Voevodsky conjectured that numerical equivalence and smash equivalence coincide on a smooth projective variety. We prove the conjecture for 1-cycles on varieties dominated by products of curves.

We show that homologically trivial algebraic cycles on a 3-dimensional abelian variety are smash-nilpotent.

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Let E be a vector bundle on a smooth complex projective curve C of genus at least two. Let Q(E, d) be the Quot scheme parameterizing the torsion quotients of E of degree d. We compute the cohomologies of the tangent bundle T Q(E,d) . In particular, the space of infinitesimal deformations of Q(E, d) is computed. Kempf and Fantechi computed the space of infinitesimal deformations of]). We also explicitly describe the infinitesimal deformations of Q(E, d).

Voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety. We prove this conjecture holds for uniruled 3-folds, 4-folds whose MRCC quotient has dimension ≤ 2 and for smooth complete intersections with very small degree.