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Generalized abundance conjecture for klt pairs

Let XX be a normal projective variety and let Δ\Delta be an effective Q\mathbb{Q}-divisor such that (X,Δ)(X,\Delta) is a klt pair. The Kodaira dimension κ(KX+Δ)\kappa(K_X+\Delta) and the numerical Kodaira dimension κσ(KX+Δ)\kappa_{\sigma}(K_X+\Delta) are defined as usual.

Generalized abundance conjecture.

κ(KX+Δ)=κσ(KX+Δ).\kappa(K_X+\Delta)=\kappa_{\sigma}(K_X+\Delta).

In particular, if KX+ΔK_X+\Delta is nef, then it is semi-ample.

This is an abundance statement for klt pairs, relating the Kodaira and numerical Kodaira dimensions. The source gives no resolution status; the conjecture is presented as a central motivation for the paper.

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 generalized abundance conjecture for klt pairs

    Let (X,B)(X,B) be a klt pair with KX+BK_X+B pseudoeffective, and let LL be a nef divisor on XX such that KX+B+LK_X+B+L is also nef. Generalized abundance conjecture. The divisor KX+B+LK_X+B+L should be numerically equivalent to some semiample Q\mathbb{Q}-divisor MM:

    KX+B+L≡M.K_X+B+L \equiv M.

    This conjecture extends abundance to the setting of a nef adjoint divisor with an additional nef summand. Its status is not resolved in the supplied source context.

    source: Priyankur Chaudhuri, “An inductive approach to generalized abundance using nef reduction”, arXiv:2110.10252 (2022).

Sources & referencesView supporting material

Primary source

Shin-ichi Matsumura, “Injectivity theorems with multiplier ideal sheaves and their applications”, arXiv:1511.04226 (2015).

abundanceabundance-conjecturebirational-geometryminimal-model-programnef-divisors

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