b-Functions

Summary

This chapter deals with b-functions associated to regular holonomic modules. Given a regular holonomic D _x -module

$$M\left( {f,s} \right) = M\mathop \otimes \nolimits_{\mathop \sigma \nolimits_X } \mathop \sigma \nolimits_X \left[ {\mathop f\nolimits^{ - 1} ,s} \right] \otimes \mathop f\nolimits^s $$

where s is a parameter. We shall assume that zero is the sole critical value of f, M = [f ⁻¹] and SS(M) does not intersect the set $\mathop c\nolimits_f = \left\{ {\left( {x,\lambda df\left( x \right)} \right):x \in X\backslash \mathop f\nolimits^{ - 1} \left( 0 \right)} \right\}$ outside the zero-section. In section 1 we study D _x [s]-submodules of M(f, s) generated by L ⊗ f ⁸ when L is a coherent Ox-submodule of M. The main result asserts that Dx[s](L ⊗f⁸) is a coherent Dx-module and $SS\left( {\mathop D\nolimits_X \left[ S \right]\left( {\tau \otimes \mathop f\nolimits^s } \right)} \right) \subset \overline {SS\left( M \right)o\mathop C\nolimits_f }$ where $\overline {SS\left( M \right)o\mathop C\nolimits_f }$is the closure in T*(X) of the fiber sum of SS(M) and Cf over π ^-1 (X\f ^-1(0)). This is applied in section 2 to study characteristic varieties of regular holonomic Dx-modules under localisations along analytic hypersurfaces. At the end of section 2 we prove that

$$SS\left( {Del\left( {\mathop o\nolimits_{X\backslash T} \otimes \tau } \right)} \right) = \mathop T\nolimits_X^ * \left( X \right) \cup \left( {\overline {\mathop C\nolimits_f } \cap \mathop \pi \nolimits^{ - 1} \left( T \right)} \right)$$

where T = f ⁻¹ (0) and L is any local system in X\T.