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The non-vanishing conjecture for smooth projective varieties

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Let XX be a smooth projective variety. If KXK_{X} is pseudo-effective, then there is an effective Q\mathbb{Q}-divisor DD such that KX∼QDK_{X} \sim_{\mathbb{Q}}D.

Non-vanishing conjecture for smooth varieties. The canonical divisor KXK_{X} has an effective Q\mathbb{Q}-linearly equivalent representative.

This is the special case of non-vanishing for which the pair has smooth underlying variety and zero boundary. The paper studies its relation to non-vanishing and log minimal model conjectures for general projective log canonical pairs; the statement remains open in higher dimension.

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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 non-vanishing conjecture for smooth projective varieties

    Let XX be a smooth projective variety. Denote its Kodaira dimension by κ(X)\kappa(X) and its numerical dimension by ν(X)\nu(X). Non-vanishing conjecture. If

    ν(X)≥0,\nu(X)\geq 0,

    then

    κ(X)≥0.\kappa(X)\geq 0.

    This conjecture asserts the existence of sections whenever the numerical dimension is nonnegative. The paper proves that, in dimensions at most dd and for varieties with ν(X)≤1\nu(X)\leq 1, it implies the existence of a good minimal model or a Mori fiber space; the conjecture itself remains open in general.

    source: Jihao Liu and Zheng Xu, “Non-vanishing implies numerical dimension one abundance”, arXiv:2505.05250 (2025).

References

Primary source

Kenta Hashizume, “On the non-vanishing conjecture and existence of log minimal models”, arXiv:1609.00121 (2017).

abundancebirational-geometryminimal-model-programnon-vanishingsmooth-varieties
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