Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings

TL;DR

The work addresses the existence and distinction of exact Lagrangian submanifolds in Milnor fibers of isolated hypersurface singularities with positive modality. By constructing Weinstein handlebody diagrams via the affine dictionary and leveraging exact Lagrangian fillings and augmentations, the authors produce infinitely many Hamiltonian non-isotopic exact Maslov-$0$ Lagrangian tori in Milnor fibers $M_f$, including $T_{p,q,r}$-type singularities, and connect these constructions to adjacencies with unimodular singularities. A key methodological advance is extending a non-vanishing criterion for symplectic homology from Legendrian augmentations to restricted augmentation frameworks on handlebody diagrams, enabling practical detection of non-vanishing SH and non-flexibility. The results bridge Milnor fiber topology, Legendrian contact algebra, and Fukaya-category considerations, with implications for the structure of wrapped Fukaya categories and their generators in positive-modality settings.

Abstract

We use exact Lagrangian fillings and Weinstein handlebody diagrams to construct infinitely many distinct exact Lagrangian tori in $4$-dimensional Milnor fibers of isolated hypersurface singularities with positive modality. We also provide a generalization of a criterion for when the symplectic homology of a Weinstein $4$-manifold is non-vanishing given an explicit Weinstein handlebody diagrams.

Figures (8)

• Figure 1: A Legendrian handle slide of $h_2$ over $h_1$.
• Figure 2: A canceling pair of $4$-dimensional $1$ and $2$-handles.
• Figure 3: A Weinstein handlebody diagram of the Milnor fiber of a $T_{p,q,r}$-singularity where $p,q,r\geq0$.
• Figure 4: A Lefschetz fibration of $T_{p,q,r}$ whose vanishing cycles are Dehn twists of the four curves $T,P,Q,R$ on the regular fiber.
• Figure 5: We simplify the Weinstein handlebody diagram of $T_{p,q,r}$ for $p,q,r\geq1$ as follows: $(A)-(B)$ perform various handle-slides; $(C)$ cancel out the $1$-handles; $(D)-(E)$ perform Legendrian Reidemeister moves; $(F)$ use the fact that one can handle-slide an $N$-copy of the unknot to a chain of $N$ unknots.

Theorems & Definitions (29)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3
• Theorem 1.4
• Corollary 1.5
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4: Ekholm_honda_kalmanKarlsson
• Definition 2.5: Definition 3.9 Casals_Ng