Commutative $B{V}_{\mathrm{\infty }}$ algebras, their morphisms and $\frac{\mathrm{\infty }}{2}$-variation of Hodge structures

We study morphisms between commutative $B{V}_{\mathrm{\infty }}$ algebras and show that, under suitable additional assumptions, a quasi-isomorphism of commutative $B{V}_{\mathrm{\infty }}$ algebras induces an identification of $\frac{\mathrm{\infty }}{2}$-variations of Hodge structures with polarizations, and consequently of Frobenius manifolds. An explicit example arising from singularity theory is provided to illustrate the result.