D-modules of pure Gaussian type and enhanced ind-sheaves
Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules...
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| Published in | Manuscripta mathematica Vol. 167; no. 3-4; pp. 435 - 467 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.03.2022
Springer Nature B.V |
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| ISSN | 0025-2611 1432-1785 |
| DOI | 10.1007/s00229-021-01281-y |
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| Abstract | Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah. |
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| AbstractList | Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah. |
| Author | Hohl, Andreas |
| Author_xml | – sequence: 1 givenname: Andreas orcidid: 0000-0002-9335-067X surname: Hohl fullname: Hohl, Andreas email: andreas.hohl@mathematik.tu-chemnitz.de organization: Fakultät für Mathematik, Technische Universität Chemnitz |
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| Cites_doi | 10.4171/RSMUP/134-4 10.1016/j.aim.2018.09.022 10.1007/978-94-017-0717-6 10.1070/IM8308 10.2140/gt.2015.19.3467 10.1007/978-3-642-31695-1 10.1007/978-3-662-12492-5 10.1007/s00029-015-0185-y 10.2977/prims/1195181610 10.1515/crelle-2016-0062 10.1007/978-0-8176-4523-6 10.1090/conm/474/09262 10.1007/s10240-015-0076-y 10.1007/s11537-016-1564-7 10.1016/j.aim.2020.107093 10.1017/CBO9781316675625 10.1007/s10240-012-0044-8 10.5802/aif.3323 10.1007/BF02698808 |
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| References | BoalchPGlobal Weyl groups and a new theory of multiplicative quiver varietiesGeom. Topol.20151934673536344710810.2140/gt.2015.19.3467 KashiwaraMSchapiraPCategories and Sheaves, Grundlehren der mathematischen Wissenschaften2006BerlinSpringer1118.18001 Kashiwara, M., Schapira, P.: Ind-sheaves. Astérisque 271, (2001) D’AgnoloAKashiwaraMEnhanced perversitiesJ. Reine Angew. Math.2019751185241395669410.1515/crelle-2016-0062 Deligne, P.: Lettre à B. Malgrange du 19/4/1978. In: Singularités irrégulières, Correspondance et documents, Documents mathématiques 5, Société Mathématique de France, Paris, pp. 25–26 (2007) HienMSabbahCThe local Laplace transform of an elementary irregular meromorphic connectionRend. Sem. Mat. Univ. Padova2015134133196342841710.4171/RSMUP/134-4 HottaRTakeuchiKTanisakiTD-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics2008BostonBirkhäuser10.1007/978-0-8176-4523-6 MalgrangeBÉquations différentielles à coefficients polynomiaux, Progress in Mathematics1991BostonBirkhëuser0764.32001 KashiwaraMD-Modules and Microlocal Calculus, Translations of Mathematical Monographs2003ProvidenceAmerican Mathematical Society KatzNMLaumonGTransformation de Fourier et majoration de sommes exponentiellesPubl. Math. Inst. Hautes Études Sci.19856214520210.1007/BF02698808 MochizukiTNote on the Stokes structure of Fourier transformActa Math. Vietnam.20103510715826421661201.32016 D’AgnoloAHienMMorandoGSabbahCTopological computation of some Stokes phenomena on the affine lineAnn. Inst. Fourier202070739808410595010.5802/aif.3323 KashiwaraMSchapiraPSheaves on Manifolds, Grundlehren der mathematischen Wissenschaften1990BerlinSpringer SabbahCDifferential systems of pure Gaussian typeIzv. Math.201680189220346268010.1070/IM8308 KashiwaraMThe Riemann-Hilbert problem for holonomic systemsPubl. Res. Inst. Math. Sci.19842031936574338210.2977/prims/1195181610 KashiwaraMSchapiraPIrregular holonomic kernels and Laplace transformSel. Math. New Ser.20162255109343783310.1007/s00029-015-0185-y KashiwaraMSchapiraPRegular and Irregular Holonomic D-modules, London Mathematical Society Lecture Note Series2016CambridgeCambridge University Press10.1017/CBO9781316675625 GelfandSIManinYIMethods of Homological Algebra20032BerlinSpringer Monographs in Mathematics. Springer10.1007/978-3-662-12492-5 BoalchPSimply-laced isomonodromy systemsPubl. Math. Inst. Hautes Études Sci.2012116168309025410.1007/s10240-012-0044-8 ItoYTakeuchiKOn irregularities of Fourier transforms of regular holonomic D-modulesAdv. Math.2020366407279710.1016/j.aim.2020.107093 D’AgnoloAKashiwaraMA microlocal approach to the enhanced Fourier-Sato transform in dimension oneAdv. Math.2018339159386689310.1016/j.aim.2018.09.022 Mochizuki, T.: