# Log minimal models according to Shokurov

@article{Birkar2008LogMM, title={Log minimal models according to Shokurov}, author={C. Birkar}, journal={Algebra \& Number Theory}, year={2008}, volume={3}, pages={951-958} }

Following Shokurov’s ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space.

#### 6 Citations

On minimal model theory for log abundant lc pairs

- Mathematics
- 2019

Abstract Under the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs ( X / Z , Δ ) {(X/Z,\Delta)} such that the log… Expand

On existence of log minimal models

- Mathematics
- Compositio Mathematica
- 2010

Abstract In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in… Expand

On existence of log minimal models II

- Mathematics
- 2011

Abstract We prove that the existence of log minimal models in dimension d essentially implies the LMMP with scaling in dimension d. As a consequence we prove that a weak nonvanishing conjecture in… Expand

On existence of log minimal models and weak Zariski decompositions

- Mathematics
- 2009

We first introduce a weak type of Zariski decomposition in higher dimensions: an $${\mathbb {R}}$$ -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be… Expand

Lectures on birational geometry

- Mathematics
- 2012

Lecture notes of a course on birational geometry (taught at College de France, Winter 2011, with the support of Fondation Sciences Math\'ematiques de Paris). Topics covered: introduction into the… Expand

On termination of log flips in dimension four

- Mathematics
- 2008

We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension κ ≥ 2.

#### References

SHOWING 1-10 OF 14 REFERENCES

On existence of log minimal models

- Mathematics
- Compositio Mathematica
- 2010

Abstract In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in… Expand

Letters of a Bi-rationalist. VII Ordered termination

- Mathematics
- 2006

To construct a resulting model in the LMMP, it is sufficient to prove the existence of log flips and their termination for some sequences. We prove that the LMMP in dimension d − 1 and the… Expand

Termination of (many) 4-dimensional log flips

- Mathematics
- 2007

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(KX+D) is numerically equivalent to an effective divisor, terminates. This implies termination of… Expand

Existence of minimal models for varieties of log general type

- Mathematics
- 2006

Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

3-FOLD LOG FLIPS

- Mathematics
- 1993

We prove that 3-fold log flips exist. We deduce the existence of log canonical and -factorial log terminal models, as well as a positive answer to the inversion problem for log canonical and log… Expand

Termination of 4-fold Canonical Flips

- Mathematics
- 2003

There does not exist an infinite sequence of 4-fold canonical flips.

Termination of many 4-fold flips

- Invent. Math
- 2007

Letters of a bi-rationalist. V. Minimal log discrepancies and termination of log flips

- (Russian) Tr. Mat. Inst. Steklova
- 2004

Letters of a bi-rationalist. V. Minimal log discrepancies and termination of log flips. (Russian) Tr

- Mat. Inst. Steklova Algebr. Geom. Metody, Svyazi i Prilozh
- 2004