Soergel bimodules and sheaves on the Hilbert scheme of points on plane

Abstract: Based on joint work with Rozansky. In my talk I outline a construction that produces a $\mathbb{C}^*\times\mathbb{C}^*$-equivariant complex of sheaves $S_b$ on $Hilb_n(\mathbb{C}^2)$ such that the space of global sections $H^*(S_b)$ of the complex are the Khovanov-Rozansky homology of the closure of the braid $b$. The construction is functorial with respect to adding a full twist to the braid. Thus we prove a weak version of the conjecture by Gorsky-Negut-Rasmussen. In the heart of our construction is a fully faithful functor from the category of Soergel bimodules to a particular category of matrix factorizations. I will keep the matrix factorization part minimal and concentrate on the main idea of the construction as well as key properties of the categories that we use.

algebraic geometry

Audience: researchers in the topic

UC Davis algebraic geometry seminar