The Borisov–Alexeev–Borisov conjecture for Fano varieties

Let $d$ be a positive integer and let $\boldsymbol{\mathcal{X}_{d,\varepsilon}}$ denote the class of $\varepsilon$-klt Fano varieties of dimension $d$, where $\varepsilon>0$ is rational. A class of projective varieties is bounded if it is parametrized, up to isomorphism over its field of definition, by a projective flat family of finite type over $\mathbb{Z}$. Borisov–Alexeev–Borisov conjecture. For any rational number $\varepsilon>0$, the class

$$\mathcal{X}_{d,\varepsilon}=\left\{X\mid X\text{ is an }\varepsilon\text{-klt Fano variety of dimension }d\right\}$$

is bounded. The conjecture is a central boundedness statement in birational geometry; it is known in characteristic $0$ by Birkar, but remains open over fields of positive characteristic.