Etats de la recherche: Arithmetic, Algebraic and Analytics Dynamics / Etats de la recherche: Dynamique arithmétique, algébrique, et analytique

Collection Etats de la recherche: Arithmetic, Algebraic and Analytics Dynamics / Etats de la recherche: Dynamique arithmétique, algébrique, et analytique

Organizer(s) Favre, Charles ; Gauthier, Thomas ; Raissy, Jasmin
Date(s) 20/01/2025 - 24/01/2025
linked URL https://conferences.cirm-math.fr/3211.html
00:00:00 / 00:00:00
3 14

Geometry and algebra of preperiodic points in $ \mathbb{P}^{N}$ - Lecture 1

By Laura Demarco

In these lectures, we will examine a series of conjectures about the geometry of preperiodic points for endomorphisms of $ \mathbb{P}^{N}$. Lecture 1 will focus on the Dynamical ManinMumford Conjecture (DMM), formulated by Shouwu Zhang in the 1990s as an extension of the well-known Manin-Mumford Conjecture (which investigated the geometry of torsion points in abelian varieties and was proved in the early 1980s by Raynaud). The DMM aims to classify the subvarieties of $\mathbb{P}^{N}$ containing a Zariski-dense set of preperiodic points. Lectures 2 and 3 will be devoted to conjectures that treat families of maps on $\mathbb{P}^{N}$. One conjecture in particular was inspired by the recently-proved ”Relative Manin-Mumford” theorem of Gao-Habegger for abelian varieties, but the dynamical version turns out to be closely related to the study of dynamical stability and to contain many previously-existing questions/conjectures/results about moduli spaces of maps on $\mathbb{P}^{N}$. These lectures are based on joint work with Myrto Mavraki.

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Citation data

  • DOI 10.24350/CIRM.V.20290103
  • Cite this video Demarco, Laura (20/01/2025). Geometry and algebra of preperiodic points in $ \mathbb{P}^{N}$ - Lecture 1. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20290103
  • URL https://dx.doi.org/10.24350/CIRM.V.20290103

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