The abundance conjecture for klt pairs

Let $(X,B)$ be a klt pair. Write $u(K_X+B)$ for the numerical dimension and $\kappa_{\iota}(K_X+B)$ for the invariant Iitaka dimension of $K_X+B$ . Abundance conjecture. One has

\kappa_{\iota}(K_X+B)=\nu(K_X+B).

This is a central conjecture in the birational geometry of singular varieties and is used in the paper as an assumption for results about foliations. The supplied text does not state whether it is known or unresolved in the stated generality.

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.

The abundance conjecture for klt pairs
Let $(X,\Delta)$ be a klt pair, meaning that $X$ has Kawamata log terminal singularities and $\Delta$ is the boundary divisor. Abundance conjecture. If
$K_X+\Delta$
is nef, then $K_X+\Delta$ is semiample: some multiple $m(K_X+\Delta)$ is basepoint free. The source identifies this as one of the most important open problems in higher-dimensional projective geometry.
source: Vladimir Lazić, Keiji Oguiso and Thomas Peternell, “The Morrison-Kawamata Cone Conjecture and Abundance on Ricci flat manifolds”, arXiv:1611.00556 (2016).

Sources & referencesView supporting material

Primary source

Jihao Liu and Zheng Xu, “Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures”, arXiv:2510.04419 (2025).

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