Curve test for enhanced ind-sheaves and holonomic D-modules. arXiv:1610.08572v3 (2018) Mochizuki, T.: Holonomic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{D}}$$\end{document}-modules with Betti structure. Mém. Soc. Math. Fr. (N.S.) 138–139, Soc. Math. France, Paris (2014) KashiwaraMRiemann-Hilbert correspondence for irregular holonomic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{D}}$$\end{document}-modulesJpn. J. Math.201611113149351068110.1007/s11537-016-1564-7 Mochizuki, T.: Stokes shells and Fourier transforms. arXiv:1808.01037v1 (2018) SabbahCAn explicit stationary phase formula for the local formal Fourier-Laplace transformContemp. Math.2008474309330245435410.1090/conm/474/09262 Sabbah, C.: Introduction to Stokes Structures. Lecture Notes in Mathematics, vol. 2060. Springer, Berlin (2013) D’AgnoloAKashiwaraMRiemann-Hilbert correspondence for holonomic D-modulesPubl. Math. Inst. Hautes Études Sci.201612369197350209710.1007/s10240-015-0076-y BjörkJ-EAnalytic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{D}}$$\end{document}-Modules and Applications, Mathematics and Its Applications1993DordrechtKluwer Academic Publishers10.1007/978-94-017-0717-6 Hohl, A.: D-Modules of Pure Gaussian Type from the Viewpoint of Enhanced Ind-Sheaves. Universität Augsburg. https://opus.bibliothek.uni-augsburg.de/opus4/79690 (2020). Accessed 20 October 2020 T Mochizuki (1281_CR24) 2010; 35 M Kashiwara (1281_CR14) 1984; 20 R Hotta (1281_CR12) 2008 P Boalch (1281_CR2) 2012; 116 Y Ito (1281_CR13) 2020; 366 1281_CR26 1281_CR27 C Sabbah (1281_CR28) 2008; 474 1281_CR29 J-E Björk (1281_CR1) 1993 B Malgrange (1281_CR23) 1991 A D’Agnolo (1281_CR5) 2016; 123 NM Katz (1281_CR22) 1985; 62 M Kashiwara (1281_CR15) 2003 A D’Agnolo (1281_CR7) 2019; 751 1281_CR25 M Kashiwara (1281_CR17) 1990 M Kashiwara (1281_CR21) 2016; 22 SI Gelfand (1281_CR9) 2003 M Kashiwara (1281_CR16) 2016; 11 P Boalch (1281_CR3) 2015; 19 A D’Agnolo (1281_CR4) 2020; 70 1281_CR18 M Kashiwara (1281_CR20) 2016 1281_CR8 M Kashiwara (1281_CR19) 2006 C Sabbah (1281_CR30) 2016; 80 M Hien (1281_CR10) 2015; 134 1281_CR11 A D’Agnolo (1281_CR6) 2018; 339 |
| References_xml | – reference: HienMSabbahCThe local Laplace transform of an elementary irregular meromorphic connectionRend. Sem. Mat. Univ. Padova2015134133196342841710.4171/RSMUP/134-4 – reference: Mochizuki, T.: Holonomic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{D}}$$\end{document}-modules with Betti structure. Mém. Soc. Math. Fr. (N.S.) 138–139, Soc. Math. France, Paris (2014) – reference: KatzNMLaumonGTransformation de Fourier et majoration de sommes exponentiellesPubl. Math. Inst. Hautes Études Sci.19856214520210.1007/BF02698808 – reference: Hohl, A.: D-Modules of Pure Gaussian Type from the Viewpoint of Enhanced Ind-Sheaves. Universität Augsburg. https://opus.bibliothek.uni-augsburg.de/opus4/79690 (2020). Accessed 20 October 2020 – reference: Mochizuki, T.: Stokes shells and Fourier transforms. arXiv:1808.01037v1 (2018) – reference: MalgrangeBÉquations différentielles à coefficients polynomiaux, Progress in Mathematics1991BostonBirkhëuser0764.32001 – reference: HottaRTakeuchiKTanisakiTD-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics2008BostonBirkhäuser10.1007/978-0-8176-4523-6 – reference: KashiwaraMSchapiraPIrregular holonomic kernels and Laplace transformSel. Math. New Ser.20162255109343783310.1007/s00029-015-0185-y – reference: GelfandSIManinYIMethods of Homological Algebra20032BerlinSpringer Monographs in Mathematics. Springer10.1007/978-3-662-12492-5 – reference: KashiwaraMSchapiraPCategories and Sheaves, Grundlehren der mathematischen Wissenschaften2006BerlinSpringer1118.18001 – reference: KashiwaraMRiemann-Hilbert correspondence for irregular holonomic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{D}}$$\end{document}-modulesJpn. J. Math.201611113149351068110.1007/s11537-016-1564-7 – reference: KashiwaraMSchapiraPRegular and Irregular Holonomic D-modules, London Mathematical Society Lecture Note Series2016CambridgeCambridge University Press10.1017/CBO9781316675625 – reference: BoalchPGlobal Weyl groups and a new theory of multiplicative quiver varietiesGeom. Topol.20151934673536344710810.2140/gt.2015.19.3467 – reference: Deligne, P.: Lettre à