Holonomic $\mathcal{D}$-modules with Betti structure

Holonomic

𝒟

-modules with Betti structure
[

𝒟

-modules holonomes munis d’une structure de Betti]

Mochizuki, Takuro

Mémoires de la Société Mathématique de France, no. 138-139 (2014) , 213 p.

Résumé
Abstract

Nous définissons la notion de structure Betti pour les $𝒟$ -modules holonomes qui ne sont pas nécessairement singuliers réguliers. Nous établissons leurs propriétés fonctorielles principales. Nous donnons également une analyse supplémentaire des fonctions holomorphes de divers types sur l’éclatement réel.

We define the notion of Betti structure for holonomic $𝒟$ -modules which are not necessarily regular singular. We establish the fundamental functorial properties. We also give auxiliary analysis of holomorphic functions of various types on the real blow up.

MR Zbl | 2 citations dans Numdam

DOI : 10.24033/msmf.448

Classification : 14F10, 32C38
Keywords: holonomic D-modules, Betti structure, Stokes structure
Mot clés : D-modules holonomiques, structure de Betti, structure de Stokes

@book{MSMF_2014_2_138-139__1_0,
     author = {Mochizuki, Takuro},
     title = {Holonomic $\mathcal{D}$-modules with {Betti} structure},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {138-139},
     year = {2014},
     doi = {10.24033/msmf.448},
     mrnumber = {3306892},
     zbl = {1327.14006},
     language = {en},
     url = {https://www.numdam.org/item/MSMF_2014_2_138-139__1_0/}
}

TY  - BOOK
AU  - Mochizuki, Takuro
TI  - Holonomic $\mathcal{D}$-modules with Betti structure
T3  - Mémoires de la Société Mathématique de France
PY  - 2014
IS  - 138-139
PB  - Société mathématique de France
UR  - https://www.numdam.org/item/MSMF_2014_2_138-139__1_0/
DO  - 10.24033/msmf.448
LA  - en
ID  - MSMF_2014_2_138-139__1_0
ER  -

%0 Book
%A Mochizuki, Takuro
%T Holonomic $\mathcal{D}$-modules with Betti structure
%S Mémoires de la Société Mathématique de France
%D 2014
%N 138-139
%I Société mathématique de France
%U https://www.numdam.org/item/MSMF_2014_2_138-139__1_0/
%R 10.24033/msmf.448
%G en
%F MSMF_2014_2_138-139__1_0

Mochizuki, Takuro. Holonomic $\mathcal{D}$-modules with Betti structure. Mémoires de la Société Mathématique de France, Série 2, no. 138-139 (2014), 213 p. doi : 10.24033/msmf.448. http://numdam.org/item/MSMF_2014_2_138-139__1_0/

Bibliographie
Cité par

[1] Y. André, Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité, Invent. Math. 170 (2007), no. 1, 147–198. | MR

[2] C. Banica, Le complété formel d’un espace analytique le long d’un sous-espace: un théorème de comparaison, Manuscripta Math. 6 (1972), 207–244. | MR | EuDML | Zbl

[3] A. Beilinson, On the derived category of perverse sheaves, in $K$ -theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math. 1289, Springer, Berlin, (1987), 27–41. | MR

[4] A. Beilinson, How to glue perverse sheaves, in $K$ -theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math. 1289, Springer, Berlin, (1987), 42–51. | MR | Zbl

[5] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque 100 (1982), 5–171. | MR

[6] A. Beilinson, S. Bloch, P. Deligne, H. Esnault, Periods for irregular connections on curves, preprint.

