Geometry of universal embedding spaces for almost complex manifolds

TL;DR

This paper explores the geometry of universal embedding spaces for compact almost complex manifolds, aiming to understand integrability and potential obstructions within a complex algebraic framework.

Contribution

It introduces a broader class of universal embedding spaces and analyzes their geometric structure related to integrability of almost complex structures.

Findings

Characterization of the integrability locus

Potential methods to identify topological obstructions

Extension of twistor space analogues to a more general setting

Abstract

We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were introduced by J-P. Demailly and H. Gaussier. Their original goal was the study of a conjecture made by F. Bogomolov, asserting the "transverse embeddability" of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of the integrability locus characterizing integrable almost complex structures. Our approach can potentially be used to investigate the existence (or non-existence) of topological obstructions to integrability.