B. Malgrange du 19/4/1978. In: Singularités irrégulières, Correspondance et documents, Documents mathématiques 5, Société Mathématique de France, Paris, pp. 25–26 (2007) – reference: KashiwaraMD-Modules and Microlocal Calculus, Translations of Mathematical Monographs2003ProvidenceAmerican Mathematical Society – reference: D’AgnoloAHienMMorandoGSabbahCTopological computation of some Stokes phenomena on the affine lineAnn. Inst. Fourier202070739808410595010.5802/aif.3323 – reference: SabbahCAn explicit stationary phase formula for the local formal Fourier-Laplace transformContemp. Math.2008474309330245435410.1090/conm/474/09262 – reference: D’AgnoloAKashiwaraMEnhanced perversitiesJ. Reine Angew. Math.2019751185241395669410.1515/crelle-2016-0062 – reference: BoalchPSimply-laced isomonodromy systemsPubl. Math. Inst. Hautes Études Sci.2012116168309025410.1007/s10240-012-0044-8 – reference: MochizukiTNote on the Stokes structure of Fourier transformActa Math. Vietnam.20103510715826421661201.32016 – reference: BjörkJ-EAnalytic D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{D}}$$\end{document}-Modules and Applications, Mathematics and Its Applications1993DordrechtKluwer Academic Publishers10.1007/978-94-017-0717-6 – reference: KashiwaraMSchapiraPSheaves on Manifolds, Grundlehren der mathematischen Wissenschaften1990BerlinSpringer – reference: Mochizuki, T.: Curve test for enhanced ind-sheaves and holonomic D-modules. arXiv:1610.08572v3 (2018) – reference: SabbahCDifferential systems of pure Gaussian typeIzv. Math.201680189220346268010.1070/IM8308 – reference: ItoYTakeuchiKOn irregularities of Fourier transforms of regular holonomic D-modulesAdv. Math.2020366407279710.1016/j.aim.2020.107093 – reference: D’AgnoloAKashiwaraMA microlocal approach to the enhanced Fourier-Sato transform in dimension oneAdv. Math.2018339159386689310.1016/j.aim.2018.09.022 – reference: Kashiwara, M., Schapira, P.: Ind-sheaves. Astérisque 271, (2001) – reference: KashiwaraMThe Riemann-Hilbert problem for holonomic systemsPubl. Res. Inst. Math. Sci.19842031936574338210.2977/prims/1195181610 – reference: D’AgnoloAKashiwaraMRiemann-Hilbert correspondence for holonomic D-modulesPubl. Math. Inst. Hautes Études Sci.201612369197350209710.1007/s10240-015-0076-y – reference: Sabbah, C.: Introduction to Stokes Structures. Lecture Notes in Mathematics, vol. 2060. Springer, Berlin (2013) – ident: 1281_CR11 – volume: 134 start-page: 133 year: 2015 ident: 1281_CR10 publication-title: Rend. Sem. Mat. Univ. Padova doi: 10.4171/RSMUP/134-4 – volume: 339 start-page: 1 year: 2018 ident: 1281_CR6 publication-title: Adv. 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Topol. doi: 10.2140/gt.2015.19.3467 – ident: 1281_CR29 doi: 10.1007/978-3-642-31695-1 – volume-title: Methods of Homological Algebra year: 2003 ident: 1281_CR9 doi: 10.1007/978-3-662-12492-5 – ident: 1281_CR18 – volume: 22 start-page: 55 year: 2016 ident: 1281_CR21 publication-title: Sel. Math. New Ser. doi: 10.1007/s00029-015-0185-y – volume: 20 start-page: 319 year: 1984 ident: 1281_CR14 publication-title: Publ. Res. Inst. Math. Sci. doi: 10.2977/prims/1195181610 – volume: 751 start-page: 185 year: 2019 ident: 1281_CR7 publication-title: J. Reine Angew. Math. doi: 10.1515/crelle-2016-0062 – ident: 1281_CR25 – volume-title: D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics year: 2008 ident: 1281_CR12 doi: 10.1007/978-0-8176-4523-6 – volume: 474 start-page: 309 year: 2008 ident: 1281_CR28 publication-title: Contemp. Math. doi: 10.1090/conm/474/09262 – ident: 1281_CR27 – volume: 123 start-page: 69 year: 2016 ident: 1281_CR5 publication-title: Publ. Math. 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Fourier doi: 10.5802/aif.3323 – volume: 62 start-page: 145 year: 1985 ident: 1281_CR22 publication-title: Publ. Math. Inst. Hautes Études Sci. doi: 10.1007/BF02698808 |
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| SubjectTerms | Algebraic Geometry Calculus of Variations and Optimal Control; Optimization Geometry Laplace transforms Lie Groups Mathematics Mathematics and Statistics Modules Number Theory Sheaves Topological Groups |
| Title | D-modules of pure Gaussian type and enhanced ind-sheaves |
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