Is there some simple proof that $\mathbb{Z}$ is not isomorphic to the fundamental group of any compact Kahler manifold? This follows from the main result of https://arxiv.org/abs/0709.4350 which states that any $3$-manifold group which is Kahler must be finite. The simplest non-finite 3-manifold group is $\mathbb{Z} = \pi_{1}(S^{2} \times S^{1})$. So I was wondering, is there an argument to rule out this particularly simple group, without applying to a lot of machinery? (also historically when were we first able to rule this group out?)

Is there some simple proof that $\mathbb{Z}$ is not isomorphic to the fundamental group of any compact Kähler manifold? This follows from the main result of https://arxiv.org/abs/0709.4350 which states that any $3$-manifold group which is Kähler must be finite. The simplest non-finite 3-manifold group is $\mathbb{Z} = \pi_{1}(S^{2} \times S^{1})$. So I was wondering, is there an argument to rule out this particularly simple group, without applying to a lot of machinery? (also historically when were we first able to rule this group out?)