Title of Seminar: Algebraic Geometry Preprint Seminar 2022
Title of Talk: Abundance Conjecture on Uniruled Varieties
Speaker: Omprokash Das, TIFR Mumbai
Date: September 20, 2022
Time: 16:00:00 Hours
Venue: AG-77
Abstract: The content of this talk is a paper by Vladimir 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 projective 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 Abundance conjecture is one of the most important outstanding conjectures 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.
Title of Talk: Abundance Conjecture on Uniruled Varieties
Speaker: Omprokash Das, TIFR Mumbai
Date: September 20, 2022
Time: 16:00:00 Hours
Venue: AG-77
Abstract: The content of this talk is a paper by Vladimir 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 projective 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 Abundance conjecture is one of the most important outstanding conjectures 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.