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Kodaira Surface, its Extended Deformation and Mirror Image

We present the extended Kuranishi space for Kodaira surface as a non-trivial 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. Our computation is done in the context of compact nilmanifolds. The calculation of extended deformation and the weak Frobenius structure is based on Merkulov's perturbation method. The approach to relating symplectic structure and complex structure is inspired by Gualtieri's work on generalized complex structures.