The Hodge conjecture for fourfolds admitting a covering by rational curves

Abstract

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.

Some aspects of the Hodge conjecture

“…The Hodge conjecture (for projective varieties) is true for rational Hodge classes of degree 4 on varieties which are swept out by rational curves (called uniruled varieties). This is a result which is due to Conte and Murre [12] in dimension 4, and which has been generalized by Bloch and Srinivas in [7].…”

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

“…This result was originally proven by Conte and Murre [26] in the case where X is a four-dimensional variety covered by rational curves. Such a variety X satisfies the hypothesis, since every point x is contained in a rational curve C x whose normalization is isomorphic to P 1 .…”

“…The difficulty here is to prove the result for integral Hodge classes. Indeed, the fact that degree 4 rational Hodge classes are algebraic for X as above can be proved by using either the results of [26] or Bloch and Srinivas [15] (see Section 3.1.2), since such an X is swept out by rational curves, hence has its CH 0 group supported on a three-dimensional closed algebraic subset, or by using the method of Zucker [115], who uses the theory of normal functions, which we have essentially followed here. Corollary 6.32 in the case of a fourfold X fibered by complete intersections of two quadrics in P 5 has been re-proved by Colliot-Thélène [22] without any assumptions on singular fibers.…”

Th�or�me de Torelli pour les cubiques de ?5

“…C'est cet isomorphisme d'Abel-Jacobi qui rend accessible le probl6me de Torelli pour la cubique; en effet, l'application des p6riodes est un isomorphisme local; c'est pour cela que la m6thode de Donagi est inapplicable, et, faute d'information infinit6simale, que l'on doit interpr6ter la structure de Hodge elle-m~me: or m6me avec une r6ponse positive/L la conjecture de Hodge pour la cubique (plus g6n6ralement pour des var~6t6s de dimension quatre balay6es par des courbes rationnelles [4]), on a peu de maniement des classes enti6res de type (2,2), tandis que la th6orie des diviseurs permet de se ramener aux objets de la g6om6trie alg6brique.…”