Some arithmetic and geometric aspects of algebraic cycles and motives

TL;DR

This thesis introduces algebraic cycles and motives, emphasizing differences between characteristic zero and positive characteristic, and presents new results on standard conjectures, p-adic periods, and motivic Galois groups using advanced mathematical techniques.

Contribution

It proves several key conjectures in algebraic geometry, including the standard conjecture of Hodge type for abelian fourfolds and the Lefschetz type for certain varieties, and constructs p-adic periods related to motivic Galois groups.

Findings

Standard conjecture of Hodge type for abelian fourfolds proved

Construction of p-adic periods controlled by motivic Galois groups

Standard conjecture of Lefschetz type for Laza–Saccà–Voisin varieties

Abstract

This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on cohomologies of algebraic varieties, periods, algebraic cycles and motives. I might one day find the energy to expand these notes and maybe translate them in English. In the meantime, please feel free to ask questions. The first sections of this text propose an introduction to the theory, I tried to present points of view and examples which are not always stressed in the literature. A leitmotif of the thesis is the difference between phenomena in characteristic zero and those in positive characteristic. The main results are: the standard conjecture of Hodge type for abelian foufolds; the construction of a class of $p$-adic periods controlled by motivic Galois groups and for which it is possible to formulate a $p$-adic analog of the Grothendieck period conjecture; the standard conjecture of Lefschetz type for the Laza--Sacc\`a--Voisin varieties and the Chow-K\"unneth conjecture for commutative group schemes. The techniques are based on $p$-adic Hodge theory, representation theory, perverse sheaves and motivic complexes \`a la Voevodsky.