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Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor. (English) Zbl 0955.14013
For a complex manifold \(X\) and a free divisor \(Y\subset X\) consider the \({\mathcal O}_X\)-modules \({\mathcal D}er(\log Y)\) of logarithmic derivations and \(\Omega ^1_X(\log Y)\) of logarithmic forms. The author proves that \({\mathcal V}_0^Y({\mathcal D}_X)={\mathcal O}_X[{\mathcal D}er(\log Y)]\) where \({\mathcal V}_{\bullet }^Y({\mathcal D}_X)\) is the \({\mathcal V}\)-filtration of Malgrange-Kashiwara relative to \(Y\) on the ring of differential operators on \(X\). This implies that \({\mathcal V}_0^Y({\mathcal D}_X)\) is a coherent sheaf. Another result is that for a left \({\mathcal V}_0^Y({\mathcal D}_X)\)-module \({\mathcal M}\) there exists a canonical isomorphism \(\Omega ^{\bullet}_X(\log Y)({\mathcal M})\cong {\mathbb R}{\mathcal H}om_{{\mathcal V}_0^Y({\mathcal D}_X)}({\mathcal O}_X,{\mathcal M})\). The author also proves that if the divisor \(Y\) is Koszul free (i.e. free for which the symbols of a basis of logarithmic derivations form a regular sequence in the graded ring associated to the filtration by the order on \({\mathcal D}_X\)), then the logarithmic de Rham complex \(\Omega ^{\bullet}_X(\log Y)\) is perverse.
Reviewer: V.P.Kostov (Nice)

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F40 de Rham cohomology and algebraic geometry
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References:
[1] J.E. BJÖRK , Rings of Differential Operators , (North Holland, Amsterdam, 1979 ). Zbl 0499.13009 · Zbl 0499.13009
[2] N. BOURBAKI , Algèbre Commutative , Chapitres 3 et 4, (Actualités Scientifiques et Industrielles, Vol. 1293, Hermann, Paris, 1967 ). · Zbl 0152.32603
[3] C. BĂNICĂ and O. STĂNĂSILĂ , Algebraic methods in the global theory of complex spaces , (John Wiley, New York, 1976 ). · Zbl 0334.32001
[4] F.J. CALDERÓN-MORENO , Quelques propriétés de la V-filtration relative à un diviseur libre , (Comptes Rendus Acad. Sci. Paris, t. 323, Série I, 1996 , pp. 377-381). MR 97i:32048 | Zbl 0862.32006 · Zbl 0862.32006
[5] F.J. CALDERÓN-MORENO , D. MOND , L. NARVÁEZ-MACARRO and F.J. CASTRO-JIMÉNEZ , Logarithmic Cohomology of the complement of a Plane Curve . (Preprint, Univ. of Warwick, 1999 ). · Zbl 1010.32016
[6] F.J. CASTRO-JIMÉNEZ , D. MOND and L. NARVÁEZ-MACARRO , Cohomology of the complement of a free divisor , (Transactions of the A.M.S., Vol. 348, 1996 , pp. 3037-3049). MR 96k:32072 | Zbl 0862.32021 · Zbl 0862.32021
[7] J. DAMON , Higher multiplicities and Almost Free Divisor and Complete Intersections , (Memoirs of the A.M.S., Vol. 589, 1996 ). MR 97d:32050 | Zbl 0867.32015 · Zbl 0867.32015
[8] P. DELIGNE , Equations Différentielles à Points Singuliers Réguliers , (Lect. Notes in Math, Vol. 163, Springer-Verlag, 1987 ). MR 54 #5232 | Zbl 0244.14004 · Zbl 0244.14004
[9] H. ESNAULT and E. VIEHWEG , Logarithmic De Rham complexes and vanishing theorems . Invent. Math., Vol. 86, 1986 , pp. 161-194). MR 87j:32088 | Zbl 0603.32006 · Zbl 0603.32006
[10] C. GODBILLON , Géométrie Différentielle et Mécanique Analytique . (Collection Méthodes, Hermann, Paris, 1969 ). MR 39 #3416 | Zbl 0174.24602 · Zbl 0174.24602
[11] M. KASHIWARA , Vanishing cycle sheaves and holonomic systems of differential equations , (Lect. Notes in Math, Vol. 1012, 1983 , pp. 134-142). MR 85e:58137 | Zbl 0566.32022 · Zbl 0566.32022
[12] B. MALGRANGE , Le polynôme de Bernstein-Sato et cohomologie évanescente , (Astérisque, Vol. 101-102, 1983 , pp. 233-267). MR 86f:58148 | Zbl 0528.32007 · Zbl 0528.32007
[13] Z. MEBKHOUT , Le formalisme des six opérations de Grothendieck pour les Dx-modules cohérents , (Travaux en cours, Vol. 35, 1989 , Hermann, Paris). MR 90m:32026 | Zbl 0686.14020 · Zbl 0686.14020
[14] F. PHAM , Singularités des systèmes de Gauss-Manin , (Progress in Math, Vol. 2, Birkhäuser, 1979 ). MR 81h:32015 | Zbl 0524.32015 · Zbl 0524.32015
[15] G.S. RINEHART , Differential forms on general commutative algebras , (Trans. A.M.S., Vol. 108, 1963 , pp. 195-222). MR 27 #4850 | Zbl 0113.26204 · Zbl 0113.26204
[16] K. SAITO , On the uniformization of complements of discriminant loci . (Preprint, Willians College, 1975 ).
[17] K. SAITO , Theory of logarithmic differential forms and logarithmic vector fields . (J. Fac. Sci. Univ. Tokyo, Vol. 27, 1980 , pp. 265-291). MR 83h:32023 | Zbl 0496.32007 · Zbl 0496.32007
[18] H. TERAO , Arrangements of hyperplanes and their freeness-I . (J. Fac. Sci. Univ. Tokyo, Vol. 27, 1980 , pp. 293-312). MR 84i:32016a | Zbl 0509.14006 · Zbl 0509.14006
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