Maxim Kontsevich - Riemann-Hilbert correspondence for q-difference modules
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2017-06-13

Abstract:
For complex number q,0 less than |q| less than 1 denote by Aq:=C〈X±1,Y±1〉/(relationY X=qXY) the corresponding quantum torus algebra.  By the q-version of Riemann-Hilbert correspondence (essentially due to Ramis, Sauloy and Zhang, 2009), the category of holonomic Aq-modules is naturally equivalent to the abelian category of coherent sheaves on elliptic curve Eq:=C×/qZ
endowed with two anti-Harder-Narasimhan filtrations.  I propose a generalization of this correspondence to the higher-dimensional case A⊗nq, n more than 1 (joint work in progress with Y. Soibelman).  The hypothetical description of the “Betti side” of RH-correspondence combines the categories of constructible sheaves with a given microlocal support, and the bounded derived category of coherent sheaves on the abelian variety Enq.