Formality of hypercommutative algebras of K\"ahler and Calabi-Yau manifolds

Any Batalin-Vilkovisky algebra with a homotopy trivialization of the BV-operator gives rise to a hypercommutative algebra structure at the cochain level which, in general, contains more homotopical information than the hypercommutative algebra introduced by Barannikov and Kontsevich on cohomology. In this paper, we use the purity of mixed Hodge structures to show that the canonical hypercommutative algebra defined on any compact Calabi-Yau manifold is formal. We also study related hypercommutative algebras associated to compact K\"ahler and Hermitian manifolds.