Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

Abstract

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers $${\mathcal{C}}^{[\bullet]}$$ and the relative Jacobian $${\mathcal{J}}$$ of a family of curves $${\mathcal{C}}/S$$ . As one of the applications, we show that in the case of a single curve C this action induces a $${\mathbb{Z}}$$ -form of a Lefschetz sl2- action on the Chow groups of C [N]. Another application gives a new grading on the ring CH 0(J) of 0-cycles on the Jacobian J of C (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in CH $$^{*}({\mathcal{C}}^{[\bullet]})$$ and in CH $$^{*}({\mathcal{J}})$$ and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in [5]. We show that our algebras of operators preserve the subrings of tautological cycles and act on them as some explicit differential operators.