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The abundance conjecture for log canonical pairs

Let (Y/Z,BY)(Y/Z,B_Y) be a log canonical (lc) pair, and suppose that KY+BYK_Y+B_Y is nef over ZZ. A divisor is semi-ample over ZZ if it defines a contraction over ZZ after a suitable positive multiple; here this means that there are a contraction h ⁣:Y→S/Zh\colon Y\to S/Z and an ample over ZZ R\mathbb R-divisor HH on SS. Abundance conjecture. The divisor KY+BYK_Y+B_Y is semi-ample over ZZ, so that

KY+BY∼Rh∗H.K_Y+B_Y\sim_{\mathbb R}h^*H.

The conjecture is stated in the source as proved through work of Miyaoka, Kawamata, and Keel–Matsuki–McKernan, although the surrounding discussion also describes higher-dimensional abundance as an important problem; the supplied parser status marks this candidate as resolved.

Equivalent formulations 2

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 abundance conjecture for log canonical pairs

    Let (X/Z,B)(X/Z,B) be a log canonical (lc) pair. A divisor is nef over ZZ if it is nef relative to ZZ, and semi-ample over ZZ if it defines a relatively semi-ample divisor.

    Abundance conjecture. If KX+BK_X+B is nef over ZZ, then it is semi-ample over ZZ.

    Abundance is a fundamental open problem in the minimal model program. The paper notes that, in higher dimensions, there has been little progress on this conjecture and identifies nonvanishing as a central conceptual obstacle.

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

  2. The abundance conjecture for log canonical pairs

    Let (X,Δ)(X,\Delta) be a log canonical pair, where Δ\Delta is a Q\mathbb Q-divisor.

    Abundance conjecture. If KX+ΔK_X+\Delta is nef, then it is semiample.

    This is a central problem in birational geometry. The conjecture is known in several cases, including projective K-trivial threefolds, but is not known in full generality.

    source: Haidong Liu and Roberto Svaldi, “Rational curves and strictly nef divisors on Calabi–Yau threefolds”, arXiv:2010.12233 (2020).

Sources & referencesView supporting material

Primary source

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

abundance-conjecturebirational-geometrylog-pairsminimal-model-program

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