Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T01:15:19.453Z Has data issue: false hasContentIssue false
  • English
  • Français

RIEMANN–HILBERT CORRESPONDENCE FOR MIXED TWISTOR ${\mathcal{D}}$-MODULES

Part of: Partial differential equations on manifolds; differential operators Analytic spaces (Co)homology theory Singularities

Published online by Cambridge University Press:  19 May 2017

Teresa Monteiro Fernandes  and
Claude Sabbah
Teresa Monteiro Fernandes
Affiliation:
Centro de Matemática e Aplicações Fundamentais – Centro de investigação Operacional e Departamento de Matemática da FCUL, Edifício C 6, Piso 2, Campo Grande, 1700, Lisboa, Portugal ([email protected])
Claude Sabbah
Affiliation:
CMLS, École polytechnique, CNRS, Université Paris-Saclay, F–91128 Palaiseau cedex, France ([email protected])http://www.math.polytechnique.fr/perso/sabbah
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the notion of regularity for a relative holonomic ${\mathcal{D}}$-module in the sense of Monteiro Fernandes and Sabbah [Internat. Math. Res. Not. (21) (2013), 4961–4984]. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative ${\mathcal{D}}$-modules underlying a regular mixed twistor ${\mathcal{D}}$-module, this functor satisfies the left quasi-inverse property.

Keywords

holonomic relative D-moduleregularityrelative constructible sheafrelative perverse sheafmixed twistor D-module

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38: Sheaves of differential operators and their modules, $D$-modules 32S40: Monodromy; relations with differential equations and $D$-modules 32S60: Stratifications; constructible sheaves; intersection cohomology 58J10: Differential complexes ; elliptic complexes
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 3 , May 2019 , pp. 629 - 672
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The research of TMF was supported by Fundação para a Ciência e Tecnologia UID/MAT/04561/2013. The research of CS was supported by the grant ANR-13-IS01-0001-01 of the Agence nationale de la recherche.

References

D’Agnolo, A., Guillermou, S. and Schapira, P., Regular holonomic D[[ℏ]]-modules, Publ. Res. Inst. Math. Sci., Kyoto Univ. 47(1) (2011), 221–255.Google Scholar
Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Volume 163 (Springer, Berlin, Heidelberg, New York, 1970).Google Scholar
Edmundo, M. J. and Prelli, L., Sheaves on 𝓣-topologies, J. Math. Soc. Japan 68(1) (2016), 347–381.Google Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T., D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, Volume 236 (Birkhäuser, Boston, Basel, Berlin, 2008). in Japanese: 1995.Google Scholar
Kashiwara, M., On the holonomic systems of differential equations II, Invent. Math. 49 (1978), 121–135.Google Scholar
Kashiwara, M., The Riemann–Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., Kyoto Univ. 20 (1984), 319–365.Google Scholar
Kashiwara, M., D-Modules and Microlocal Calculus, Translations of Mathematical Monographs, Volume 217 (American Mathematical Society, Providence, RI, 2003).Google Scholar
Kashiwara, M. and Kawai, T., On the holonomic systems of differential equations (systems with regular singularities) III, Publ. Res. Inst. Math. Sci., Kyoto Univ. 17 (1981), 813–979.Google Scholar
Kashiwara, M. and Schapira, P., Sheaves on Manifolds, Grundlehren Math. Wiss, Volume 292 (Springer, Berlin, Heidelberg, 1990).Google Scholar
Kashiwara, M. and Schapira, P., Moderate and Formal Cohomology Associated with Constructible Sheaves, Mém. Soc. Math. Fr. (NS), Volume 64 (Société Mathématique de France, Paris, 1996).Google Scholar
Kashiwara, M. and Schapira, P., Ind-sheaves, Astérisque, Volume 271 (Société Mathématique de France, Paris, 2001).Google Scholar
Kashiwara, M. and Schapira, P., Categories and Sheaves, Grundlehren Math. Wiss., Volume 332 (Springer, Berlin, Heidelberg, 2006).Google Scholar
Mebkhout, Z., Le formalisme des six opérations de Grothendieck pour les D-modules cohérents, Travaux en cours, Volume  35 (Hermann, Paris, 1989).Google Scholar
Mebkhout, Z., Le théorème de positivité, le théorème de comparaison et le théorème d’existence de Riemann, Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, Volume  8 pp. 165–310 (Société Mathématique de France, Paris, 2004).Google Scholar
Mochizuki, T., Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor D-Modules, Mem. Amer. Math. Soc., Volume 185, no. 869–870 (American Mathematical Society, Providence, RI, 2007). arXiv:math.DG/0312230 & arXiv:math.DG/0402122.Google Scholar
Mochizuki, T., Mixed Twistor D-Modules, Lecture Notes in Mathematics, Volume 2125 (Springer, Heidelberg, New York, 2015).Google Scholar
Monteiro Fernandes, T. and Prelli, L., Relative subanalytic sheaves, Fund. Math. 226(1) (2014), 79–100.Google Scholar
Monteiro Fernandes, T. and Sabbah, C., On the de Rham complex of mixed twistor D-modules, Int. Math. Res. Not. (IMRN) (21) (2013), 4961–4984.Google Scholar
Prelli, L., Sheaves on subanalytic sites, Rend. Semin. Mat. Univ. Padova 120 (2008), 167–216.Google Scholar
Prelli, L., Microlocalization of Subanalytic Sheaves, Mém. Soc. Math. Fr. (NS), Volume 135 (Société Mathématique de France, Paris, 2013).Google Scholar
Sabbah, C., Proximité évanescente, I. La structure polaire d’un D-module, Appendice en collaboration avec F. Castro, Compos. Math. 62 (1987), 283–328.Google Scholar
Sabbah, C., Polarizable Twistor D-Modules, Astérisque, Volume 300 (Société Mathématique de France, Paris, 2005).Google Scholar
Schapira, P. and Schneiders, J.-P., Index Theorem for Elliptic Pairs, Astérisque, Volume 224 (Société Mathématique de France, Paris, 1994).Google Scholar
Wang, L., The constructibility theorem for differential modules, PhD thesis, University of Illinois at Chicago (2008) http://indigo.uic.edu/handle/10027/13547.Google Scholar