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In this paper, we introduce the notion of generalized rationalOkamoto-Painlev\'e pair (S, Y) by generalizing the notion of the spaces ofinitial conditions of Painlev\'e equations. After classifying those pairs, wewill establish an algebro-geometric approach to derive the Painlev\'edifferential equations from the deformation of Okamoto-Painlev\'e pairs byusing the local cohomology groups. Moreover the reason why the Painlev\'eequations can be written in Hamiltonian systems is clarified by means of theholomorphic symplectic structure on S - Y. Hamiltonian structures forOkamoto-Painlev\'e pairs of type $\tilde{E}_7 (= P_{II})$ and $\tilde{D}_8 (=P_{III}^{\tilde{D}_8})$ are calculated explicitly as examples of our theory.Comment: 38 pages, 4 figures, minor corrections, typo

Deformation of Okamoto–Painlevé pairs and Painlevé equations