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We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.

We introduce the notion of regularity for a relative holonomic ${\mathcal{D}}$-module in the sense of Monteiro Fernandes and Sabbah [Internat. Math. Res. Not. (21) (2013), 4961–4984]. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative ${\mathcal{D}}$-modules underlying a regular mixed twistor ${\mathcal{D}}$-module, this functor satisfies the left quasi-inverse property.

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Relative regular Riemann–Hilbert correspondence II

RIEMANN–HILBERT CORRESPONDENCE FOR MIXED TWISTOR ${\mathcal{D}}$-MODULES

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Relative regular Riemann–Hilbert correspondence II

RIEMANN–HILBERT CORRESPONDENCE FOR MIXED TWISTOR ${\mathcal{D}}$-MODULES