Almost complex manifolds from the point of view of Kodaira dimension

TL;DR

This paper explores the extension of Kodaira dimension, a key invariant in complex geometry, to almost complex manifolds, focusing on its behavior under deformations and implications for meromorphic functions.

Contribution

It introduces the concept of Kodaira dimension for non-integrable almost complex manifolds and examines its properties and behavior under deformations.

Findings

Kodaira dimension can be generalized to almost complex manifolds.

Behavior under deformations varies between integrable and non-integrable cases.

Speculations on the role of meromorphic functions in almost complex geometry.

Abstract

In complex geometry a classical and useful invariant of a complex manifold is its Kodaira dimension. Since its introduction by Iitaka in the early 70's, its behavior under deformations was object of study and it is known that Kodaira dimension is invariant under holomorphic deformations for a smooth projective manifolds, while there are examples of holomorphic deformations of non-projective manifolds for which the Kodaira dimension is non-constant. Recently this concept has been generalized to almost complex manifolds, we want to present here some of its main features in the non-integrable case, mainly with respect to deformations. At the end we conclude with some speculations on the theory of meromorphic functions on almost complex manifolds.