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A strong KT (SKT) manifold consists of a Hermitian structure whose torsionthree-form is closed. We classify the invariant SKT structures onfour-dimensional solvable Lie groups. The classification includes solutions ongroups that do not admit compact four-dimensional quotients. It also shows thatthere are solvable groups in dimension four that admit invariant complexstructures but have no invariant SKT structure.

Invariant strong KT geometry on four-dimensional solvable Lie groups