|Speaker:||Anton Fonarev (HSE Moscow)|
|Title:||On the generalised Dubrovin's conjecture for Grassmannians|
|Date:||Tue, Nov 13, 2018, 13:15 - 14:30|
|Place:||Seminar Room A|
A celebrated conjecture by Dubrovin gives an intriguing relation between the quantum cohomology of a smooth projective variety and the bounded derived category of coherent sheaves on it. More precisely, it says that having generically semisimple big quantum cohomology is equivalent to admitting a full exceptional collection in the derived category. Several years ago S. Galkin, A. Melit, and M. Smirnov and independently N. Perrin showed that the conjecture does not hold if big quantum cohomology is replaced by small quantum cohomology. However, in their recent preprint A. Kuznetsov and M. Smirnov suggested addendum to Dubrovin's conjecture which suggests that semisimplicity of the small quantum cohomology of a smooth Fano variety with Picard rank equal to 1 implies existence of an exceptional collection with some exceptionally nice properties in its derived category. They also did some work towards the proof of their conjecture for classical Grassmannians.
It the first part of the talk we will broadly cover the history of Dubrovin's conjecture and its modifications. In the second part of the talk we will explain how to check that the Kuznetsov-Smirnov conjecture in the surprisingly nontrivial case of the classical Grassmannian varieties.