Numerical and Homological Equivalence of Algebraic Cycles on Hodge Manifolds

Abstract

with rational coefficients). We denote by 3h (M), (resp. (3(M)) the cycles which are homologous, (resp. numerically equivalent) to zero. In general g n D3-hand 3m, = m. Matsusaka's work [5] implies 3l,= h. We show that 3 n (M) 3h (M) if and only if the ring of algebraic cohomology classes on M satisfies several equivalent criteria. These conditions are shown to be fulfilled if M is an abelian variety or if M has complex dimension ? 4. Given an algebraic cohomology class on a product M X N of Hodge manifolds it is not in general known whether its Kunneth components are also algebraic classes. If n((M X N) = (MX N) then we show that the ring of algebraic classes is invariant under Kunneth decomposition. (This last result was obtained after reading an announcement of a similar result in a letter from Grothendieck to Serre.) The author expresses his gratitude to Professor Arthur P. Mattuck for his suggestions and his encouragement. 1. In discussing Hodge manifolds we employ the notation used in [7]. Let Mm denote a lodge manifold of real dimension 2m. The exterior algebra of complex valued, C: forms on M is denoted by E (M). Letting PP: & (M) ---,P (31) denote the natural projection, we set X-z (-1) ppP p