Infinitesimal invariants for cycles modulo algebraic equivalence and 1-cycles on Jacobians

Abstract

We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers. We apply this construction to the Ikeda family, which gives optimal results for the Beauville decomposition of the 1-cycle of a very general plane curve in its Jacobian. IntroductionThis paper is devoted first of all to the construction of infinitesimal invariants for (families of) cycles modulo algebraic equivalence, similar to the infinitesimal invariants used in [30] for the study of families of cycles modulo rational equivalence, and secondly to a geometric application of these infinitesimal invariants to the study of the length of the Beauville decomposition of the 1-cycle of a curve in its Jacobian.Infinitesimal invariants appear in the following situation: Let π : Y → B be a smooth projective morphism, and let Z be a codimension n cycle in Y. We assume that the cohomology class [Z] ∈ H 2n (Y, Q) belongs to the s-th level of the Leray filtration on H 2n (Y, Q). Then we produce for any point b ∈ B an infinitesimal invariant δ[Z] b,alg which depends only on the first order neighborhood Z b,2 = Z |Y b,2 of the restricted cycle Z b of the fibre X b (hence the name "infinitesimal invariant"). More precisely, Z has a Dolbeault cohomology class [Z] n,n ∈ H n (Y, Ω n X ) and this infinitesimal invariant will depend only on the restricted classThis infinitesimal invariant has the property that it allows to decide whether the restricted cycle Z b is algebraically equivalent to 0 at the very general point b ∈ B. More precisely, we will prove the following result:

Computations with modified diagonals

“…Gross and Schoen [6] proved that if X is a (smooth projective) hyperelliptic curve and a is a fixed point of a hyperelliptic involution then Γ 3 (X; a) represents a torsion class in the Chow group of X 3 . On the other hand it is known that if X is a generic complex smooth plane curve and m is small compared to its genus then Γ m (X; a) is not algebraically equivalent to 0, whatever a is, see [11] (for the link between vanishing of Γ m (X; a) and Voisin's result on the Beauville decomposition of the Abel-Jacobi image of a curve see the proof of Prop.4.3 of [3]). Let X be a complex projective K3 surface: Beauville and Voisin [3] have proved that there exists c ∈ X such that the rational equivalence class of Γ 3 (X; c) is torsion.…”

On the dimension of Voisin sets in the moduli space of abelian varieties

“…where π(t) = y. The following is well-known by the experts and is a version of the classical results by Mumford and Roitman on zero-cycles (see [11,Lemma 2.2] and also [4]): Proposition 3.1 If for any t ∈ C B , the restricted cycle Z t is rationally equivalent to 0, then there is a dense Zariski open set V ⊂ C B such that…”

Some new results on modified diagonals

“…which is by definition p * 12 ∆ + X • p * 23 ∆ − X , thus proving formula (17). In order to prove formula (18), we use the fact that the 0-cycle b can be written as j n j x j , where the x j 's are i-invariant.…”

“…By Proposition 2.5, and using the fact that b satisfies property (27), (that is, condition (10) in Proposition 2.5), we conclude that Γ d(m−1)+1 (X, b) = 0 in CH(X d(m−1)+1 )/R. For nonvanishing results concerning the cycles Z s (when X is very general) and its decomposition, let us mention [7], [17] (in the later paper, it is proved that if g ≥ s 2 /2, then Z s = 0 modulo algebraic equivalence for a very general curve X of genus g). Let us show the following:…”