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 ${\displaystyle X}$ with Kawamata log terminal singularities over a field ${\displaystyle k}$ if the canonical bundle ${\displaystyle K_{X}}$ is nef, then ${\displaystyle K_{X}}$ is semi-ample, i.e. ${\displaystyle mK_{X}}$ is base-point free for some ${\displaystyle m>0}$. In particular, if abundance holds, then one is able to define a model ${\displaystyle X\rightarrow Y=\mathrm {Proj} \bigoplus _{l\geqslant 0}H^{0}(X,lK_{X}).}$

Important cases of the abundance conjecture have been proven by Caucher Birkar.^[1]