  The purpose of this paper is to explain and to generalize, in a homotopical
way, the result of Barannikov-Kontsevich and Manin which states that the
underlying homology groups of some Batalin-Vilkovisky algebras carry a
Frobenius manifold structure. To this extent, we first make the minimal model
for the operad encoding BV-algebras explicit. Then we prove a homotopy transfer
theorem for the associated notion of homotopy BV-algebra. The final result
provides an extension of the action of the homology of the
Deligne-Mumford-Knudsen moduli space of genus 0 curves on the homology of some
BV-algebras to an action via higher homotopical operations organized by the
cohomology of the open moduli space of genus zero curves. Applications in
Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry
conjecture are given.
