YUAN Xinyi and XIE Junyi’s Collaborative Paper Finally Proves All Cases of the Geometric Bogomolov Conjecture

Professor YUAN Xinyi, a member of the “golden generation”of mathematicians from Peking University, specializes in number theory and arithmetic geometry. As one of the leaders of the SRMC research team, recently, YUAN Xinyi and XIE Junyi's paper, "Geometric Bogomolov Conjecture in Arbitrary Characteristics," has been accepted by the top mathematics journal InventionesMathematicae.

Diophantine geometry, as a long-standing and still active branch of number theory, aims to study the rational solutions of polynomial equations with rational coefficients.

In the language of modern algebraic geometry, the goal of Diophantine geometry is to study the rational points of algebraic varieties, and it is also natural to extend rational points to algebraic points.

The underlying philosophy here is that the algebraic points on an abelian variety are "dense," but on subvarieties of other types, the algebraic points are "sparse."

These concepts may seem vague, but they can be rigorously stated using the canonical height of algebraic points. This conjecture is the famous Arithmetic Bogomolov Conjecture.

In 1998, Ullmo and ZHANG Shouwu proved the Arithmetic Bogomolov Conjecture, using as their main tool the equidistribution theorem for small-height points, which they had previously proven in collaboration with Szpiro.

In the 21st century, by drawing analogies between number fields and function fields, Gubler and Yamaki proposed the Geometric Bogomolov Conjecture and proved several important cases of it.

In 2018, Cantat, GAO Ziyang, Habegger, and XIE Junyi, using tools from complex analysis, proved the conjecture in the case of characteristic zero.

In 2021, XIE Junyi and YUAN Xinyi 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 the conclusion by using intersection theory in algebraic geometry and the Manin-Mumford Conjecture, which had been resolved by Raynaud and Hrushovski.