Legendrian Fronts for Affine Varieties

TL;DR

This paper develops a method to analyze Weinstein structures with Lefschetz fibrations using Legendrian front projections, enabling new insights into symplectic topology and affine varieties.

Contribution

It introduces a systematic recipe for translating Weinstein Lefschetz bifibrations into Legendrian handlebodies and applies this to various problems in symplectic topology.

Findings

Detection of flexibility and rigidity in Weinstein manifolds

Existence of closed exact Lagrangian submanifolds

Koras--Russell cubic is Stein deformation equivalent to affine 3-space

Abstract

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras--Russell cubic is Stein deformation equivalent to affine complex 3-space and verify the affine parts of the algebraic mirrors of two Weinstein 4-manifolds.