Microfunction solutions of holonomic systems with irregular singularities

One important feature of a holonomic system with regular singularities is that, if we consider the problem in the “complex holomorphic” category, its solution complex is cohomologically ℂ constructible. For example, for a pair of complex manifolds Y ↪ X and a holonomic ε x module M with regular singularities, the solution complex 0.0 $$\hom {\varepsilon _x}(M,C_{Y|X}^{,f}$$ has ℂ constructible cohomologies where C Y|X ℝ f denotes the sheaf of tempered holomorphic microfunctions. However, for a holonomic module with irregular singularities, the solution complex (0.0) is no longer ℂ constructible in general. Such a breakdown of complex structure is deeply connected with Stokes lines of ordinary differential equations with irregular singularities. The purpose of this article is to investigate the relations between solutions of a system and the classical Stokes lines.