On the boundedness of anti-canonical volumes of singular Fano $3$-folds in characteristic $p>5$

Abstract

In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0,1)$ 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 $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.

Some organizational changes and some minor mathematical changes. Final version. To appear in the International Mathematics Research Notices