Skip to main content Accessibility help


  • Alberto Castaño Domínguez (a1) and Christian Sevenheck (a2)


We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric ${\mathcal{D}}$ -modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.



Hide All

The authors are partially supported by the project SISYPH: ANR-13-IS01-0001-01/02 and DFG grant SE 1114/5-1.



Hide All
1. Adolphson, A., Hypergeometric functions and rings generated by monomials, Duke Math. J. 73(2) (1994), 269290.
2. Arinkin, D., Rigid irregular connections on ℙ1 , Compos. Math. 146(5) (2010), 13231338.
3. Borisov, L. A. and Horja, R. P., Mellin-Barnes integrals as Fourier–Mukai transforms, Adv. Math. 207(2) (2006), 876927.
4. Berkesch, C., Matusevich, L. F. and Walther, U., On normalized Horn systems, preprint, 2018, arXiv:1806.03355 [math.AG].
5. Brieskorn, E., Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), 103161.
6. Castaño Domínguez, A., Reichelt, T. and Sevenheck, C., Examples of hypergeometric twistor ${\mathcal{D}}$ -modules, Algebra Number Theory, preprint, 2018,arXiv:1803.04886 [math.AG], to appear.
7. Corti, A. and Golyshev, V., Hypergeometric equations and weighted projective spaces, Sci. China Math. 54(8) (2011), 15771590.
8. Cox, D. A. and Katz, S., Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, Volume 68 (American Mathematical Society, Providence, RI, 1999).
9. D’Agnolo, A. and Eastwood, M., Radon and Fourier transforms for D-modules, Adv. Math. 180(2) (2003), 452485.
10. Dimca, A., Sheaves in topology, in Universitext (Springer, Berlin, 2004).
11. Denef, J. and Loeser, F., Weights of exponential sums, intersection cohomology, and Newton polyhedra, Invent. Math. 106(2) (1991), 275294.
12. Douai, A. and Sabbah, C., Gauss–Manin systems, Brieskorn lattices and Frobenius structures. I, Ann. Inst. Fourier (Grenoble) 53(4) (2003), 10551116.
13. Douai, A. and Sabbah, C., Gauss–Manin systems, Brieskorn lattices and Frobenius structures. II, Frobenius manifolds, Aspects of Mathematics, Volume E36, pp. 118 (Vieweg, Wiesbaden, 2004).
14. Dettweiler, M. and Sabbah, C., Hodge theory of the middle convolution, Publ. Res. Inst. Math. Sci. 49(4) (2013), 761800.
15. Esnault, H., Sabbah, C. and Yu, J.-D., E 1 -degeneration of the irregular Hodge filtration, J. Reine Angew. Math. 729 (2017), 171227.
16. Fedorov, R., Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles, Int. Math. Res. Not. IMRN 18 (2018), 55835608.
17. Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, in Mathematics: Theory & Applications (Birkhäuser Boston Inc, Boston, MA, 1994).
18. de Gregorio, I., Mond, D. and Sevenheck, C., Linear free divisors and Frobenius manifolds, Compos. Math. 145(5) (2009), 13051350.
19. Hertling, C. and Sevenheck, C., Nilpotent orbits of a generalization of Hodge structures, J. Reine Angew. Math. 609 (2007), 2380.
20. Hotta, R., Takeuchi, K. and Tanisaki, T., D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, Volume 236 (Birkhä user Boston Inc, Boston, MA, 2008). Translated from the 1995 Japanese edition by Takeuchi.
21. Katz, N. M., Exponential Sums and Differential Equations, Annals of Mathematics Studies, Volume 124 (Princeton University Press, Princeton, NJ, 1990).
22. Katzarkov, L., Kontsevich, M. and Pantev, T., Bogomolov–Tian–Todorov theorems for Landau–Ginzburg models, J. Differential Geom. 105(1) (2017), 55117.
23. Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math. 32(1) (1976), 131.
24. Martin, N., Middle multiplicative convolution and hypergeometric equations, preprint, 2018, arXiv:1809.08867 [math.AG].
25. Mochizuki, T., Wild harmonic bundles and wild pure twistor D-modules, Astérisque (340) (2011), x+607.
26. Mochizuki, T., Mixed twistor D-modules, Lecture Notes in Mathematics, Volume 2125 (Springer, Cham, 2015).
27. Mochizuki, T., Twistor property of GKZ-hypergeometric systems, preprint, 2015, arXiv:1501.04146 [math.AG].
28. Reichelt, T., Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules, Compos. Math. 150 (2014), 911941.
29. Reichelt, T. and Sevenheck, C., Hypergeometric Hodge modules, Algebraic Geom., preprint, 2015, arXiv:1503.01004 [math.AG] 2015, to appear.
30. Reichelt, T. and Sevenheck, C., Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules, J. Algebraic Geom. 24(2) (2015), 201281.
31. Reichelt, T. and Sevenheck, C., Non-affine Landau-Ginzburg models and intersection cohomology, Ann. Sci. Éc. Norm. Supér. 50(3) (2017), 665753.
32. Sabbah, C., Déformations isomonodromiques et variétés de Frobenius, in Savoirs Actuels (EDP Sciences, Les Ulis, 2002). Mathématiques.
33. Sabbah, C., Hypergeometric periods for a tame polynomial, Port. Math. (NS) 63(2) (2006), 173226.
34. Sabbah, C., Fourier–Laplace transform of a variation of polarized complex Hodge structure, J. Reine Angew. Math. 621 (2008), 123158.
35. Sabbah, C., Irregular Hodge theory (with the collaboration of Jeng-Daw Yu), Mém. Soc. Math. Fr. (NS) (156) (2018), vi+126.
36. Sabbah, C., Some properties and applications of Brieskorn lattices, J. Singul. 18 (2018), 239248.
37. Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24(6) (1988), 849995. 1989.
38. Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26(2) (1990), 221333.
39. Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211319.
40. Shamoto, Y., Hodge–Tate conditions for Landau–Ginzburg models, Publ. Res. Inst. Math. Sci. 54(3) (2018), 469515.
41. Simpson, C. T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3(3) (1990), 713770.
42. Sabbah, C. and Yu, J.-D., On the irregular Hodge filtration of exponentially twisted mixed Hodge modules, Forum Math. Sigma 3(e9) (2015), 71 pp.
43. Sabbah, C. and Yu, J.-D., Irregular Hodge numbers of confluent hypergeometric differential equations, preprint, 2018, arXiv:1812.00755 [math.AG], 2018.
44. Jeng-Daw, Y., Irregular Hodge filtration on twisted de Rham cohomology, Manuscripta Math. 144(1–2) (2014), 99133.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


  • Alberto Castaño Domínguez (a1) and Christian Sevenheck (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed