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

• Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Conjecture 3.12, p. 81, ISBN 978-0-521-63277-5, https://books.google.com/books?id=YrysxvPbBLwC&pg=PA81 
• Lehmann, Brian (2017), "A snapshot of the minimal model program", in Coskun, Izzet; de Fernex, Tommaso; Gibney, Angela, Surveys on recent developments in algebraic geometry: Papers from the Bootcamp for the 2015 Summer Research Institute on Algebraic Geometry held at the University of Utah, Salt Lake City, UT, July 6–10, 2015, Proceedings of Symposia in Pure Mathematics, 95, Providence, RI: American Mathematical Society, pp. 1–32, https://www2.bc.edu/brian-lehmann/papers/snapshot.pdf