Regular holonomic D-modules
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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.
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.
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.
KeywordsExact Sequence Spectral Sequence Complex Manifold Left Ideal Chapter Versus
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