The abundance conjecture for smooth projective varieties

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 smooth projective varieties

   Let $X$ be a smooth projective variety such that $K_X$ is nef.

   Abundance conjecture. Then $K_X$ is semi-ample.

   The conjecture predicts that a nef canonical class is generated by global sections after taking a positive tensor power. It is used here to rule out varieties with infinite quasi-projective Galois covers and non-almost-abelian Galois groups; the general statement remains open.

   source: Benoît Claudon, Andreas Hoering and János Kollár, “Algebraic varieties with quasi-projective universal cover”, arXiv:1102.2762 (2011).

2. The abundance conjecture for smooth projective varieties

   Let $X$ be a smooth projective variety. Its Kodaira dimension is denoted by $\kappa(X)$ and its numerical dimension by $\nu(X)$. Abundance conjecture.

   $$\kappa(X)=\nu(X).$$

   The conjecture predicts equality of these two dimensions for every smooth projective variety. It is known in dimension at most five under the additional assumptions described in the paper, and the general case remains open.

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