Geometry Seminar

 

University of Tor Vergata, Department of Mathematics

6^th of November 2023, 14:30-16:00, room “Dal Passo”

 

 

 

 

 

 

 

 

On the smoothing problem for cycles in the Whitney range

 

 

Claire Voisin

Institut de Mathématiques de Jussieu-Paris rive gauche

 

 

Borel and Haefliger asked whether the group of cycle classes on a smooth projective variety  X is generated by classes of smooth subvarieties (such cycle classes will be said "smoothable").  Outside the Whitney range, that is, when the codimension c of the cycles is not greater than  the dimension d, there are  many counterexamples, the most recent ones being due to Olivier Benoist. In the Whitney range where c>d, it is known that (c-1)!z is smoothable for any cycle z of dimension d. Also Hironaka proved that cycles of dimension at most 3 are smoothable.

I study the cycles obtained by pushing-forward  products of divisors under a flat projective map from a smooth variety.  I show they are smoothable in the Whitney range and I conjecture that any cycle can be constructed this way. I prove that, for any cycle z of dimension d, (d-6)!z  can be constructed this way, which implies that  (d-6)!z is smoothable if d<c. In particular,   cycles  of dimension d at most 7 are smoothable if d<c.

.

 

 

 

 

 

 

 

 

 

 

 

 

This talk is part of the activity of the MIUR Excellence Department Projects MathMod@TOV, and the PRIN 2022 Moduli Spaces and Birational Geometry