Integral representations for solutions of exponential Gauß-Manin systems

Marco Hien; Céline Roucairol

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 4, page 505-532
  • ISSN: 0037-9484

Abstract

top
Let f , g : U → 𝔸 1 be two regular functions from the smooth affine complex variety U to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system 𝒪 U e g with respect to f . We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.

How to cite

top

Hien, Marco, and Roucairol, Céline. "Integral representations for solutions of exponential Gauß-Manin systems." Bulletin de la Société Mathématique de France 136.4 (2008): 505-532. <http://eudml.org/doc/272310>.

@article{Hien2008,
abstract = {Let $f,g:U \rightarrow \{\mathbb \{A\}\}^1 $ be two regular functions from the smooth affine complex variety $U $ to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system $\mathcal \{O\}_U e^g $ with respect to $f $. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.},
author = {Hien, Marco, Roucairol, Céline},
journal = {Bulletin de la Société Mathématique de France},
keywords = {gauß-Manin systems; $\mathcal \{D\} $-modules},
language = {eng},
number = {4},
pages = {505-532},
publisher = {Société mathématique de France},
title = {Integral representations for solutions of exponential Gauß-Manin systems},
url = {http://eudml.org/doc/272310},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Hien, Marco
AU - Roucairol, Céline
TI - Integral representations for solutions of exponential Gauß-Manin systems
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 4
SP - 505
EP - 532
AB - Let $f,g:U \rightarrow {\mathbb {A}}^1 $ be two regular functions from the smooth affine complex variety $U $ to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system $\mathcal {O}_U e^g $ with respect to $f $. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.
LA - eng
KW - gauß-Manin systems; $\mathcal {D} $-modules
UR - http://eudml.org/doc/272310
ER -

References

top
  1. [1] S. Bloch & H. Esnault – « Homology for irregular connections », J. Théor. Nombres Bordeaux16 (2004), p. 357–371. Zbl1075.14016MR2143558
  2. [2] A. Borel – Algebraic 𝒟 -modules, Perspectives in Math., vol. 2, Academic Press, 1987. Zbl0642.32001
  3. [3] A. Dimca, F. Maaref, C. Sabbah & M. Saito – « Dwork cohomology and algebraic 𝒟 -modules », Math. Ann.318 (2000), p. 107–125. Zbl0985.14007MR1785578
  4. [4] A. Dimca & M. Saito – « On the cohomology of a general fiber of a polynomial map », Compositio Math.85 (1993), p. 299–309. Zbl0824.14016MR1214449
  5. [5] M. Hien – « Periods for irregular singular connections on surfaces », Math. Ann.337 (2007), p. 631–669. Zbl1118.14024MR2274546
  6. [6] N. M. Katz – « On the calculation of some differential Galois groups », Invent. Math.87 (1987), p. 13–61. Zbl0609.12025MR862711
  7. [7] F. Maaref – « Sur un analogue irrégulier de la connexion de Gauss-Manin », Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), p. 117–124. Zbl0965.32020MR1721566
  8. [8] H. Majima – Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Mathematics, vol. 1075, Springer, 1984. Zbl0546.58003MR757897
  9. [9] B. Malgrange – Équations différentielles à coefficients polynomiaux, Progress in Mathematics, vol. 96, Birkhäuser, 1991. Zbl0764.32001MR1117227
  10. [10] F. Pham – Singularités des systèmes différentiels de Gauss-Manin, Progress in Mathematics, vol. 2, Birkhäuser, 1979. Zbl0524.32015
  11. [11] —, « La descente des cols par les onglets de Lefschetz, avec vues sur Gauss-Manin », Astérisque130 (1985), p. 11–47. Zbl0597.32012MR804048
  12. [12] —, Intégrales singulières, Savoirs Actuels, EDP Sciences/CNRS Éditions, 2005. Zbl1125.42001
  13. [13] C. Roucairol – « Irregularity of an analogue of the Gauss-Manin systems », Bull. Soc. Math. France134 (2006), p. 269–286. Zbl1122.32019MR2233709
  14. [14] —, « The irregularity of the direct image of some 𝒟 -modules », Publ. Res. Inst. Math. Sci.42 (2006), p. 923–932. Zbl1132.32005MR2289081
  15. [15] —, « Formal structure of direct image of holonomic 𝒟 -modules of exponential type », Manuscripta Math.124 (2007), p. 299–318. Zbl1140.32021MR2350548
  16. [16] C. Sabbah – « Équations différentielles à points singuliers irréguliers en dimension 2 », Ann. Inst. Fourier (Grenoble) 43 (1993), p. 1619–1688. Zbl0803.32005MR1275212
  17. [17] —, « Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2 », Astérisque 263 (2000), p. 190. Zbl0947.32005
  18. [18] R. G. Swan – « The theory of sheaves », 1964. Zbl0119.25801
  19. [19] J.-L. Verdier – « Classe d’homologie associée à un cycle », in Séminaire de géométrie analytique (École Norm. Sup., Paris, 1974-75) Astérisque 36–37 (1976), p. 101–151. Zbl0346.14005MR447623
  20. [20] —, « Stratifications de Whitney et théorème de Bertini-Sard », Invent. Math.36 (1976), p. 295–312. Zbl0333.32010MR481096

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.