Symplectic embeddings of 4-manifolds via Lefschetz fibrations

TL;DR

This paper demonstrates that certain 4-manifolds with Lefschetz fibrations can be symplectically embedded into specific 6-manifolds, expanding understanding of symplectic embeddings in higher dimensions.

Contribution

It establishes conditions under which 4-manifolds with Lefschetz fibrations embed symplectically into 6-manifolds, introducing new embedding techniques and identifying suitable ambient spaces.

Findings

4-manifolds with Lefschetz fibrations embed into ( ext{CP}^1 imes ext{CP}^1 imes ext{CP}^1)

Existence of a sub-critical Weinstein 6-manifold for iso-symplectic embeddings

All finite type Weinstein 4-manifolds embed in a specific Weinstein 6-manifold

Abstract

In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, \omega_{pr}),$ where $\omega_{pr}$ is the product symplectic form on $\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$--manifold in which all finite type Weinstein $4$--manifolds admit iso-symplectic embeddings.