Algebraic embeddings of smooth almost complex structures

TL;DR

This paper proves that every compact almost complex manifold can be embedded into an affine algebraic variety as a transverse structure to an algebraic distribution, with universal embedding spaces whose dimensions grow quadratically.

Contribution

It introduces a method to realize almost complex structures via algebraic distributions and constructs universal embedding spaces with quadratic dimension growth.

Findings

Every almost complex structure can be realized by algebraic distributions.

Existence of universal embedding spaces with quadratic dimension growth.

Discussion on embedding holomorphic manifolds as smooth subvarieties transverse to algebraic foliations.

Abstract

The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F.Bogomolov: can one embed every compact holomorphic manifold as a C infinity smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?