Frobenius Manifolds from Hyperk

We construct a dGBV algebra from Dolbeault complex of any closed hyperkk ahler manifold. A Frobenius manifold structure on an neighborhood of the origin in Dolbeault cohomology then arises via Manin's generalization of Barannikov-Kontsevich's construction of formal Frobenius manifold structure on formal extended moduli space of a Calabi-Yau manifold. It is explained why these two kinds of formal Frobenius manifolds might be mirror images of each other under the conjectured mirror symmetry. The complexity in the theory of quantum cohomology has inspired us to nd easier methods of obtaining deformation of ring structure on de Rham cohomology. For KK ahler manifolds, we have achieved this goal in two diierent ways. The basic idea is to deform the de Rham complex, then identify the cohomology of the deformed complex with the de Rham cohomology as vector spaces, and check the ring structure is deformed. The rst approach, announced in 2], with detailed account in 3], is to deene explicitly a deformation of the wedge product of exterior forms, the exterior diierential is also deformed to a new diierential which is a derivation for the deformed product. We deene the so-called quantum de Rham cohomology as the cohomology of the deformed operator. A remarkable fact is the above construction can be carried out for any Poisson manifold. Only when one comes to identify the quantum de Rham cohomology with the original de Rham cohomology as vector spaces does one need the technical condition that the manifold is KK ahler and closed. Complex projective spaces provide simple examples to see that one does get nontrivial deformation of the ring structure this way. Quantum de Rham cohomology is diierent from quantum cohomology in symplectic geometry. This can be seen in the case of the complex tori, for which quantum cohomology does not provide any deformation for the cohomology ring, since there is no nontriv-ial pseudo-holomorphic rational curve. But quantum de Rham cohomology does provide nontrivial deformations. Recently, a new method of deforming the cohomology of a diierential algebra (A;) has been developed in Barannikov-Kontsevich 1] and Manin 9] which are based on ideas from earlier works of Tian 11] and Todorov 12]. The idea is to have another diierential on on A with the same cohomology which deenes a bracket ], and nd a 2 A such that d a = d + a ] is a derivation with 2 a = 0, whose …