Almost complex manifolds with small Nijenhuis tensor

Luis Fernandez,Scott O Wilson,Tobias Shin

doi:10.1515/coma-2020-0122

Abstract

Abstract We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C 0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

Highlights

The purpose of the following short note is to give several explicit examples of compact manifolds with a parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C, or supremum, norm
The -dimensional examples possess no complex structure, whereas the -dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown. The idea that such examples might exist was inspired by e orts of the second author to place the work of Demailly and Gaussier [3] in the context of Gromov’s h-principle, whereby an integrable complex structure was interpreted as a holonomic solution of a locally closed di erential relation
The lack of homotopy obstructions to formal solutions of this di erential relation led the second author to attempt various h-principle techniques, to try to deform a “formal integrable complex structure” into a genuine one. This naturally led to the question of whether every almost complex manifold has almost complex structures that are arbitrarily close to an integrable one

Summary

Introduction

The purpose of the following short note is to give several explicit examples of compact manifolds with a parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C , or supremum, norm. The lack of homotopy obstructions to formal solutions of this di erential relation led the second author to attempt various h-principle techniques, to try to deform a “formal integrable complex structure” into a genuine one. This naturally led to the question of whether every almost complex manifold has almost complex structures that are arbitrarily close to an integrable one. Some recent results in this direction, particular to dimension , appear in the work of Fei et al [5], establishing su cient conditions for some symplectic manifolds

Nilmanifold examples

Solvmanifold examples

Then the Nijenhuis tensor is determined by