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Mirror Symmetry for hyperkähler manifolds

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form, also known as complex manifolds of hyperkähler type. We obtain that a complex manifold of hyperkähler type is mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as the equivalence of certain algebraic structures related to variations of Hodge structures. We compute the canonical flat coordinates on the moduli space of Calabi-Yau manifolds of hyperkähler type, introduced to Mirror Symmetry by Bershadsky, Cecotti, Ooguri and Vafa.