Splitting symplectic monodromy

TL;DR

The paper investigates when symplectic monodromy representations $\rho: \pi_1(\mathcal{M}_X)\to \pi_0\mathrm{Symp}(X,\omega)$ split, by linking $\rho$ to autoequivalences of Fukaya-type categories through homological mirror symmetry. It provides concrete splittings in several rich geometries, notably type $A$ Milnor fibres, quiver-based CY$_3$-settings from marked bordered surfaces, and both the double bubble and conifold plumbings, using cluster categories, stability conditions, and Deligne-type groupoid arguments. The work also connects to birational geometry via the log CY/birational plane setting and explores numerous further examples, including tori, surfaces, log Calabi–Yau surfaces, and del Pezzo surfaces, while highlighting higher-dimensional non-splitting phenomena and heading toward $C^0$-category considerations. Overall, the results illustrate that monodromy splitting is a robust but delicate phenomenon, orchestrated by deep interactions between symplectic topology, algebraic geometry, and categorical mirror symmetry, with many cases still open for precise criteria and general principles.

Abstract

We discuss some examples in which symplectic monodromy (provably or conjecturally) splits off the symplectic mapping class group, hoping to illustrate different techniques and inputs to the arguments. Along the way we formulate several open questions and conjectures.

Figures (4)

• Figure 1: The $A_k$-quiver
• Figure 2: Vanishing cycles for Morse-Bott-Lefschetz fibration $X \to \mathbb{C}$, fibre $(\mathbb{C}^*)^2$
• Figure 3: Morse--Bott--Lefschetz fibration on $(W_0, \omega_0)$. The crosses in the base denote singular values, and the dot a regular one.
• Figure 4: Example of disjoint Lagrangian matching spheres in $(X, \omega_{s,t})$.

Theorems & Definitions (37)

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