Commutative $BV_\infty$ algebras, their morphisms and $\frac{\infty}{2}$-variation of Hodge structures

TL;DR

This paper explores the relationships between commutative $BV_$ algebras and $rac{}$-variations of Hodge structures, demonstrating how quasi-isomorphisms induce isomorphisms of associated geometric structures, with an example from singularity theory.

Contribution

It establishes conditions under which quasi-isomorphisms of commutative $BV_$ algebras induce equivalences of $rac{}$-variations of Hodge structures and Frobenius manifolds.

Findings

Quasi-isomorphisms induce isomorphisms of $rac{}$-variations of Hodge structures.

Explicit example from singularity theory illustrating the theoretical results.

Connection between algebraic morphisms and geometric structures in Hodge theory.

Abstract

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