Descripción del proyecto
Ampliar la conjetura de Bogomolov más allá de las variedades abelianas
El equipo del proyecto DiophGeo, financiado por las Acciones Marie Skłodowska-Curie, tiene previsto ampliar el uso de la conjetura de Bogomolov más allá de las variedades abelianas. Las investigaciones del último decenio indican que esto podría ser posible. Por ejemplo, se han demostrado análogos relativos de la conjetura de Manin-Mumford en varias familias de variedades abelianas y un análogo relativo de la conjetura de Bogomolov para secciones en un producto fibrado de familias elípticas. En última instancia, la investigación ha demostrado una conjetura dinámica de Bogomolov para los mapas racionales divididos. El proyecto utilizará un análogo de la conjetura de equidistribución en familias de variedades abelianas sobre una curva base.
Objetivo
"This project proposes research with a view towards extensions of the Bogomolov conjecture beyond the original setting of abelian varieties. In the past decade, there have been some indications that this may be possible: (a) Masser and Zannier have proven ""relative'' analogues of the Manin-Mumford conjecture in various families of abelian varieties, (b) DeMarco and Mavraki have shown a ""relative'' analogue of the Bogomolov conjecture for sections in a fibered product of elliptic families, and (c) Ghioca, Nguyen, and Ye have proven a ""dynamical'' Bogomolov conjecture for split rational maps.
A prominent tool in almost all proofs of the Bogomolov conjecture are equidistribution techniques (i.e. Yuan's equidistribution theorem). However, there are two problems with this approach when it comes to ""relative'' generalizations.
First, the Néron-Tate local height in families of abelian varieties exhibits b-singularities nearby degenerate fibers, preventing a direct use of Yuan's theorem if the family has degenerate fibers. Recently, I have overcome these problems and proven a satisfactory analogue of the equidistribution conjecture in families of abelian varieties over a base curve. Part of the research proposed here is to generalize and exploit this result further.
Second, equidistribution techniques usually fall short of ""relative'' Bogomolov-type results -- in stark contrast to the case of abelian varieties. Similar problems arise in the ""dynamical"" setting, indicating a profound conceptional obstacle. For this reason, it is proposed here to adapt a method of David and Philippon, who gave an equidistribution-free direct proof of the Bogomolov conjecture for abelian varieties, to the relative setting. Such a method, if successful, should shed some light on an ""ultimate'' Bogomolov conjecture encompassing virtually all the Bogomolov-type results known up to the present day."
Ámbito científico
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Régimen de financiación
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinador
1165 Kobenhavn
Dinamarca