[7] J. Bingener, Über Formale Komplexe Räume, Manuscripta Math. 24 (1978), 253–293. | MR | EuDML | Zbl

[8] S. Bloch, H. Esnault, Homology for irregular connections, J. théorie des nombres, Bordeaux 16 (2004), 357–371. | MR | EuDML | Zbl | Numdam

[9] A. Borel, P. O. Grivel, B. Kaup, A. Haefliger, B. Malgrange, F. Ehlers, Algebraic $𝒟$ -modules, Perspectives in Mathematics 2, Academic Press, Inc., Boston, MA, 1987. | MR

[10] A. D’Agnolo, M. Kashiwara, On a reconstruction theorem for holonomic systems, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), 178–183. | MR | Zbl

[11] A. D’Agnolo, M. Kashiwara, Riemann-Hilbert correspondence for holonomic $D$ -modules; arXiv:1311.2374. | MR

[12] P. Deligne, Équations différentielles à points singuliers réguliers, Lectures Notes in Maths., vol. 163, Springer, 1970. | MR | Zbl

[13] P. Deligne, B. Malgrange, J-P. Ramis, Singularités irrégulières, Documents mathématiques 5, Société Mathématique de France, 2007. | MR

[14] R. Hartshorne, On the De Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5–99. | MR | EuDML | Zbl | Numdam

[15] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, (1977). | MR | Zbl

[16] M. Hien, Periods for flat algebraic connections, Invent. Math. 178 (2009), 1–22. | MR | Zbl

[17] R. Hotta, K. Takeuchi and T. Tanisaki, $D$ -modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser Boston, Inc., Boston, MA, 2008. | MR | Zbl

[18] B. Iversen, Cohomology of sheaves, Springer-Verlag, Berlin, 1986. | MR | Zbl

[19] M. Kashiwara, On the maximally overdetermined system of linear differential equations I, Publ. Res. Inst. Math. Sci. 10 (1974/75), 563–579. | MR | Zbl

[20] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), 319–365. | MR | Zbl

[21] M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, in Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, (1983), 134–142. | MR | Zbl

[22] M. Kashiwara, $D$ -modules and microlocal calculus, Translations of Mathematical Monographs, 217, Iwanami Series in Modern Mathematics, American Mathematical Society, 2003. | MR

[23] M. Kashiwara and P. Schapira, Sheaves on manifolds, Springer-Verlag, Berlin, 1990. | MR

[24] M. Kashiwara, P. Schapira, Ind-sheaves, Astérisque 271 (2001). | MR | Zbl | Numdam

[25] L. Katzarkov, M. Kontsevich and T. Pantev, Hodge theoretic aspects of mirror symmetry, in From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math. 78, Amer. Math. Soc., Providence, RI, (2008), 87–174; math:0806.0107. | Zbl

[26] K. Kedlaya, Good formal structures for flat meromorphic connections, I: Surfaces, Duke Math. J. 154 (2010), 343–418; arXiv:0811.0190. | MR | Zbl

[27] K. Kedlaya. Good formal structures for flat meromorphic connections, II: Excellent schemes, J. Amer. Math. Soc. 24 (2011), 183–229; arXiv:1001.0544. | MR | Zbl

[28] M. Kontsevich, Holonomic $D$ -modules and positive characteristic, Japan J. Math. 4 (2009), 1–25. | MR | Zbl

[29] B. A. Krasnov, Formal Modifications. Existence Theorems for modifications of complex manifolds, Math. USSR. Izvestija 7 (1973), 847–881. | MR | Zbl

[30] G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210. | MR | EuDML | Zbl | Numdam

[31] A. Levelt, Jordan decomosition for a class of singular differential operators, Ark. Math. 13 (1975), 1–27. | MR | Zbl

[32] R. Macpherson, K. Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84 (1986), 403–435. | MR | EuDML | Zbl

[33] H. Majima, Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Mathematics 1075, Springer-Verlag, Berlin, 1984. | MR | Zbl

[34] B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics 3, Tata Institute of Fundamental Research, Bombay, Oxford University Press, London, 1967. | MR

[35] B. Malgrange, La classification des connexions irrégulières à une variable, in Mathematics and physics (Paris, 1979/1982), Progr. Math., 37, Birkhäuser Boston, Boston, MA, (1983), 381–399. | MR

[36] B. Malgrange, Polynômes de Bernstein-Sato et cohomologie évanescente, in Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque 101–102 (1983), 243–267. | MR | Zbl | Numdam

