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The minimal model conjecture for log canonical pairs

Let (X/Z,B)(X/Z,B) be a log canonical (lc) pair, meaning that (X/Z,B)(X/Z,B) is a pair with log canonical singularities over ZZ. A log minimal model is a birational model on which KY+BYK_Y+B_Y is nef over ZZ, while a Mori fibre space is a contraction of a (KY+BY)(K_Y+B_Y)-negative extremal ray with dim⁡Y>dim⁡T\dim Y>\dim T. Minimal model conjecture. Every lc pair (X/Z,B)(X/Z,B) has a log minimal model or a Mori fibre space. This is one of the central assertions of the log minimal model program. It is settled in dimensions 33 and 44, most cases in dimension 55 are known, and pairs of general type are covered in every dimension; the general case remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The minimal model conjecture for log canonical pairs

    Let (X/Z,B)(X/Z,B) be a log canonical (lc) pair. The pair is pseudo-effective if KX+BK_X+B is pseudo-effective over ZZ, and it is effective if KX+B≡M/ZK_X+B \equiv M/Z for some M≥0M\ge 0.

    Minimal model conjecture. If (X/Z,B)(X/Z,B) is pseudo-effective, then it has a log minimal model; if it is not pseudo-effective, then it has a Mori fibre space.

    This is one of the central open problems in birational geometry and the classification theory of algebraic varieties. The paper studies how the existence of log minimal models relates to the log minimal model program with scaling and to nonvanishing.

    source: Caucher Birkar, “On existence of log minimal models II”, arXiv:0907.4170 (2009).

Sources & referencesView supporting material

Primary source

Caucher Birkar, “Lectures on birational geometry”, arXiv:1210.2670 (2012).

birational-geometrylog-pairsminimal-model-program

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