Tautological systems, homogeneous spaces and the holonomic rank problem

Paul Görlach; Thomas Reichelt; Christian Sevenheck; Avi Steiner; Uli Walther

doi:10.5802/jep.332

Tautological systems, homogeneous spaces and the holonomic rank problem
[Systèmes tautologiques, espaces homogènes et le problème du rang holonome]

Paul Görlach ¹ ; Thomas Reichelt ² ; Christian Sevenheck ³ ; Avi Steiner ³ ; Uli Walther ⁴

¹Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Institut für Algebra und Geometrie, Universitätsplatz 2, 39106 Magdeburg, Germany
²Lehrstuhl für Mathematik VI, Institut für Mathematik, Universität Mannheim, A 5, 6, 68131 Mannheim, Germany
³Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
⁴Purdue University, Dept. of Mathematics, 150 N. University St., West Lafayette, IN 47907, USA

Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 519-591

Abstract (VO)
Résumé

Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure a of mixed Hodge module. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by giving a functorial construction for them. As an application, we solve the holonomic rank problem for such tautological systems in full generality.

De nombreux systèmes différentiels hypergéométriques d’origine géométrique peuvent être munis d’une structure de module de Hodge mixte. Nous généralisons ce résultat fondamental aux systèmes tautologiques associés aux espaces homogènes en en donnant une construction fonctorielle. Nous appliquons ce résultat pour résoudre, en toute généralité, le problème du rang holonome pour ces systèmes tautologiques.

Reçu le : 2025-01-11
Accepté le : 2026-02-11
Publié le : 2026-03-06

DOI : 10.5802/jep.332

Classification : 32C38, 14F10, 32S40
Keywords: Tautological system, Fourier–Laplace transformation, mixed Hodge module, Lie group, homogeneous space
Mots-clés : Système tautologique, transformation de Fourier-Laplace, module de Hodge mixte, groupe de Lie, espaces homogène

Affiliations des auteurs :

Paul Görlach ¹ ; Thomas Reichelt ² ; Christian Sevenheck ³ ; Avi Steiner ³ ; Uli Walther ⁴

¹ Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Institut für Algebra und Geometrie, Universitätsplatz 2, 39106 Magdeburg, Germany
² Lehrstuhl für Mathematik VI, Institut für Mathematik, Universität Mannheim, A 5, 6, 68131 Mannheim, Germany
³ Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
⁴ Purdue University, Dept. of Mathematics, 150 N. University St., West Lafayette, IN 47907, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Paul G\"orlach and Thomas Reichelt and Christian Sevenheck and Avi Steiner and Uli Walther},
     title = {Tautological systems, homogeneous spaces and the holonomic rank problem},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     year = {2026},
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     doi = {10.5802/jep.332},
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     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.332/}
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Paul Görlach; Thomas Reichelt; Christian Sevenheck; Avi Steiner; Uli Walther. Tautological systems, homogeneous spaces and the holonomic rank problem. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 519-591. doi: 10.5802/jep.332

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