# On existence of log minimal models

@article{Birkar2010OnEO, title={On existence of log minimal models}, author={C. Birkar}, journal={Compositio Mathematica}, year={2010}, volume={146}, pages={919 - 928} }

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 dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log… Expand

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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

Finite generation of the log canonical ring in dimension four

- Mathematics
- 2010

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance… Expand

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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

Log minimal models according to Shokurov

- Mathematics
- 2008

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… Expand

Termination of (many) 4-dimensional log flips

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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

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In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs $(X/Z,B)$ with $B$ big$/Z$. This then implies existence of klt log flips, finite generation of… Expand

Minimal models, flips and finite generation : a tribute to V.V. SHOKUROV and Y.-T. SIU

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In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs (X/Z, B) with B big/Z. This then implies existence of klt log flips, finite generation of klt… Expand

Mld's vs thresholds and flips

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Abstract Minimal log discrepancies (mld's) are related not only to termination of log flips [Shokurov, Algebr. Geom. Metody 246: 328–351, (2004)] but also to the ascending chain condition (ACC) of… Expand

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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

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