Formality of differential graded algebras and complex Lagrangian submanifolds

TL;DR

This paper proves the formality of a specific differential graded algebra associated with compact Lagrangian submanifolds in holomorphic symplectic varieties using deformation quantisation.

Contribution

It demonstrates the formality of endomorphism DGAs for Lagrangians and generalizes to pairs of Lagrangians, with results on A-infinity modules.

Findings

Endomorphism DGAs are formal for compact Lagrangians in holomorphic symplectic varieties.

Generalization to pairs of Lagrangians is established.

Auxiliary results on the behavior of formality in families of A-infinity modules.

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.