Strongly homotopy algebras of a K\"ahler manifold

TL;DR

The paper constructs two canonical strongly homotopy algebras from a compact K"ahler manifold, relating to its Hodge theory, and introduces a third algebra for Calabi-Yau manifolds connected to moduli space deformations.

Contribution

It establishes the existence of two natural strongly homotopy algebras associated with K"ahler manifolds and a third for Calabi-Yau cases, linking algebraic structures to geometric properties.

Findings

Harmonic forms' product remains harmonic in these algebras

Two strongly homotopy algebras are associated with de Rham and Dolbeault complexes

A third algebra relates to Calabi-Yau moduli space

Abstract

It is shown that any compact K\"ahler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If $M$ happens to be a Calabi-Yau manifold, there exists a third strongly homotopy algebra closely related to the Barannikov-Kontsevich extended moduli space of complex structures.