Microlocal Riemann–Hilbert Correspondence

Abstract

We construct the global microlocal Riemann-Hilbert correspondence as an explicit equivalence between the abelian stack of microlocal perverse sheaves defined in [W] and the abelian stack of regular holonomic microdifferential modules of [KK]. The theory of analytic ind-sheaves and its microlocalization is crucial for our construction since it allows us to define solution complexes with values in the (ind-)ring of microlocal holomorphic functions (resp. microlocal tempered holomorphic functions).

Constructibility and duality for simple holonomic modules on complex symplectic manifolds

“…Sketch of proof. (i) is well-known and follows from [10] (see [21] for further developments). The idea of the proof is as follows.…”

Morita Classes of Microdifferential Algebroids

“…For example, it makes sense to consider coherent modules over an E-algebroid, and in particular regular modules along complex involutive subvarieties of X. The Lagrangian case is of particular interest, since these modules are the counterpart of microlocal perverse sheaves in the Riemann-Hilbert correspondence (see [17,39,12,13]). A very interesting example of non-coherent module is the (twisted) sheaf of microfunctions along a totally real, I-symplectic Lagrangian submanifold (see [17, §4]).…”