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We first obtain eigenvalue estimates for the Hodge Laplacian on Fanomanifolds, which follow from the Bochner-Kodaira formula. Then we apply it tostudy the geometry of the Kuranishi family of deformations of Fano manifolds.We show that the original K\"ahler form remains to be a K\"ahler form for othermembers of the Kuranishi family, and give an explicit formula of the Riccipotential. We also show that our set-up gives another account for theDonaldson-Fujiki picture.

Hodge Laplacian and geometry of Kuranishi family of Fano manifolds