On the Smoothing Problem for Cycles in the Whitney Range

Explore a lecture on the smoothing problem for cycles in the Whitney range, presented by Claire Voisin at the Hausdorff Center for Mathematics. Delve into the question posed by Borel and Haefliger regarding whether the group of cycle classes on a smooth projective variety X is generated by classes of smooth subvarieties. Examine counterexamples outside the Whitney range and learn about known results within it, including Hironaka's proof for cycles of dimension at most 3. Investigate Voisin's study of cycles obtained by pushing-forward products of divisors under a flat projective map from a smooth variety. Discover her conjecture that any cycle can be constructed this way and her proof that for any cycle z of dimension d, (d - 6)!z can be constructed using this method. Understand the implications of this result, particularly for cycles of dimension d at most 7 when d is smaller than the codimension c.