Algebraic Cycles as Residues of Meromorphic Forms

Abstract

According to a classical result of Weil [15], a divisor α of a smooth n-dimensional projective variety X is homologous to zero if and only if it is the residue of a closed meromorphic 1-form on X. Griffiths proved recently [9, pp. 3-8] that a 0-cycle α of X is homologous to zero if and only if it is the Grothendieck residue of a meromorphic n-form ώ on X having poles in the union of a family of complex hypersurfaces Y1 . . . . . Yn, of X, such that ∩ Yi is 0-dimensional and contains the support of α. We show in this paper (Theorem 3.7) that, in fact, any q-dimensional algebraic cycle α of X, 0≦ q ≦ n, is the analytic residue of a semimeromorphic (n-q)-form ώ on X, having poles in the union of a family F = {Y1 . . . . . Yn-q} of hypersurfaces in X such that ∩ F contains the support of α.