Analytic *D*-Modules and Applications
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# Regular holonomic *D*-modules

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## Summary

*D*-modules is studied. A holonomic complex

*M*on a complex manifold

*X*is regular holonomic if its formal solution complex is equal to its analytic solution complex at every point, i.e. if

*x*

_{0}∈ Supp(

*M*). The class of regular holonomic complexes is denoted by D

^{b}

_{r.h}(

*D*

_{ X }). A holonomic module is regular holonomic if its single degree complex is regular holonomic. The class of regular holonomic modules is denoted by RH(

*D*

_{ X }).

In Section 3 we prove that every regular holonomic complex is *fully regular* in the sense that every cohomology module is regular holonomic, as well as any of its holonomic subquotients. The full regularity is not at all obvious and its proof requires several steps. In particular we need special results in the case when dim(*X*) = 1. For this reason the first two sections treat *D*-module theory in dimension one, where the regular holonomicity is related to Fuchsian differential equations.

*Riemann-Hilbert correspondence*which asserts that the de Rham functor gives an equivalence of categories between

*D*

^{b}

_{r.h}(

*D*

_{ X }) and

*D*

^{b}

_{c}(C

_{ X }). This amounts to prove that the natural map

The Riemann-Hilbert correspondence implies that regular holonomic complexes may be defined by constructible sheaves. Special cases are analyzed in section 5 and 6. There is also the abelian category RH(*D* _{ X }) which by the Riemann-Hilbert correspondence gives an abelian subcategory of *D* ^{b} _{c}(C_{ X }). This subcategory is denoted by Perv(C_{ X }) and its objects are constructible sheaf complexes satisfying the perversity condition, expressed by upper bounds on the dimensions of its cohomology modules of the complex and its dual.

*V*. If

*d*=

*d*

_{ X }−

*d*

_{ V }there exists a unique largest holonomic D

_{ X }-submodule ℒ(

*V*) of

*H*

^{d}

_{[V]}(

*O*

_{ X }) whose holonomic dual has no torsion in the sense that the dimension of the support of any non-zero section is

*d*

_{ V }. We set

*IC*

^{•}(

*V*) of Perv(C

_{ X }) is the

*intersection complex*of

*V*. We prove that it is self-dual. Introducing local systems on complex analytic strata in

*X*one extends the construction of intersection complexes, where the strategy is to exhibit special regular holonomic

*D*

_{ X }-modules which occur in section 5.

Equivalent *regularity conditions* are established in section 6. We prove that a holonomic module is regular holonomic if and only if its inverse image to a curve is regular holonomic. Regularity can be interpretated by comparison properties and we expose a result by Mebkhout which relates the irregularity of a holonomic module along every analytic hypersurface *T* to a certain perverse sheaf on *T*.

In section 7 we construct the *L* ^{2}-lattice in regular holonomic modules and extend results from Chapter 4. Of particular interest is the fact that generators of ℒ(*V*) come from fundamental class sections. For hypersurfaces with isolated singularities Theorem 5.7.21 gives a necessary and sufficient condition in order that a section of *H* ^{1} _{[T]}(*0* _{ X }) belongs to ℒ(*T*).

Section 8 treats algebraic *D*-modules and the interplay with regular holonomic modules on the analytic manifold associated with an algebraic manifold to *D*-modules which are regular holonomic in the algebraic sense.

## Keywords

Exact Sequence Spectral Sequence Complex Manifold Left Ideal Chapter Versus## Notes

- Beilinson, A. A., Bernstein, J., and Deligne, P., Faisceux Pervers, Astérisque
**100**, 1982.Google Scholar - Katz, N., The regularity theorem in algebraic geometry, Actes Congres Int. Nice. Vol 1, 1970, pp. 437–443.Google Scholar
- [2]Björk, J-E., Non-commutative noetherian rings and their use in homological algebra, J. of pure and appl. Algebra
**38**(1985), 111–119.zbMATHCrossRefGoogle Scholar - Ramis, J-P., Variations sur le thème “GAGA”, Lecture Notes in Math. 894, Springer, 1978, pp. 228–289.Google Scholar
- [7]Mebkhout, Z., Une équivalence de catégories; une autre équivalence de catégories, Compositio Math.
**51**, 55–62 and 63–69.Google Scholar - Kashiwara, M. and Tanisaki, T., The characteristic cycles of holonomic systems on a flag manifold related to the Weyl group algebra, Invent. Math.
**77**(1984), 185–198.MathSciNetzbMATHGoogle Scholar - [8]Mebkhout, Z., La formalisme des six operations de Grothendieck pour les D-modules cohérents, Hermann, Paris, 1989.Google Scholar
- [4]Mebkhout, Z., Remarque sur l’irrégularite des DX-modules holonômes, ibid.
**303**(1986), 803–806.MathSciNetzbMATHGoogle Scholar - [2]Borel A., Algebraic D-modules, Academic Press, 1987.Google Scholar
- [1]Barlet, D., Fonctions de type trace, Ann. l’Inst. Fourier
**33**(1983), 43–76.MathSciNetzbMATHCrossRefGoogle Scholar