Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Weinstein domains in symplectic manifolds

Thomas E. Mark, Bülent Tosun

TL;DR

The paper establishes a symplectic analogue of rational convexity by proving that a Weinstein domain embedded in a closed symplectic 4-manifold forces the existence of symplectic hypersurfaces in its complement after contraction. Using this, it derives obstructions for a closed 3-manifold to bound a Weinstein domain in positive rational surfaces, yielding non-embeddability results for Brieskorn spheres in small rational surfaces like $S^2\times S^2$ or ${\mathbb CP}^2\#k\overline{\mathbb{CP}}^2$ with $k\le 7$, while showing that embeddings exist in some larger rational surfaces with constrained topology. The core mechanism combines Donaldson’s asymptotically holomorphic technique with Heegaard Floer theory, notably the reduced invariant $c_{red}(\xi)$, to produce the obstruction and to relate ambient positivity, divisors in the complement, and boundary contact invariants. The authors also extend the framework to non-integral symplectic classes and provide a broad suite of applications, including families of Brieskorn spheres with restricted embedding behavior and concrete examples arising from rational cuspidal curves and Mazur manifolds.

Abstract

We prove that a Weinstein domain symplectically embedded in a closed symplectic manifold always admits symplectic hypersurfaces in its complement, possibly after a deformation. As a consequence, we obtain an obstruction for a closed 3-dimensional manifold to arise as the boundary of a Weinstein domain in a class of symplectic 4-manifolds that includes many symplectic rational surfaces. A particular application is that no Brieskorn homology sphere bounds a Weinstein domain symplectically embedded in a rational surface diffeomorphic to $S^2\times S^2$ or to ${\mathbb C} P^2\# k \overline{{\mathbb C}P}^2$, for any $k\leq 7$, despite the fact that many Brieskorn spheres bound Stein domains holomorphically embedded in these rational surfaces. Several families of Brieskorn spheres are obtained that do not bound a Weinstein domain in any 4-manifold with a ``positive'' symplectic structure. Such Weinstein domains do exist in certain positive symplectic rational surfaces when $k\geq 8$, though their topology is significantly constrained.

On Weinstein domains in symplectic manifolds

TL;DR

The paper establishes a symplectic analogue of rational convexity by proving that a Weinstein domain embedded in a closed symplectic 4-manifold forces the existence of symplectic hypersurfaces in its complement after contraction. Using this, it derives obstructions for a closed 3-manifold to bound a Weinstein domain in positive rational surfaces, yielding non-embeddability results for Brieskorn spheres in small rational surfaces like or with , while showing that embeddings exist in some larger rational surfaces with constrained topology. The core mechanism combines Donaldson’s asymptotically holomorphic technique with Heegaard Floer theory, notably the reduced invariant , to produce the obstruction and to relate ambient positivity, divisors in the complement, and boundary contact invariants. The authors also extend the framework to non-integral symplectic classes and provide a broad suite of applications, including families of Brieskorn spheres with restricted embedding behavior and concrete examples arising from rational cuspidal curves and Mazur manifolds.

Abstract

We prove that a Weinstein domain symplectically embedded in a closed symplectic manifold always admits symplectic hypersurfaces in its complement, possibly after a deformation. As a consequence, we obtain an obstruction for a closed 3-dimensional manifold to arise as the boundary of a Weinstein domain in a class of symplectic 4-manifolds that includes many symplectic rational surfaces. A particular application is that no Brieskorn homology sphere bounds a Weinstein domain symplectically embedded in a rational surface diffeomorphic to or to , for any , despite the fact that many Brieskorn spheres bound Stein domains holomorphically embedded in these rational surfaces. Several families of Brieskorn spheres are obtained that do not bound a Weinstein domain in any 4-manifold with a ``positive'' symplectic structure. Such Weinstein domains do exist in certain positive symplectic rational surfaces when , though their topology is significantly constrained.

Paper Structure

This paper contains 17 sections, 17 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and statement of some results
  3. Additional context and main structural theorem
  4. Organization and acknowledgements
  5. Proof of Theorems \ref{['nonembthm']} and \ref{['obstrthm']}
  6. Obstruction to Weinstein boundaries
  7. Suitable 3-manifolds
  8. Constraints on Brieskorn spheres in rational surfaces
  9. Applications and examples
  10. Non-suitable Brieskorn spheres
  11. Rational cuspidal curves
  12. Brieskorn spheres in larger rational surfaces
  13. Additional examples
  14. Proof of Theorem \ref{['ratconvthm']}
  15. Background and main argument
  16. ...and 2 more sections

Key Result

Theorem 1.4

Suppose $Y = \Sigma(p_1,\ldots,p_n)$ is a Brieskorn integer homology sphere that embeds as a contact type hypersurface in a positive symplectic rational surface $(X,\omega)$, such that the convex side $(W, \omega|_W)$ is a Weinstein domain. Then:

Figures (2)

  • Figure 1: Surgery diagram for $\Sigma(p,q,npq-1) = M(-2; \frac{q^*}{p}, \frac{p^*}{q}, \frac{(pq-1)n-1}{pqn-1})$
  • Figure 2: The plumbing graph $\Gamma_{p, pk+1, 1}$. Note $q^*=1$ and $p^*=(p-1)k+1$

Theorems & Definitions (36)

  • Conjecture 1.2: weimin, Conjecture 5.1
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.11
  • Theorem 1.12
  • proof : Proof of Theorem \ref{['obstrthm']}(a)
  • proof : Proof of Theorem \ref{['obstrthm']}(b)
  • ...and 26 more