  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.
