Symmetries of almost complex structures and pseudoholomorphic foliations

Abstract

Contrary to complex structures, a generic almost complex structure J has no local symmetries. We give a criterion for finite-dimensionality of the pseudogroup G of local symmetries for a given almost complex structure J. It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some integrable structure, like a pseudoholomorphic foliation. We classify the sub-maximal (from the viewpoint of the size of G) symmetric structures J. Almost complex structures in dimensions 4 and 6 are discussed in greater details in this paper.