[37] B. Malgrange, Équations différentielles à coefficients polynomiaux, Progress in Mathematics 96, Birkhäuser, Boston, 1991. | MR

[38] B. Malgrange, Connexions méromorphies 2, Le réseau canonique, Invent. Math. 124 (1996), 367–387. | MR

[39] B. Malgrange, On irregular holonomic $D$ -modules, Éléments de la théorie des systèmes différentiels géométriques, 391–410, Sémin. Congr. 8, Soc. Math. France, Paris, 2004. | MR | Zbl

[40] Z. Mebkhout, Une équivalence de catégories. Compositio Math. 51 (1984), 51–62. | MR | EuDML | Zbl | Numdam

[41] Z. Mebkhout, Une autre équivalence de catégories, Compositio Math. 51 (1984), 63–88. | MR | EuDML | Zbl | Numdam

[42] Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les $𝒟_{X}$ -modules cohérents, Hermann, Paris, 1989. | MR | Zbl

[43] Z. Mebkhout, Le théorème de positivité de l’irrégularité pour les $𝒟_{X}$ -modules, in The Grothendieck Festschrift, Vol. III, 83–132, Progr. Math. 88, Birkhäuser Boston, Boston, MA, 1990. | MR

[44] T. Mochizuki, Good formal structure for meromorphic flat connections on smooth projective surfaces, in Algebraic Analysis and Around, Advanced Studies in Pure Mathematics 54 (2009), 223–253; math:0803.1346. | MR | Zbl

[45] T. Mochizuki, On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles, Ann. Inst. Fourier (Grenoble) 59 (2009), 2819–2837. | MR | EuDML | Zbl | Numdam

[46] T. Mochizuki, Note on the Stokes structure of Fourier transform, Acta Math. Vietnam 35 (2010), 107–158 | MR | Zbl

[47] T. Mochizuki, Wild harmonic bundles and wild pure twistor $D$ -modules, Astérisque 340 (2011); math:0803.1344. | MR | Zbl | Numdam

[48] T. Mochizuki, Asymptotic behaviour of variation of pure polarized TERP structures, Publ. Res. Inst. Math. Sci. 47 (2011), 419–534; math:0811.1384. | MR | Zbl

[49] T. Mochizuki, The Stokes structure of good meromorphic flat bundle, J. Inst. Math. Jussieu 10 (2011), 675–712. | MR | Zbl

[50] T. Mochizuki, Mixed twistor $D$ -modules; arXiv:1104.3366. | MR

[51] G. Morando, An existence theorem for tempered solutions of $D$ -modules on complex curves, Publ. Res. Inst. Math. Sci. 43 (2007), 625–659. | MR | Zbl

[52] C. Sabbah, Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension $2$ , Astérisque 263, Société Mathématique de France, Paris, 2000. | MR | Numdam

[53] C. Sabbah, Polarizable twistor $D$ -modules, Astérisque 300, Société Mathématique de France, Paris, (2005). | MR | Zbl | Numdam

[54] C. Sabbah, Introduction to Stokes structures, Lecture Notes in Mathematics 2060 Springer, Heidelberg, 2013; arXiv:0912.2762. | MR | Zbl

[55] M. Saito, Modules de Hodge polarisables, Publ. RIMS 24 (1988), 849–995. | MR | Zbl

[56] M. Saito, Duality for vanishing cycle functors, Publ. Res. Inst. Math. Sci. 25 (1989), 889–921. | MR | Zbl

[57] M. Saito, Mixed Hodge modules, Publ. RIMS 26 (1990), 221–333. | MR | Zbl

[58] M. Saito, Induced $D$ -modules and differential complexes, Bull. Soc. Math. France 117 (1989), 361–387. | MR | EuDML | Zbl | Numdam

[59] J.-L. Verdier, Extension of a perverse sheaf over a closed subspace, in Differential systems and singularities (Luminy, 1983), Astérisque 130 (1985), 210–217. | MR | Numdam

Cité par Sources :