A Desingularization Theorem for Systems of Microdifferential Equations

Summary

Centro de Matemática e Aplicações Fundamentais and Departamento de Matemática da Universidade de Lisboa

Abstract We prove that the blow up of a regular holonomic system of microdifferential equations is regular holonomic and calculate its support. We prove desingularization theorems for Lagrangian curves and for regular holonomic systems with support on a Lagrangian curve.

1: INTRODUCTION

In [N] we introduced a notion of blow up of a holonomic ε_x-module. In Section 4 we prove that, under reasonable assumptions, the blow up of a regular holonomic system is regular holonomic and calculate its support. These results motivate a desingularization game for Lagrangian subvarieties of a contact manifold. In Section 5 we show how to win the game when the contact manifold has dimension 3. As a consequence we get a desingularization theorem for regular holonomic ε_x-modules when the dimension of X equals 2. In Sections 2 and 3 we recall the main results of [N].

The author would like to thank M. Kashiwara for useful discussions.

2: LOGARITHMIC CONTACT MANIFOLDS

Let X be a complex manifold. A subset Y of X is called a normal crossings divisor if for any x^° ∈ Y there is an open neighbourhood U of x^°, a system of local coordinates (x₁,…x_n) defined on U and an integer v such that