Formality of hypercommutative algebras of K\"ahler and Calabi-Yau manifolds

TL;DR

This paper proves the formality of hypercommutative algebras on compact Calabi-Yau manifolds using mixed Hodge structures and explores related structures on Kähler and Hermitian manifolds.

Contribution

It demonstrates the formality of hypercommutative algebras on Calabi-Yau manifolds and analyzes related algebraic structures on Kähler and Hermitian manifolds.

Findings

Hypercommutative algebra on Calabi-Yau manifolds is formal.

Uses mixed Hodge structures to establish formality.

Studies related hypercommutative algebras on Kähler and Hermitian manifolds.

Abstract

Any Batalin-Vilkovisky algebra with a homotopy trivialization of the BV-operator gives rise to a hypercommutative algebra structure at the cochain level which, in general, contains more homotopical information than the hypercommutative algebra introduced by Barannikov and Kontsevich on cohomology. In this paper, we use the purity of mixed Hodge structures to show that the canonical hypercommutative algebra defined on any compact Calabi-Yau manifold is formal. We also study related hypercommutative algebras associated to compact K\"ahler and Hermitian manifolds.