Complex manifolds as families of homotopy algebras

TL;DR

This paper establishes a categorical equivalence between formal complex structures and homotopy algebras, extending it from formal settings to entire complex manifolds, linking geometric structures to algebraic families.

Contribution

It introduces a new equivalence of categories connecting complex geometry with homotopy algebraic structures, broadening the understanding of complex manifolds through algebraic methods.

Findings

Equivalence between formal complex structures and homotopy algebras.

Extension of this equivalence to entire complex manifolds.

Representation of complex structures as families of algebras.

Abstract

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex structure on a smooth manifold corresponds in this way to a family of algebras indexed by the points of the manifold.