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We present the extended Kuranishi space for a Kodaira surface as a nontrivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image in the sense of Merkulov. The calculations of extended deformation and the weak Frobenius structure are based on Merkulov's perturbation method. Our computation of cohomology is done in the context of compact nilmanifolds.

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.

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.

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.