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We classify non-nilpotent complex structures on 6-nilmanifolds and theirassociated invariant balanced metrics. As an application we find a large familyof solutions of the heterotic supersymmetry equations with non-zero flux,non-flat instanton and constant dilaton satisfying the anomaly cancellationcondition with respect to the Chern connection.

Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry