| Résume||In the algebraic theory of differential equations, the classical Riemann—Hilbert correspondence tells us that perverse sheaves are the topological counterpart of holonomic D-modules with regular singularities. It was a long-standing problem to establish a similar result in the case of (possibly) irregular singularities. Such a result has been achieved by D’Agnolo—Kashiwara in 2013, providing a topological and sheaf-theoretic framework for computations in this more general context.
In the first part of the talk, I will give an introduction to this subject, motivating in particular the use of the theory of ind-sheaves of Kashiwara—Schapira, on which the construction of enhanced perverse sheaves relies.
In the second part, I will report on a recent joint work with D. Barco, M. Hien and C. Sevenheck, where hypergeometric differential equations are studied using these techniques. It is shown that under appropriate symmetry conditions on the parameters determining such a system, the associated enhanced perverse sheaf (a priori defined over the complex numbers) has a structure over some smaller field. Such a question is motivated, for instance, by Hodge theory, where perverse sheaves over the rationals play an important role.|