# Algebraic embeddings of smooth almost complex structures

### Jean-Pierre Demailly

Université Grenoble Alpes, Gières, France### Hervé Gaussier

Université Grenoble Alpes, Gières, France

## 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 complex manifold as a $C_{∞}$ smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?

## Cite this article

Jean-Pierre Demailly, Hervé Gaussier, Algebraic embeddings of smooth almost complex structures. J. Eur. Math. Soc. 19 (2017), no. 11, pp. 3391–3419

DOI 10.4171/JEMS/742