A Characterization of Rationality and Borel Subgroups

This is joint work with Andriy Regeta and Christian Urech. In this talk, we focus on the following two questions about the group of birational transformations, $\mathrm{Bir}(X)$, of an irreducible variety $X$:
1. If $\mathrm{Bir}(X)$ and $\mathrm{Bir}(\mathbb{P}^n)$ are isomorphic, does this imply that $X$ and $\mathbb{P}^n$ are birational?
2. What are the Borel subgroups of $\mathrm{Bir}(X)$?

The first question was answered affirmatively in 2014 by Serge Cantat under the additional assumption that $\mathrm{dim} X \leq n$. We prove that the first question has an affirmative answer without this extra assumption (and we do not use the result of Serge Cantat).
Regarding the second question, Jean-Philippe Furter and Isac Hedén completely classified the Borel subgroups of $\mathrm{Bir}(\mathbb{P}^n)$ in 2023 for the case $n = 2$. We prove that any Borel subgroup of $\mathrm{Bir}(X)$ has derived length at most twice the dimension of $X$, and if equality holds, then $X$ is rational, and the Borel subgroup is conjugate to the standard Borel subgroup in $\mathrm{Bir}(\mathbb{P}^n)$. Moreover, we provide examples of Borel subgroups in $\mathrm{Bir}(\mathbb{P}^n)$ of derived length less than $2n$ for any $n \geq 2$ (the case $n = 2$ was treated by Furter and Hedén). This answers affirmatively a conjecture of Vladimir Popov.