Title of Seminar: Algebraic Geometry Preprint Seminar 2022
Title of Talk: Strictly nef divisors on K-trivial fourfolds
Speaker: Priyankur Chaudhury, TIFR
Date: January 18, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: This talk will be based on the paper "Strictly nef divisors on K-trivial fourfolds" by Haidong Liu and Shinichi Matsumura : [2105.07259] Strictly nef divisors on K-trivial fourfolds (arxiv.org) A Cartier divisor L on a projective variety X is called strictly nef if its intersection number with every curve in X is strictly positive. Such divisors need not be ample; there are examples due to Mumford, Ramanujam and Subramanian. A conjecture of Serrano from the 90's predicts however that if X is smooth and projective, then the canonical divisor K of X added to any large enough multiple of L will be ample. The hardest case of this conjecture is the K-trivial case, where it is not even completely known in dimension 3. In the above paper, the authors confirm this conjecture for K-trivial fourfolds with vanishing irregularity, i.e any strictly nef divisor on such a fourfold must be ample. The proof consists of two main parts: 1. showing that any strictly nef divisor on a K-trivial fourfold has non-negative Kodaira dimension. This part uses analytic methods: the calculus of currents and multiplier ideal sheaves. 2. showing that any effective strictly nef divisor on a K-trivial fourfold is ample. This uses some standard techniques of the minimal model program.
Title of Talk: Strictly nef divisors on K-trivial fourfolds
Speaker: Priyankur Chaudhury, TIFR
Date: January 18, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: This talk will be based on the paper "Strictly nef divisors on K-trivial fourfolds" by Haidong Liu and Shinichi Matsumura : [2105.07259] Strictly nef divisors on K-trivial fourfolds (arxiv.org) A Cartier divisor L on a projective variety X is called strictly nef if its intersection number with every curve in X is strictly positive. Such divisors need not be ample; there are examples due to Mumford, Ramanujam and Subramanian. A conjecture of Serrano from the 90's predicts however that if X is smooth and projective, then the canonical divisor K of X added to any large enough multiple of L will be ample. The hardest case of this conjecture is the K-trivial case, where it is not even completely known in dimension 3. In the above paper, the authors confirm this conjecture for K-trivial fourfolds with vanishing irregularity, i.e any strictly nef divisor on such a fourfold must be ample. The proof consists of two main parts: 1. showing that any strictly nef divisor on a K-trivial fourfold has non-negative Kodaira dimension. This part uses analytic methods: the calculus of currents and multiplier ideal sheaves. 2. showing that any effective strictly nef divisor on a K-trivial fourfold is ample. This uses some standard techniques of the minimal model program.