On the Abundance Problem for $3$-folds in characteristic $p>5$

Abstract

In this article we prove two cases of the abundance conjecture for 3-folds in characteristic p > 5: (i) (X, ∆) is klt and κ(X, K X + ∆) = 1, and (ii) (X, ∆) is klt, K X + ∆ ≡ 0 and X is not uniruled.

On the relative Minimal Model Program for threefolds in low characteristics

“…Note that the case of dim Pic 0 (X) = 0 and p > 5 was solved in [Zha17] (cf. the appendix to [DW16]) using F-splittings and techniques of the positive characteristic generic vanishing theorem ([HP16]).…”

On the canonical bundle formula and log abundance in positive characteristic

“…The abundance conjecture is one of the most fundamental open problems left in the study of the birational geometry of threefolds in characteristic p ą 5, and it has gathered much attention recently, leading to very interesting results by Das, Hacon, Waldron, and Zhang (see [DW16,Wal17a,Zha17]). It predicts that a log minimal model is either of general type or admits a "log Calabi-Yau" fibration over a lower dimensional variety.…”

“…Some log canonical variants of the base-point-free theorem have been obtained in [MNW15] (for L big over F p ), [NW17] (for p ą 5 over F p ), and [Wal17b] (lc-MMP). Certain important special cases of the abundance conjecture have been proven in [Wal17a] (Kodaira dimension two), [DW16] (Kodaira dimension one and partially for nef dimension zero), and [Zha17] (when the Albanese dimension is non-zero and the boundary is empty and partially for klt boundaries).…”