# b-Functions

• Jan-Erik Björk
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 247)

## 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.

## Keywords

Exact Sequence Irreducible Component Complex Manifold Cohomology Class Period Integral
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

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