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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, making the trivializing homotopy and the   $L_\infty $   isomorphism explicit.

Formality of Kapranov's Brackets in Kähler Geometry via Pre-Lie Deformation Theory