Lagrangian homology spheres in $(A_{m})$ Milnor fibres via $\mathbb{C}^{*}$–equivariant $A_{\infty}$–modules

Abstract

We establish restrictions on Lagrangian embeddings of spheres, and more generally rational homology spheres, into certain open symplectic manifolds, namely the $\left(A_m\right)$ Milnor fibres of odd complex dimension. This relies on general considerations about equivariant objects in module categories (which may be applicable in other situations as well), as well as results of Ishii–Ueda–Uehara concerning the derived categories of coherent sheaves on the resolutions of $\left(A_m\right)$ surface singularities.