Is there an algebraic cycle corresponding to the Weil pairing on an abelian variety (of dim>1)? Ideally I'd like to see an example as explicit as possible, e.g.
an explicitly given variety of dim>1 and an explicitly given subvariety.

Do you mean the Poincare pairing between the cohomology of an abelian variety and its dual, or the pairing on the cohomology of an abelian variety induced by the choice of a polarization? I would think the latter is just the polarization pairing attached to the ample line bundle that's giving you the polarization (at least up to sign).
– Keerthi Madapusi PeraApr 17 '12 at 21:17

2

OP: Can you say more precisely what you are looking for? The Weil pairing is usually thought of as a pairing only on $n$-torsion points taking values in the group of $n^\text{th}$ roots of unity. In what sense would you like to see this as an algebraic cycle? Are you asking how to interpret this as a pairing of the corresponding torsion invertible sheaves (on the respective dual Abelian varieties)? For that, see Deligne's expose in SGA 4.
– Jason StarrApr 18 '12 at 10:36