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 groupinvariant setting the automorphism group is compact if and only if the almost complex structure is not integrable. Is this a general phenomenon?
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3$\begingroup$ You have misstated Wolf's result. What he shows is that, in the 'irreducible' case (appropriately defined), the automorphism group of a compact homogeneous almostcomplex manifold is compact if and only if the almostcomplex structure is NOT integrable. For example, the automorphism group of $\mathbb{CP}^n$ is not compact while the almostcomplex automorphism group of the nonintegrable $S^6=\mathrm{G}_2/\mathrm{SU}(3)$ is compact. $\endgroup$– Robert BryantDec 9, 2021 at 14:11

1$\begingroup$ Thanks, I have made an edit to change the statement $\endgroup$– deepfloeDec 9, 2021 at 15:23
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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.

2$\begingroup$ 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. $\endgroup$ Dec 9, 2021 at 15:48

2$\begingroup$ That's why 'irreducible' is so important in Wolf's result. $\endgroup$ Dec 9, 2021 at 16:23

$\begingroup$ Good point, in the general case we should at least add the irreducibility assumption that the almost complex manifold is not a (nontrivial) product of two other almost complex structures $\endgroup$– deepfloeDec 10, 2021 at 8:14