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Failure of the Brauer-Manin obstruction for a simply connected fourfold, and an orbifold version of the Mordell theorem
Post-edited
Authors : Kebekus, Stefan (Author of the conference)
CIRM (Publisher )
11G35 - Varieties over global fields
14F22 - Brauer groups of schemes
CIRM (Publisher )
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| Simply connected surfaces over global fields Failure of the Hasse principle Mordell theorem for orbifold curves Height bounds |
Abstract :
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.
This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
Joint work with Pereira and Smeets.
Keywords : Brauer-Manin obstruction; Hasse principle; orbifold; Mordell-type theorem
MSC Codes : This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
Joint work with Pereira and Smeets.
Keywords : Brauer-Manin obstruction; Hasse principle; orbifold; Mordell-type theorem
11G35 - Varieties over global fields
14F22 - Brauer groups of schemes
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 17/12/2019 Conference Date : 25/11/2019 Subseries : Research talks arXiv category : Algebraic Geometry ; Number Theory Mathematical Area(s) : Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Video Time : 00:51:18 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-11-25_Kebekus.mp4 
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Information on the Event
Event Title : Algebraic Geometry and Complex Geometry / Géométrie algébrique et géométrie complexeEvent Organizers : Benoist, Olivier ; Floris, Enrica Dates : 25/11/2019 - 29/11/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/2074.html
Citation Data
DOI : 10.24350/CIRM.V.19580903Cite this video as: Kebekus, Stefan (2019). Failure of the Brauer-Manin obstruction for a simply connected fourfold, and an orbifold version of the Mordell theorem. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19580903 URI : http://dx.doi.org/10.24350/CIRM.V.19580903 |
See Also
- [Multi angle] The essential skeletons of pairs and the geometric P=W conjecture / Author of the conference Mauri, Mirko.
- [Multi angle] Representation varieties of fundamental groups of complex algebraic varieties and mixed Hodge structures / Author of the conference Lefèvre, Louis-Clément.
- [Multi angle] Rational curves on K3 surfaces / Author of the conference Gounelas, Frank.
- [Multi angle] Momentum polytopes of Kähler compact multiplicity-free manifolds / Author of the conference Cupit-Foutou, Stéphanie.
Bibliography
- KEBEKUS, Stefan, PEREIRA, Jorge Vitório, et SMEETS, Arne. Brauer-Manin failure for a simply connected fourfold over a global function field, via orbifold Mordell. arXiv preprint arXiv:1905.02795, 2019. - https://arxiv.org/abs/1905.02795
