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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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References:

[1] Hironaka, H., Resolution of singularities of an algebraic variety over a field of charac- teristic 0, Annals of Math., 79 (1964), 109-326. · Zbl 0122.38603
[2] - , Subanalytic sets, Number theory, algebraic geometry and commutative algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo (1973), 453-493. · Zbl 0259.00006
[3] - Introduction aux ensembles sous-analytiques, Asterisque 1 et 8, Societe math. de France (1973), 13-20, redige par A. Hirschowitz and P. LeBarz.
[4] Kashiwara, M., Faisceaux constructibles et systemes holonomes d’equations aux derivees partielles lineaires a points singuliers reguliers, Sdm. Goulaouic-Schwartz, 1979-80, expose 19. · Zbl 0444.58014
[5] - , On the maximally overdetermined system of linear differential equations, I, Publ. RIMS, Kyoto Univ., 10 (1975) 563-579. · Zbl 0313.58019
[6] - , .^-functions and holonomic systems, Invent. Math., 38 (1976), 33-53. · Zbl 0354.35082
[7] Kashiwara, M. and Kawai, T., On holonomic systems of microdifferential equations III- system with regular singularities, Publ. RIMS, Kyoto Univ., 17 (1981), 813-979. · Zbl 0505.58033
[8] Lojasiewicz, S., Sur le probleme de la division, Stadia Math., 8 (1959), 87-136.
[9] Martineau, A., Distributions et valeurs au bord des fonctions holomorphes, Oeuvre de Andre Martineau, Paris, CNRS, U977), 439-582.
[10] Mebkout, Z., Sur le probleme de Riemann-Hilbert, Lecture Notes in Physics, 126, Berlin-Heidelberg-New York, Springer (1980), 99-110.
[11] Schwartz, L., Theorie des distributions, Paris, Hermann (1950). · Zbl 0037.07301
[12] Sato, M., Kawai, T. and Kashiwara, M., Microfunctions and pseudo-differential equations, Lecture Notes in Math., 649, Berlin-Heidelberg-New York, Springer (1978), 228-289. · Zbl 0277.46039
[13] Verdier, J. L., Dualite dans la cohomologie des espaces localement compacts, Sim. Bourbaki, expose 300 (1965). Added in proof’. The proof of Theorem 2.8 is not complete because we assumed dim Fz<oo for an -K-constructible sheaf F but not for an S-constructible sheaf F. This difficulty can be overcome by one of the following methods. The first method relies on the fact that Theorem 2. 8 is proven if we replace ”12-constructible” with ”weakly i^-constructible” and that the functor TH is a well-defined exact functor on the category of weakly J2-construc- tible sheaves. The second method is to prove Theorem 2. 8 in the original form by using the following lemma which can be easily shown. Lemma. If F’ is a bounded complex of S-constructive sheaves on &such that dimJfj:(F’)x <C^\circ ^\circ , then F’ is quasi-isomorphic to a bounded complex F’  of S-cons true tible sheaves such that
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