Formality of differential graded algebras and complex Lagrangian submanifolds

Abstract

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