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An almost Abelian group is a non-Abelian Lie group with a codimension 1Abelian subgroup. This paper investigates invariant Hermitian and K\"{a}hlerstructures on connected complex almost Abelian groups. We find explicitformulas for the left and right Haar measures, the modular function, and leftand right generator vector fields on simply connected complex almost Abeliangroups. From the generator fields, we obtain invariant vector and tensor fieldframes, allowing us to find an explicit form for all invariant tensor fields.Namely, all such invariant tensor fields have constant coefficients in theinvariant frame. From this, we classify all invariant Hermitian forms oncomplex simply connected almost Abelian groups, and we prove the nonexistenceof invariant K\"{a}hler forms on all such groups. Via constructions involvingthe pullback of the quotient map, we extend the explicit description ofinvariant Hermitian structures and the nonexistence of K\"{a}hler structures toconnected almost Abelian groups.

Invariant Geometric Structures on Almost Abelian Lie Groups