Formality of Kapranov's brackets in K\"ahler geometry via pre-Lie deformation theory

TL;DR

This paper explores the algebraic structures underlying Kähler geometry, demonstrating that Kapranov's $L_$ algebra on the Dolbeault complex is homotopy abelian and independent of the Kähler metric, using pre-Lie deformation theory.

Contribution

It introduces a pre-Lie algebra perspective to derive properties of Kapranov's brackets, providing explicit constructions of homotopy trivializations and isomorphisms.

Findings

Kapranov's $L_$ algebra is homotopy abelian.

Independence of the $L_$ structure from the Kähler metric.

Explicit formulas for trivializing homotopies and isomorphisms.

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 K\"ahler manifold is homotopy abelian and independent on the choice of K\"ahler metric up to an $L_\infty$ isomorphism, by making the trivializing homotopy and the $L_\infty$ isomorphism explicit.