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. $mK_{X}$ is base-point free for some $m>0$. In particular, if abundance holds, then one is able to define a model $X\rightarrow Y=\mathrm {Proj} \bigoplus _{l\geqslant 0}H^{0}(X,lK_{X}).$ Important cases of the abundance conjecture have been proven by Caucher Birkar.