×

zbMATH — the first resource for mathematics

dg-methods for microlocalization. (English) Zbl 1223.35019
Let \(X\) be a complex manifold, and let \({\mathcal E}_X\) be the ring of microdifferential operators. It acts on the microlocalization \(\mu \hom(F,{\mathcal O}_X)\), for \(F\) in the derived category of sheaves on \(X\). As a consequence of their new microlocalization, M. Kashiwara, P. Shapira, F. Ivorra and I. Waschkies proved [in: J. Bernstein (ed.) et al., Studies in Lie theory. Dedicated to A. Joseph on his sixtieth birthday. Basel: Birkhäuser. Progress in Mathematics 243, 171–221 (2006; Zbl 1098.35008)] that \(\mu\hom(F,{\mathcal O}_X)\) can be defined as an object of \(D({\mathcal E}_X)\) (and this follows from the fact that \(\mu_X{\mathcal O}_X\) is concentrated in one degree).
In this paper, the author proves that the tempered microlocalization \(T\)-\(\mu \hom(F,{\mathcal O}_X)\) and \(\mu_X{\mathcal O}_X^t\) are also objects of \(D({\mathcal E}_X)\). To accomplish this, since one does not know if \(\mu_X{\mathcal O}_X^t\) is concentrated in one degree, the author constructs resolutions of \({\mathcal E}_X\) and \(\mu_X{\mathcal O}_X^t\) such that the action of \({\mathcal E}_X\) is realized in the category of complexes, and then builds a microlocalization in this framework.

MSC:
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. Andronikof, Microlocalisation tempérée, Mém. Soc. Math. France (N.S.) 57 (1994). · Zbl 0805.58059 · numdam:MSMF_1994_2_57__1_0 · eudml:94909
[2] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer, Berlin, 1994. · Zbl 0808.14038 · doi:10.1007/BFb0073549
[3] E. Bierstone and P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5-42. · Zbl 0674.32002 · doi:10.1007/BF02699126 · numdam:PMIHES_1988__67__5_0 · eudml:104032
[4] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), 319-365. · Zbl 0566.32023 · doi:10.2977/prims/1195181610
[5] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss. 292, Springer, Berlin, 1990. · Zbl 0709.18001
[6] , Moderate and formal cohomology associated with constructible sheaves, Mém. Soc. Math. France (N.S.) 64 (1996). · Zbl 0881.58060 · numdam:MSMF_1996_2_64__1_0 · eudml:94916
[7] , Ind-sheaves, Astérisque 271 (2001). · Zbl 0993.32009
[8] , Categories and sheaves, Grundlehren Math. Wiss. 332, Springer, Berlin, 2006. · Zbl 1118.18001 · doi:10.1007/3-540-27950-4
[9] M. Kashiwara, P. Schapira, F. Ivorra and I. Waschkies, Microlocalization of ind-sheaves, in Studies in Lie theory, Progr. Math. 243, Birkhäuser, 2006, 171-221. · Zbl 1098.35008 · arxiv:math/0407371
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.