  We first obtain eigenvalue estimates for the Hodge Laplacian on Fano
manifolds, which follow from the Bochner-Kodaira formula. Then we apply it to
study 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 other
members of the Kuranishi family, and give an explicit formula of the Ricci
potential. We also show that our set-up gives another account for the
Donaldson-Fujiki picture.
