Regular Lagrangians in Lefschetz fibrations

TL;DR

The paper proves a Weinstein-domain analogue of Giroux–Pardon by showing that any regular Lagrangian L in a Weinstein domain W is realizable, up to a 1-Weinstein homotopy, as an arc in the base of an admissible Weinstein Lefschetz fibration p:W\to\mathbb{D}^2, with p(L) projecting to a Morse-typed arc relative to a chosen efficient function f. The authors develop the framework of coupled Weinstein handles to build such Lefschetz fibrations, and establish that regularity of L is equivalent to the existence of a coupled Lefschetz handle decomposition compatible with the fibration. They apply these results to visualize exact Lagrangian fillings of Legendrian links in (S^3, ξ_st), showing that mutations of a given filling produce Lagrangians arc-admissible for the same fibration, and they construct CAL-skeleta for decomposable fillings, connecting to cluster structures in microlocal sheaf theory. The work advances both the Morse-theoretic and convex-hypersurface approaches to Weinstein topology and provides a concrete toolkit for understanding Lagrangian mutations and skeleta via Lefschetz projections, with potential implications for symplectic fillings and cluster algebras.

Abstract

We characterize regularity of Lagrangian submanifolds in Weinstein Lefschetz fibrations, establishing a conjecture of Giroux and Pardon. Our main result is the Weinstein analogue of a closed symplectic Lefschetz pencil result of Auroux, Muñoz, and Presas. As an application, given a Legendrian link in tight $S^3$ and an exact filling which is part of an arboreal skeleton for the $4$-ball, we build a Lefschetz fibration such that the image of the filling and all of its mutations are arcs in the base.

Figures (14)

• Figure 1: Comparison of results and techniques for closed symplectic manifolds and Weinstein domains.
• Figure 2: A positive braid $\beta$ and its rainbow closure $\Lambda_{\beta}$.
• Figure 3: Each row depicts a $4$-dimensional Weinstein $2$-handle $h_2^4 = \mathbb{D}^2 \times \mathbb{D}^2$ with Liouville vector field $X_{\lambda}$ in black. In each row, the highlighted portions specify a $2$-dimensional Lagrangian $\ell$-handle $h_{\ell}^2$ for $\ell=0$ in blue, $\ell=2$ in red, and $\ell=1$ in green. Note that the blue and red Lagrangian handles correspond to the co-core and core of the Weinstein handle, respectively.
• Figure 4: The first row depicts a $4$-dimensional coupled Weinstein $1$-handle, and the second row depicts, in the language of \ref{['def:coupled_lefschetz_handle']}, a $4$-dimensional coupled Lefschetz $1$-handle with the decomposition described by the proof of \ref{['lemma:coupled_weinstein_lefschetz_handle_htpy']}.
• Figure 5: The statement of \ref{['thm:main_alt']} for two different coupled Lefschetz handlebodies. In both figures, the disk is the base of the Weinstein Lefschetz fibration $p_i:W_i \to \mathbb{D}^2$ and stars are critical values. On the left, $L_1$ is a closed manifold (for instance, diffeomorphic to $T^2$) and $(W_1,L_1)$ is a strict coupled Lefschetz handlebody. On the right, $L_2$ is a manifold with boundary and $(W_2,L_2)$ is a (non-strict) coupled Lefschetz handlebody.

Theorems & Definitions (60)

• Definition 1.1
• Theorem 1.1
• Conjecture 1.1: Giroux-Pardon
• Definition 1.2
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Proposition 1.1
• Corollary 1.1
• Remark 2.1