Existence of log canonical flips and a special LMMP

Caucher Birkar

Existence of log canonical flips and a special LMMP

Algebraic Geometry 2012-04-25 v3

Authors: Caucher Birkar

Abstract

Let $(X/Z,B+A)$ be a $\Q$ -factorial dlt pair where $B,A\ge 0$ are $\Q$ -divisors and $K_X+B+A\sim_\Q 0/Z$ . We prove that any LMMP $/Z$ on $K_X+B$ with scaling of an ample $/Z$ divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.

Keywords

log canonical flips LMMP dlt pairs log minimal model Mori fibre space ACC for lc thresholds

Cite

@article{arxiv.1104.4981,
  title  = {Existence of log canonical flips and a special LMMP},
  author = {Caucher Birkar},
  journal= {arXiv preprint arXiv:1104.4981},
  year   = {2012}
}

Comments

40 pages. Reorganised, exposition improved, also includes preprint arXiv:1104.4979v1. The main result is now unconditional thanks to Hacon and Xu who generalise a result of Fujino and Gongyo on semi-log canonical pairs that is used in this paper. To appear in Pub. Math. de l'IH\'ES

Existence of log canonical flips and a special LMMP

Abstract

Keywords

Cite

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