Motivation for Hodge cycles

Details Motivation for Hodge cycles Motivation for Hodge cycles

2005-01-21

arxiv.org/abs/math/0501348v3

Mathematics

Algebraic Geometry

18 pages, latex; Final revision (to appear in Adv. in Math)

Scientific paper

Given two smooth projective varieties X and Y over a field, we say that X motivates Y if the (suitably defined) motive of Y is contained in the category generated from X by taking sums, summands and products. This notion has appeared implicitly in many places, but it seems useful to make it explicit. Some techniques are given for checking this condition, but in a nutshell it involves building a correspondence which "dominates" Y. Among the more interesting examples dealt with are moduli spaces. We show that in number of cases moduli spaces of sheaves over curves or surfaces are motivated by underly curve or surface. This allows us to check (or recheck) the (generalized) Hodge and Lefschetz standard conjectures for some of these examples. The dominating correspondence in these examples is built from the universal sheaf, and can, in some instances, be realized as a kind of Fourier-Mukai transform.

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