Abstract. We generalize the construction of Raynaud [R78] of smooth projective surfaces of general type in positive characteristic that violate the Kodaira vanishing theorem. This corrects an earlier paper [T10] of the same purpose. These examples are smooth surfaces fibered over a smooth curve whose direct images of the relative dualizing sheaves are not nef, and they violate Kollár's vanishing theorem. Further pathologies on these examples include the existence of non-trivial vector fields and that of non-closed global differential 1-forms.
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