In a recent paper Zucker has proved the Hodge conjecture for cubic fourfolds [9]. Subsequently, one of the authors has given another proof, valid for any unirational fourfold [6]. The methods used there, combined with results by Bloch [2], give also a proof of the Hodge conjecture for quartic fourfolds.Let X be a smooth projective variety defined over the complex numbers tlS. Consider the Hodge decomposition: H'(X, ff;)= ~ HI',q(X). p+q=i Let CHP(X) denote the Chow group of algebraic cycles on X, modulo rational equivalence, of codimension p. Consider the standard map: ;.~ : COP(X)®~--, HZ.(X, ~)~n",.(X) .The Hodge (p, p)-conjecture states that the map 2~ is surjective, i.e. that every rational cohomology class of type (p, p) comes from an algebraic cycle with rational coefficients. If dimX = n, the Hodge (p, p)-conjecture is always true for p =0, 1, n-1, n (see [5], Propositions 3.1 and 3.4).In this paper we will show that the Hodge (2, 2)-conjecture holds for any fourfold admitting an algebraic family of rational curves covering it (Theorem 1). The proof is based on two lemmas already given in [6] and on a third lemma stating that the Hodge (2, 2)-conjecture holds for any fourfold of the form X × Y, where X is a smooth threefold and Ya smooth rational curve. The rest of the paper is devoted to show that the condition under which Theorem 1 holds is satisfied for smooth quintic fourfolds in IP 5 (Theorem 2) and for quadric bundles of dimension 4 (Theorem 3), so that the Hodge (2, 2)-conjecture holds for them. In particular, we prove that the generic smooth quintic fourfold in IP 5 contains a three-dimensional family of conics such that through a generic point of it go finitely malay of them. The proof of this result is inspired by the methods used by Predonzan [7] to study the family of lines lying on a smooth hypersurface in IP" and by Tennison [8] to study the family of conics lying on the quartic threefold in IP 4. Throughout the paper we will work over the field ¢ of complex numbers.
