# Automorphism group of compact almost complex manifold

Does the automorphism group of a compact almost complex manifold carry a (canonical) Lie group structure? Part 3 of Theorem 4.1 in *"The automorphism group of a homogeneous almost complex manifold" by J. Wolf (link at AMS site) says that in a specific group-invariant setting the automorphism group is compact if and only if the almost complex structure is not integrable. Is this a general phenomenon?

• You have misstated Wolf's result. What he shows is that, in the 'irreducible' case (appropriately defined), the automorphism group of a compact homogeneous almost-complex manifold is compact if and only if the almost-complex structure is NOT integrable. For example, the automorphism group of $\mathbb{CP}^n$ is not compact while the almost-complex automorphism group of the non-integrable $S^6=\mathrm{G}_2/\mathrm{SU}(3)$ is compact. Dec 9, 2021 at 14:11
• Thanks, I have made an edit to change the statement Dec 9, 2021 at 15:23

See Kobayashi, Transformation Groups, Theorem 4.1 page 16, where the theorem is proved that the group of automorphisms of a smooth compact almost complex manifold is a finite dimensional Lie group acting smoothly.

• Clearly a product like $\mathbb{CP}^n\times S^6$, with the product almost complex structure coming from the usual complex structure on $\mathbb{CP}^n$, has noncompact automorphism group, is homogeneous, and not a complex manifold. So noncompact automorphism group is not enough to ensure integrability. Dec 9, 2021 at 15:48
• That's why 'irreducible' is so important in Wolf's result. Dec 9, 2021 at 16:23
• Good point, in the general case we should at least add the irreducibility assumption that the almost complex manifold is not a (non-trivial) product of two other almost complex structures Dec 10, 2021 at 8:14