EXTENSION OF FUNCTORS FOR ALGEBRAS OF FORMAL DEFORMATION

MARTINS, ANA RITA ; FERNANDES, TERESA MONTEIRO ; RAIMUNDO, DAVID

Glasgow mathematical journal, Tome 56 (2014) no. 1, pp. 103-141

Résumé

Suppose we are given complex manifolds X and Y together with substacks $\mathcal{S}$ and $\mathcal{S}'$ of modules over algebras of formal deformation $\mathcal{A}$ on X and $\mathcal{A}'$ on Y, respectively. Also, suppose we are given a functor Φ from the category of open subsets of X to the category of open subsets of Y together with a functor F of prestacks from $\mathcal{S}$ to $\mathcal{S}'\circ\Phi$. Then we give conditions for the existence of a canonical functor, extension of F to the category of coherent $\mathcal{A}$-modules such that the cohomology associated to the action of the formal parameter $\hbar$ takes values in $\mathcal{S}$. We give an explicit construction and prove that when the initial functor F is exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of $\mathcal{D}[[\hbar]]$-modules. We also obtain the Cauchy–Kowalewskaia–Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic $\mathcal{D}[[\hbar]]$-modules and a coherency criterion for proper direct images of good $\mathcal{D}[[\hbar]]$-modules.

Détail

DOI : 10.1017/S0017089513000116

Mots-clés : Primary: 32C38, 46L65, Secondary: 18E30, 46A13

Bibliographie

[1] 1.Brylinski, J. L., Malgrange, B. and Verdier, J. L., Transformation de Fourier geométrique I, II, C.R. Acad. Sci. 297 (1983), 55–58; (1986), 193–198. Google Scholar

[2] 2.D'Agnolo, A., Guillermou, S. and Schapira, P., Regular holonomic -modules (RIMS, Kyoto University, 2010). Google Scholar

[3] 3.Kashiwara, M., On the holonomic systems of linear differential equations II, Invent. Math. 49 (1978) 121–135. Google Scholar | DOI

[4] 4.Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations, Lecture Notes in Mathematics, vol. 1016 (Springer-Verlag, Berlin, Germany, 1983). Google Scholar | DOI

[5] 5.Kashiwara, M., -modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217 (American Mathematical Society, Providence, RI, 2003). Google Scholar

[6] 6.Kashiwara, M. and Schapira, P., Sheaves on manifolds, Grundlehren der Math. Wiss. vol. 292 (Springer-Verlag, Berlin Germany, 1990). Google Scholar

[7] 7.Kashiwara, M. and Schapira, P., Categories and sheaves, Grundlehren der Math. Wiss. vol. 332 (Springer-Verlag, Berlin Germany, 2006). Google Scholar

[8] 8.Kashiwara, M. and Schapira, P., Deformation quantization modules, Astérisque (Soc. Math. France) 345 (2012) arXiv:1003.3304v2. Google Scholar

[9] 9.Laurent, Y. and Malgrange, B., Cycles proches, spécialisation et -modules, Annales de l'institut Fourier 45 (n5) (1995), 1353–1405. Google Scholar

[10] 10.Laurent, Y. and Schapira, P., Images inverses des modules différentiels, Compositio Math. 61 (1987), 229–251. Google Scholar

[11] 11.Maisonobe, P. and Mebkhout, Z., Le théorème de comparaison pour les cycles évanescents, Séminaires et Congrès, 8 (2004), 311–389 (Societe Mathîmatique de France). Google Scholar

[12] 12.Mebkhout, Z., Le formalisme des six opérations de Grothendieck pour les -modules cohérents, III, § 4. (en collaboration avec C. Sabbah; French edn.) Travaux en Cours collection No. 35 (Hermann, Paris, France, 1989). Google Scholar

[13] 13.Fernandes, T. Monteiro, Microlocalisation of -modules along a submanifold, Bull. Soc. Math. France 123 (1995), 293–327. Google Scholar