Abundance Conjecture on Uniruled Varieties

Abstract: This content of this talk is a paper of Valdimir Lazic. Abundance conjecture says that if X is a smooth projective variety such that its canonical divisor K_X is nef, i.e. K_X intersects every curve non-negatively, then there is a positive integer m such that the m-th tensor power of the canonical line bundle \omega_X^{\otimes m}\cong \mathcal{O}_X(mK_X) has non-zero global sections, and moreover, these global sections generate the line bundle \omega_X^{\otimes m}. In particular, there is a projective morphism f:X\to \mathbb{P}^N to a porjective space determined by global sections of \omega_X^{\otimes m}. This morphism allows X to be seen as a fibration of Calabi-Yau varieties (i.e. varieties whose canonical classes are trivial). The Abudance conjecture is one of most important outstanding conjecture in the minimal model program. In the paper titled ‘’Abundance for Uniruled Varieties which are not Rationally Connected’’, Lazic shows that if (X, B) is a klt pair of dimension n such that X is uniruled but not rationally connected, and if we assume that the minimal model program holds in dimension n-1, then the Abundance conjecture holds for (X, B) is dimension n. In my talk I will explain the main ideas and techniques of Lazic’s proof.