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.

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    $\begingroup$ 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). $\endgroup$ – Keerthi Madapusi Pera Apr 17 '12 at 21:17
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    $\begingroup$ 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. $\endgroup$ – Jason Starr Apr 18 '12 at 10:36

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