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

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∞ algebra structure on the Dolbeault complex of a K"ahler manifold is homotopy abelian and independent on the choice of Kaehler metric up to an L∞ isomorphism, by making the trivializing homotopy and the L∞ isomorphism explicit.