  We study the positive Hermitian curvature flow on the space of left-invariant
metrics on complex Lie groups. We show that in the nilpotent case, the flow
exists for all positive times and subconverges in the Cheeger-Gromov sense to a
soliton. We also show convergence to a soliton when the complex Lie group is
almost abelian. That is, when its Lie algebra admits a (complex) co-dimension
one abelian ideal. Finally, we study solitons in the almost-abelian setting. We
prove uniqueness and completely classify all left-invariant, almost-abelian
solitons, giving a method to construct examples in arbitrary dimensions, many
of which admit co-compact lattices.
