  It is shown that any compact K\"ahler manifold $M$ gives canonically rise to
two strongly homotopy algebras, the first one being associated with the Hodge
theory of the de Rham complex and the second one with the Hodge theory of the
Dolbeault complex. In these algebras the product of two harmonic differential
forms is again harmonic. If $M$ happens to be a Calabi-Yau manifold, there
exists a third strongly homotopy algebra closely related to the
Barannikov-Kontsevich extended moduli space of complex structures.
