Hodge Cycles, Motives, and Shimura Varieties

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…[12]). The Tannakian formalism, see [65], singles out the categories which are equivalent to categories of representations of affine groups schemes and it gives a precise prescription for reconstructing the group scheme from its category of representations. The geometrization of the Satake isomorphism essentially states that the category P G(O) is equivalent to the category of representations Repr( L G) of the Langlands dual group L G, so it yields a recipe to re-construct this dual group.…”

Integral motives and special values of zeta functions

“…For tensor structures on categories we use the terminology of [10]. Thus a tensor category C is a category together with a bifunctor ⊗ : C × C → C and compatible associativity and commutativity constraints for which there exists an identity object .…”

The Hodge conjecture for general Prym varieties

“…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 1 (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 symplectic group Sp(2g); where the class in 2 ∧ H 1 (P (C), Q) = H 2 (P (C), Q) which is stabilised by this group is the first Chern class of the natural polarisation on the Prym variety. Invariant theory (see [9] or [5] and [6]) then implies that the only Hodge cycles on P are powers (under cup-product) of this polarisation class.…”

“…The results on Mumford-Tate groups are applied to this larger family of Prym varieties in section three. In addition we use the semi-simplicity of the Mumford-Tate group (see [2]) and the result of Pirola (see [8]) to reduce the problem to an elementary lemma on subgroups of the symplectic group.…”