Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type

TL;DR

The paper demonstrates that there exist infinite families of diffeomorphic Weinstein manifolds that are nevertheless not Weinstein equivalent, by comparing Milnor fibers of ADE-type singularities with end-connected sums. The authors construct explicit exotic pairs $(X^{2n+2}_k,Y^{2n+2}_k)$ using Lefschetz fibrations on $A^{2n}_2$, proving $WFuk(X^{2n+2}_k)$ and $WFuk(Y^{2n+2}_k)$ have non-equivalent structures, with $X^{2n+2}_k$ indecomposable and $Y^{2n+2}_k$ decomposable; they also establish diffeomorphism results for certain indices. The work extends to two generalizations: multi-$A$-type end sums and cross-type ADE comparisons, yielding diffeomorphic yet symplectically distinct families; these are distinguished using decomposability criteria and symplectic cohomology calculations. Overall, the results provide a systematic method to generate and distinguish exotic Weinstein manifolds beyond prior examples, highlighting the finer symplectic invariants encoded in wrapped Fukaya categories and symplectic cohomology.

Abstract

In this paper, we give infinitely many diffeomorphic families of different Weinstein manifolds. The diffeomorphic families consist of the Milnor fibers of $ADE$-type, and a Weinstein manifold constructed by gluing a cotangent bundle to the Milnor fiber of $A$-type. The last-mentioned members of the families have decomposable wrapped Fukaya categories.

Figures (8)

• Figure 1: Three star marked points are critical values of $\rho$, the blue (resp. red) curve is the image of $\alpha$ (resp. $\beta$) under $\rho$.
• Figure 2: $a)$ describes the handle decomposition of $\mathbb{D}^2$, consisting of three $0$ handles and two $1$ handles. The dashed lines are boundaries of handles, the dots are centers of handles, and the arrows describe the Liouville vector field. $b)$ describes the decomposition of $A_1 \cup A_2 \cup B = \mathbb{D}^2$. The part of $\partial B$, which is located inside the circle, is given by a union of unstable manifolds of the centers of two $1$ handles, with respect to the Liouville vector flow.
• Figure 3: The big circle means the base of $\pi: Y^{2n+2}_k \to \mathbb{C}$, the small dotted circle means the boundary of radius $1$ disk $\mathbb{D}_1^2$ which is the union of $S_1, S_2$, and $T$, i.e., the subcritical part is given as the inverse image of the dotted disk. The star marked points are singular values of $\pi$ decorated with the vanishing cycles, i.e., $H_1, \cdots, H_{2k+1}$ (resp. $H_{2k+2}, H_{2k+3}$) are attached to the 'left' or $S_1$ (resp. 'right' or $S_2$) side of the subcritical part.
• Figure 4: $a)$ is a conceptual picture describing the Lagrangian skeleton of the subcritical part $A_2^{2n} \times \mathbb{D}^2 \subset Y^{2n+2}_k$. We note that we are considering the product Weinstein structure for $a)$. The black, red, and blue lines correspond to $\alpha$ in $A_2^{2n}$, $\beta$ in $A_2^{2n}$, and the Lagrangian skeleton of $S_1 \cup S_2 \cup T = \mathbb{D}^2$, respectively. We note that the Lagrangian skeleton of $Y^{2n+2}_k$ is constructed by attaching $2k+3$ Lagrangian disks to $a)$, which correspond to the $2k+3$ critical handles. The $2k+1$ Lagrangian disks corresponding to $H_1, \cdots, H_{2k+1}$ are attached to the thick black line, and the other two Lagrangian disks corresponding to $H_{2k_2}, H_{2k+3}$ are attached to the thick red line. $b)$ is the Lagrangian skeleton of the subcritical part in the modified Weinstein domain. We note that there are no changes on the thick black and red lines.
• Figure 5: The red and blue curves on base of $\rho$ are $\gamma_1$ and $\gamma_2$, respectively.

Theorems & Definitions (53)

• Theorem 1.1: Technical statement is Theorem \ref{['thm main']}
• Theorem 1.2: Technical statement is Theorem \ref{['thm exotic families of Z']}
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof