On the Periods of Certain Rational Integrals: I

In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how Macaulay's theorem 4.11 is essentially equivalent to a suitable vanishing theorem for sheaf cohomology. We shall also give a proof of the de Rham algebraic theorem used in the proof of Theorem 5.3. However, our principal motivation is to be able to discuss rational integrals in case our hypersurface V c P, has rather simple singularities, and the localization technique of sheaf theory seems to be the best method for doing this (cf. ?? 15, 16 below). Let then V c P, be a non-singular hypersurface. We want to give a sheaf-theoretic version of the Hodge filtration of Hq(V, C) (cf. ? 8). For this we let Qv be the sheaf on V of holomorphic q-forms and & c QV the subsheaf of closed forms. The Poincare' lemma for holomorphic differentials gives an exact sequence