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

TL;DR

This paper advances the understanding of the abundance conjecture for three-dimensional algebraic varieties in characteristic p>5, proving specific cases related to the positivity of canonical divisors.

Contribution

It proves two new cases of the abundance conjecture for 3-folds in characteristic p>5, focusing on KLT pairs with certain canonical divisor properties.

Findings

Proves abundance for KLT 3-folds with Kodaira dimension 1.

Establishes abundance for KLT 3-folds with numerically trivial canonical divisor.

Advances the minimal model program in positive characteristic.

Abstract

In this article we prove two cases of the abundance conjecture for $3$-folds in characteristic $p>5$: $(i)$ $(X, \Delta)$ is KLT and $\kappa(X, K_X+\Delta)=1$, and $(ii)$ $(X, 0)$ is KLT, $K_X\equiv 0$ and $X$ is not uniruled.