Abundance conjecture

In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety $X$ with Kawamata log terminal singularities over a field $k$ if the canonical bundle $K_X$ is nef, then $K_X$ is semi-ample, i.e. $m{K}_X$ is base-point free for some $m>0$. In particular, if abundance holds, then one is able to define a model $X\to Y=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}\underset{l⩾0}{⨁}{H}^0(X,l{K}_X).$

Important cases of the abundance conjecture have been proven by Caucher Birkar.

References

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