= Compact connected complex Lie groups are complex tori
{title2=$G\cong\mathbb C^n/B,\quad B\cong\mathbb Z^{2n}$}
Every scalar <holomorphic function> on a compact connected <complex manifold> is constant by the <maximum modulus principle>. Apply this to each entry of the holomorphic <Adjoint representation of a Lie group>; its value is constantly the identity. Differentiation makes the complex <Lie algebra> abelian, hence connectedness makes the <group> abelian. The holomorphic <Lie exponential map> identifies it with $\mathbb C^n$ modulo a <discrete subgroup>. Compactness forces that <subgroup> to span the underlying <real vector space>, giving a full-rank <Euclidean lattice> and a <complex torus>.
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