A Minimal Model Program for $\mathbb{Q}$-Gorenstein varieties

TL;DR

This paper revisits and simplifies proofs of known results in the minimal model program for $Q$-Gorenstein varieties, and establishes the existence of flips and finiteness of flip sequences in $Q$-Gorenstein spherical varieties.

Contribution

It provides a streamlined proof approach for $Q$-Gorenstein MMP results and confirms the existence of flips and finiteness in spherical varieties.

Findings

Rewritten proofs of known $Q$-Gorenstein MMP results

Positive answer on existence of flips in spherical varieties

Finiteness of flip sequences in spherical varieties

Abstract

The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$\mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal model problem, 1985) of the same results in $\mathbb{Q}$-factorial MMP. And, in the family of $\mathbb{Q}$-Gorenstein spherical varieties, we answer positively to the questions of existence of flips and of finiteness of sequences of flips. I apologize for the first version of this paper, which I wrote without knowing that these results already exist.