Formality of Kapranov's brackets in Kähler geometry via pre-Lie deformation theory

Abstract

We recover some recent results by Dotsenko, Shadrin and Vallette on the Deligne groupoid of a pre-Lie algebra, showing that they follow naturally by a pre-Lie variant of the PBW Theorem. As an application, we show that Kapranov's $L_{\infty }$ algebra structure on the Dolbeault complex of a Kähler manifold is homotopy abelian and independent on the choice of Kähler metric up to an $L_{\infty }$ isomorphism, by making the trivializing homotopy and the $L_{\infty }$ isomorphism explicit.

arXiv commentThis is a new version of the old "Formality of Kapranov's brackets on pre-Lie algebras". To appear in Int. Math. Res. Not