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The Riemann-Hilbert problem for holonomic systems. (English) Zbl 0566.32023
Let X be a paracompact complex manifold of dimension n, \(X_{{\mathbb{R}}}\) the underlying real analytic manifold and \(\bar X\) the complex conjugate of X. Let \({\mathcal D}_ X\) and \({\mathcal O}_ X\) be the sheaf of differential operators and holomorphic functions. The ring \({\mathcal A}_{X_{{\mathbb{R}}}}\) and \({\mathcal D}_{X_{{\mathbb{R}}}}\) coincides with the restriction of \({\mathcal O}_{X\times \bar X}\) and \({\mathcal D}_{X\times \bar X}\). For a \({\mathcal D}_{X_{{\mathbb{R}}}}\)-module \({\mathcal M}\), R\({\mathcal H}om_{{\mathcal D}_{\bar X}}({\mathcal O}_{\bar X},{\mathcal M})\) is quasi- isomorphic to the Dolbeault complex \[ M\to^{{\bar \partial}}\Omega_{X_{{\mathbb{R}}}}^{(0,1)}\otimes_{{\mathcal A}}{\mathcal M}\to^{{\bar \partial}}\Omega_{X_{{\mathbb{R}}}}^{(0,2)}\otimes_{{\mathcal A}}{\mathcal M}\to...\to \Omega_{X_{{\mathbb{R}}}}^{(0,n)}\otimes_{{\mathcal A}}{\mathcal M}, \] where \(\Omega_{X_{{\mathbb{R}}}}^{(p,q)}\) is the sheaf of real analytic (p,q)-forms. Let Mod(X) be the category of sheaves of \({\mathbb{C}}\)- vector spaces on X, D(X) its derived category and \(D_ c^ b(X)\) the full sub-category of D(X) consisting of bounded complexes whose cohomology groups are constructible. Let Mod(\({\mathcal D}_ X)\) be the category of \({\mathcal D}_ X\)-modules, D(\({\mathcal D}_ X)\) its derived category and \(D^ b_{rh}({\mathcal D}_ X)\) the full sub-category of D(\({\mathcal D}_ X)\) consisting of bounded complexes whose cohomology groups are regular holonomic. Replace ”regular holonomic” with ”holonomic” and obtain Mod(\({\mathcal D}_ X^{\infty})\), D(\({\mathcal D}_ X^{\infty})\) and \(D^ b_ h({\mathcal D}_ X^{\infty})\) similarly. The author defines the functors \(J_ X: D^ b_{rh}({\mathcal D}_ X)\to D^ b_ h({\mathcal D}_ X^{\infty})\), \(\Phi_ X: D^ b_{rh}({\mathcal D}_ X)\to D_ c^ b(X)\), \(\Phi_ X^{\infty}: D_ h^ b({\mathcal D}_ X^{\infty})\to D_ c^ b(X)^ 0\), \(\Psi_ X: D_ c^ b(X)^ 0\to D({\mathcal D}_ X)\) and \(\Psi_ X^{\infty}: D_ c^ b(X)^ 0\to D({\mathcal D}_ X^{\infty})\) naturally. He proves that \(\Psi (D_ c^ b(X)^ 0)\subset D^ b_{rh}({\mathcal D}_ X)\) and \(\Psi^{\infty}=J\circ \Psi\), that J, \(\Phi\), \(\Phi\) \({}^{\infty}, \Psi\) and \(\Psi^{\infty}\) give the equivalence between the categories \(D_ c^ b(X)\), \(D^ b_{rh}({\mathcal D}_ X)\) and \(D_ h^ b({\mathcal D}_ X^{\infty})\) and that \(\Phi\) and \(\Psi\) are inverse to each other. Thus he proves the equivalence of the derived category of holonomic systems and that of constructible sheaves. This was announced by the author in Sémin. Goulaouic-Schwartz, Équations Dériv. Partielles 1979/80, exposé No.19 (1980; Zbl 0444.58014), and by Z. Mebkhout [in Complex analysis, microlocal calculus and relativistic quantum theory, Lect. Notes Phys. 126, 99-110 (1980; Zbl 0444.32003)] was given another proof. The key of the author’s proof is to reduce the problem to a simpler case by Hironaka’s desingularization.
Reviewer: J.Kajiwara

MSC:
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
35Q15 Riemann-Hilbert problems in context of PDEs
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
58J10 Differential complexes
32K15 Differentiable functions on analytic spaces, differentiable spaces
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