# compact almost complex submanifolds of complex Lie groups

Does there exist any complex Lie group $G$ such that there are some positive-dimensional compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$?

I want to get some examples.

This is motivated by the following Corollary 1.21 from Complex Manifolds, Lecture Notes written by Clemens Koppensteiner (link)

Proposition 1.20 (Generalization of Liouville's theorem). Let $M$ be a compact [complex] manifold and $f$ a holomorphic function on $M$. Then $f$ is constant.

Corollary 1.21 There exist no compact complex submanifolds of $\mathbb{C}^n$ of positive dimension

Scan including the proofs:

• It's best to cite the source of anything you quote or refer to. I was able to find it by google in this case caramdir.at/uploads/math/piii-cm/complex-manifolds.pdf – j.c. Sep 28 '15 at 12:45
• What do you mean by an "almost complex submanifold" of a complex manifold such as $G$? – Jason Starr Sep 28 '15 at 12:45
• It's hard to interpret the question in a nontrivial way (take any positive-dimensional compact complex Lie group $G$, then $G$ itself works...). Whether $\mathbb{P}^n$ embeds as a complex submanifold of a compact complex Lie group is possibly more reasonable (although I'm not a specialist, maybe the answer is obvious or well-known). – YCor Jan 23 '17 at 21:08

Whatever almost complex submanifolds are, complex submanifolds should be a subclass. There are compact complex tori which admit complex sub-tori. One (natural) class of examples are given as follows: Consider a Riemann surface $\Sigma$ and its Jacobian $Jac,$ and a assume that there is a holomorphic double covering $\pi\colon \Sigma\to \Sigma'.$ wThen, by pulling back, you have a subtorus $Jac'\subset Jac$ and and also its $'complement'$ the Prym variety (of $\pi.$)