Cyclic $BV_\infty$ algebra and Frobenius manifold

We describe the construction of Frobenius manifold out of a cyclic (commutative) $BV_\infty$ algebra $(A,\Delta)$ under the assumption of a Hodge-to-de Rham degeneration property and the existence of a compatible homotopy retract of $A$ onto its cohomology. We then apply it to Jacobi manifolds and Hermitian manifolds, generalizing known results in literature.