Time: Sep
30 , 2024

Recently, Professor YUAN Xinyi and Professor XIE Junyi's paper entitled "Geometric Bogomolov Conjecture in Arbitrary Characteristics" has been accepted by the top mathematical journal Inventiones Mathematicae . YUAN Xinyi, a member of the “golden generation” of mathematicians from Peking University, specializes in number theory and arithmetic geometry.

As a long-standing and active branch of number theory, Diophantine geometry aims to study the rational solutions of polynomial equations with rational coefficients. In modern algebraic geometry, the goal of Diophantine geometry is to study rational points of algebraic clusters, and naturally generalize rational points to algebraic points.

In 1980, Bogomolov proposed the Arithmetic Bogomolov Conjecture. In 1998, Ullmo and ZHANG Shouwu proved the Arithmetic Bogomolov Conjecture, mainly using the equal distribution theorem for small-height points that they had previously proved in collaboration with Szpiro.

In the 21st century, through the analogy of number fields and function fields, Gubler and Yamaki proposed the Geometric Bogomolov Conjecture and proved some important cases of this conjecture. In 2018, Cantat, in collaboration with GAO Ziyang, Habegger and XIE Junyi, proved the conjecture in the case of characteristic zero, by using tools of complex analysis.

In 2021, YUAN Xinyi and XIE Junyi collaborated to finally prove all cases of the Geometric Bogomolov Conjecture. Their proof followed the path laid out by Ullmo, ZHANG Shouwu, Gubler and Yamaki, and ultimately reached its end by using the intersection theory of algebraic geometry and the Manin-Mumford Conjecture, which had been solved by Raynaud and Hrushovski.