manifold admits infinitely many concave symplectic toric fillings that are mutually not equivariantly symplectomorphic and not related by blow ups. The concave symplectic toric structure is constructed on certain linear and cyclic plumbings over spheres.

Concave symplectic toric fillings

TL;DR

The paper proves that every contact toric 3-manifold admits infinitely many concave symplectic toric fillings that are mutually not equivariantly symplectomorphic and not related by blow ups. It achieves this by constructing toric structures on linear and cyclic plumbings over spheres, using moment-map data and -equivalence to distinguish fillings, and by separating the analysis into non-free and free toric actions. For non-free actions, distinct cases determined by the angle between moment-cone rays yield infinite families of linear plumbings (and sometimes cyclic ones) with distinct self-intersection data; for free actions, cyclic plumbings are built from elongated linear ones, producing fillings of with distinct toric data. The results unify toric and contact-topological methods to significantly broaden the known landscape of concave toric fillings, with concrete consequences for lens spaces and the 3-torus.

Abstract

As shown by Etnyre and Honda in [EH], every contact 3-manifold admits infinitely many concave symplectic fillings that are mutually not symplectomorphic and not related by blow ups. In this note we refine this result in the toric setting by showing that every contact toric 3-manifold admits infinitely many concave symplectic toric fillings that are mutually not equivariantly symplectomorphic and not related by blow ups. The concave symplectic toric structure is constructed on certain linear and cyclic plumbings over spheres.

Paper Structure

This paper contains 9 sections, 2 theorems, 9 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on contact toric manifolds
  3. Contact 3-manifolds with a non-free toric action
  4. Contact 3-manifolds with a free toric action
  5. A symplectic toric structure on linear and cyclic plumbings
  6. Cyclic plumbings
  7. Proof of Theorem \ref{['thm1']}
  8. Contact 3-manifolds with a non-free toric action
  9. Contact 3-manifolds with a free toric action

Key Result

Corollary 1.2

There are three contact structures on any lens space $L(k, l)$, a universally tight one, half- and full-Lutz twist of the universally tight one, that admit a concave symplectic filling by infinitely many distinct linear plumbings over spheres. All tight contact structures on $T^3$ admit a concave sy

Figures (10)

  • Figure 1: Linear and cyclic plumbing graphs.
  • Figure 2: A moment cone of a contact toric 3-manifold.
  • Figure 3: Tight contact toric structure $\xi_t$ on $L(k,l)$ (left) and $S^1\times S^2$ (right).
  • Figure 4: Overtwisted contact toric structure $\xi_{ot1}$.
  • Figure 5: Overtwisted contact toric structure $\xi_{ot2}$.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Corollary 1.2
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • Remark 3.5
  • Remark 4.1