Symmetries of almost complex structures and pseudoholomorphic foliations (Q2925307)

Let \((M,J)\) be a \(2n\)-dimensional almost-complex manifold. This paper is concerned with the pseudogroup of local (pseudoholomorphic) symmetries of \((M,J)\). This contains the group of (global) automorphisms of \((M,J)\), and the main theme of the paper is that a large symmetry pseudogroup implies that \((M,J)\) carries some integrable structure. One of the main results is that if \(J\) is nondegenerate (which here means that the Nijenhuis tensor, viewed as a map \(\Lambda^2 TM\to TM\), is surjective when \(n>2\) and that its image is a non-integrable distribution when \(n=2\)), then the pseudogroup of symmetries is finite-dimensional. In particular, if \(J\) is nondegenerate in at least one point then the automorphism group of \((M,J)\) is a finite-dimensional Lie group.NEWLINENEWLINEFurthermore, if \(n=2\) then the dimension of the symmetries pseudogroup is at most \(4\), with equality holding only for left-invariant almost-complex structures on Lie groups, and if \(n=3\) then the dimension is at most \(14\), with equality holding only for the standard almost-complex structure on \(S^6\) coming from the octonions, and for the similar unit sphere \(S^{2,4}\) in the purely imaginary split octonions.