On the Boundedness of Anti-Canonical Volumes of Singular Fano3-Folds in Characteristicp> 5

Abstract

AbstractIn this article, we prove the following version of the weak Borisov-Alexeev-Borisov (Weak-BAB) conjecture for $3$-folds in $\operatorname{char} p>5$: fix a set $I\subseteq [0, 1)$ which satisfies the descending chain condition (DCC) and an algebraically closed field $k$ of characteristic $p>5$. Let ${\mathfrak{D}}$ be a collection of klt pairs $(X, \Delta )$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $\Delta $ is an ${\mathbb{R}}$-divisor with coefficients in $I$, (3) $K_X+\Delta \equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\operatorname{vol}_X(-K_X) \ | \ (X, \Delta )\in{\mathfrak{D}}\mbox{ for some }\Delta \}$ is bounded from above.