  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.
