Vector fields and differential forms on generalized Raynaud surfaces

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We consider the tangent and cotangent sheaves of generalized Raynaud surfaces, which have cuspidal fibrations and which are mostly of general type.In particular, we compute the dimensions of the space of vector fields and of non-closed differential 1-forms.Let /: V-+B be a fibration from a smooth projective surface to a smooth projective curve over an algebraically closed field k.In the case of characteristic zero, almost all fibres of / are nonsingular.In the case of positive characteristic, however, there exists fibrations whose general fibres have singularities.Generalized Raynaud surfaces are typical examples of surfaces which have such fibrations.Moreover, these surfaces have interesting geometry in positive characteristic.In the present article, we compute the dimension of the space of vector fields and give an estimate for the difference of the dimension of the space of all global differential 1-forms from that of the space of closed differential 1-forms on generalized Raynaud surfaces.LEMMA 1.1 (Kurke [1]).Let ω be an exact differential on a smooth projective curve C. Suppose that the divisor of ω has the form pnD, where D is a nonzero effective divisor and n is a positive integer with nψO (mod/?) and n>\.Then there exist local sections 1991 Mathematics Subject Classification.Primary 14J29.