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We recover some recent results by Dotsenko, Shadrin and Vallette on theDeligne groupoid of a pre-Lie algebra, showing that they follow naturally by apre-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 ishomotopy 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.

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