This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Khlerian complex structures on six-dimensional nilpotent algebras whose differential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classication of pseudo-Khler structures on six-dimensional nilpotent algebras whose mirror images are themselves.

Foundation Compositio Mathematica, 1990, tous droits rservs. Laccs aux archives de la revue Compositio Mathematica(http://http://www. compositio. nl/) implique laccord avec les conditions gnrales dutilisation (http://www. numdam. org/legal. php). Toute utilisation commerciale ou impression systmatique est constitutive dune infraction pnale. Toute copie ou impression de ce fichier doit contenir la prsente mention de copyright.

It is shown that an HKT space with closed parallel potential 1-form has D (2, 1; 1) symmetry. Every locally conformally hyper-Khler manifold generates this type of geometry. The HKT spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic condition on their torsion.