Absolute sets of rigid local systems
Annales de l'Institut Fourier, Online first, 44 p.

The absolute sets of local systems on a smooth complex algebraic variety are the subject of a conjecture of Budur–Wang based on an analogy with special subvarieties of Shimura varieties. An absolute set should be the higher-dimensional generalization of a local system of geometric origin. We show that the conjecture for absolute sets of simple cohomologically rigid local systems reduces to the zero-dimensional case, that is, to Simpson’s conjecture that every such local system with quasi-unipotent monodromy at infinity and determinant is of geometric origin. In particular, the conjecture holds for this type of absolute sets if the variety is a curve or if the rank is two.

Les ensembles absolus de systèmes locaux sur une variété algébrique complexe lisse font l’objet d’une conjecture de Budur–Wang basée sur une analogie avec les sous-variétés spéciales des variétés de Shimura. Un ensemble absolu devrait être la généralisation en dimension supérieure d’un système local d’origine géométrique. Nous montrons que la conjecture pour les ensembles absolus de systèmes locaux simples cohomologiquement rigides se réduit au cas de la dimension zéro, c’est-à-dire à la conjecture de Simpson selon laquelle chaque système local de ce type avec monodromie quasi-unipotente à l’infini, ainsi que le déterminant, est d’origine géométrique. En particulier, la conjecture est vraie pour ces ensembles absolus si la variété est une courbe ou si le rang est deux.

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DOI: 10.5802/aif.3551
Classification: 14F05,  14F10,  14F45
Keywords: Absolute set, local system, logarithmic connection, rigid local system.
Budur, Nero 1, 2; Lerer, Leonardo A. 3; Wang, Haopeng 4

1 Department of Mathematics KU Leuven Celestijnenlaan 200B, 3001 Leuven (Belgium)
2 BCAM Mazarredo 14, 48009 Bilbao (Spain)
3 Département de Mathématiques d’Orsay Université Paris-Saclay F-91405 Orsay (France)
4 Department of Mathematics KU Leuven Celestijnenlaan 200B, 3001 Leuven, (Belgium)
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Budur, Nero; Lerer, Leonardo A.; Wang, Haopeng. Absolute sets of rigid local systems. Annales de l'Institut Fourier, Online first, 44 p.

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