# Regular holonomic D-modules

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

## Summary

This is a central chapter of this book where the class of regular holonomic 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
$$RHo{m_{Dx({x_0})}}(M({x_0}),{\widehat O_X}({x_0})/{O_X}({x_0})) = 0$$
for every x 0 ∈ Supp(M). The class of regular holonomic complexes is denoted by Db 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.

Once the full regularity is proved for regular holonomic complexes we use results about Deligne modules to prove that regular holonomicity is preserved under various opertions. A conclusive list of results occurs in Theorem 5.4.1. In section 5 we establish the 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
$${\text{R}}Ho{m_{Dx}}(M,N) \to {\text{R}}Hom{c_X}({\text{D}}{{\text{R}}_X}(M),{\text{D}}{{\text{R}}_X}(N))$$
is bijective for every pair of regular holonomic complexes and that every constructible sheaf complex is the de Rham complex of a regular holonomic complex.

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.

Of particular interest is the intersection complex of a pure-dimensional analytic set 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
$$DRx(\ell (V)) = I{C^ \bullet }(V)$$
The object 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
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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