Almost complex manifolds are (almost) complex

Tobias Shin

Published 2019-03-24Version 1

We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. We prove that the relation has formal solutions up to complex dimension 77, and that, given a formal solution, there is always a holonomic section approximating it. The latter result implies in particular that, up to complex dimension 77, any almost complex manifold admits a sequence of almost complex structures so that the pointwise supremum norms of the Nijenhuis tensors become arbitrarily small.