Chow Motives of 3-folds and Fiber Spaces, by Pedro L. del Angel and Stefan J. Mueller-Stach

Let k be a field of characteristic zero. For every smooth projective k-variety Y of dimension n which admits a proper, connected morphism f of relative dimension one onto a smooth projective k-variety S, we construct idempotent correspondences (projectors) in the product of Y with itself, i.e. pure Chow motives in the sense of Grothendieck. If n=3 and a precise condition on the second cohomology group of Y is satisfied, we can prove that there exists a Chow-Kuenneth decomposition of the diagonal on Y (Murre's Conjecture). We give examples, even 3-folds of general type, where our method applies.

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