Symmetries of almost complex structures and pseudoholomorphic foliations

TL;DR

This paper investigates the symmetry properties of almost complex structures, identifying conditions for small symmetry pseudogroups and linking large symmetry groups to integrable structures like pseudoholomorphic foliations, especially in dimensions 4 and 6.

Contribution

It classifies sub-maximal symmetric almost complex structures in dimensions 4 and 6 and provides estimates on automorphism group dimensions for non-degenerate cases.

Findings

Small symmetry pseudogroups are typical, large ones indicate integrability.

Classification of sub-maximal symmetric structures in 4 and 6 dimensions.

Estimate on automorphism group dimension for non-degenerate structures.

Abstract

Generically an almost complex structure has no symmetries at all, but there exist symmetric structures. In this paper we describe how to guarantee that the pseudogroup of local symmetries is small (finite-dimensional). It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some integrable structure, like a pseudoholomorphic foliation. We are mostly concerned with almost complex structures in dimensions 4 and 6, where we classify the sub-maximal symmetric structures, and we briefly discuss the higher dimensions. For non-degenerate almost complex structures we give an estimate on the dimension of the automorphism group.