The Hodge Conjecture for General Prym Varieties

We work over C, the field of complex numbers. The Prym variety of a double cover C → D of a smooth connected projective curve D by a smooth connected curve C is defined (see [7]) as the identity component of the kernel of the norm homomorphism N : J(C) → J(D) between the Jacobians of the curves. This is an abelian variety polarised by the restriction of the canonical polarisation on J(C); we denote this variety by P (C → D) or simply P when there is no possibility of ambiguity. A Hodge class on a variety X is an integral singular cohomology class on the complex manifold X(C) which is represented by a closed differential form of type (p, p). The Hodge conjecture (see [3]) asserts that some multiple of such a class is the cohomology class of an algebraic cycle on X. Let A be an abelian variety. The Künneth decomposition implies that the rational singular cohomology of A × · · · × A is a direct sum of subquotients of tensor products of H(A(C),Q). Hence we have an action of a linear automorphism of this vector space on these cohomology groups. The Mumford-Tate group H(A) of A can thus be defined (see [2]) as the group of all linear automorphisms of H(A(C),Q) which stabilise all Hodge cycles on the varieties A× · · · ×A. The aim of this note is to show that the Mumford-Tate group H(P ) of a general Prym variety P (C → D) is isomorphic to the full sym-