  In the theory of deformation of Okamoto-Painlev\'e pair (S,Y), a local
cohomology group $H^1_D(\Theta_S(-\log D))$ plays an important role. In this
paper, we estimate the local cohomology group of pair (S,Y) for several types,
and obtain the following results. For a pair (S,Y) corresponding to the space
of initial conditions of the Painlev\'e equations, we show that the local
cohomology group $H^1_D(\Theta_S(-\log D))$ is at least 1 dimensional. This
fact is the key to understand Painlev\'e equation related to (S,Y). Moreover we
show that, for the pairs (S,Y) of type $\tilde{A_8}$, the local cohomology
group $H^1_D(\Theta_S(-\log D))$ vanish. Therefore in this case, there is no
differential equation on S-Y in the sense of the theory.
