The BAB conjecture for log Fano varieties

Fix an algebraically closed field $k$, an integer $d>0$, and a rational number $\varepsilon>0$. Let $X$ be a $d$-dimensional normal projective variety over $k$, and let $B$ be a $\mathbb{Q}$-divisor on $X$. The minimal log discrepancy of $(X,B)$ is denoted by $\operatorname{mld}(X,B)$. BAB conjecture. The varieties $X$ such that

$$-(K_X+B)\text{ is nef and big}$$

for some $B$ and

$$\operatorname{mld}(X,B)>\varepsilon$$

form a bounded family. This is the boundedness conjecture for varieties of bounded dimension with log Fano-type positivity and a uniform lower bound on minimal log discrepancies; the paper gives a partial affirmative answer for weak Fano threefolds in characteristic $p>5$ under additional hypotheses.