Formality of differential graded algebras and complex Lagrangian submanifolds

Abstract

Let $i: \mathrm{L} \hookrightarrow \mathrm{X}$ be a compact K\"{a}hler Lagrangian in a holomorphic symplectic variety $\mathrm{X}/\mathbf{C}$. We use deformation quantisation to show that the endomorphism differential graded algebra $\mathrm{RHom}\big(i_*\mathrm{K}_\mathrm{L}^{1/2},i_*\mathrm{K}_\mathrm{L}^{1/2}\big)$ is formal. We prove a generalisation to pairs of Lagrangians, along with auxiliary results on the behaviour of formality in families of $\mathrm{A}_\infty$-modules.