CM-minimizers and standard models of Fano fibrations over curves

12/04/2023 10:30 am - 11:30 am
CMSA Room G10
Address: CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Algebraic Geometry in String Theory Seminar

Speaker: Maksym Fedorchuk (Boston College)

Title: CM-minimizers and standard models of Fano fibrations over curves

Abstract: A recent achievement in K-stability of Fano varieties is an algebro-geometric construction of a projective moduli space of K-polystable Fanos. The ample line bundle on this moduli space is the CM line bundle of Tian. One of the consequences of the general theory is that given a family of K-stable Fanos over a punctured curve, the polystable filling is the one that minimizes the degree of the CM line bundle after every finite base change. A natural question is to ask what are the CM-minimizers without base change. In answering this question, we arrive at a theory of Koll\’ar stability for fibrations over one-dimensional bases, and standard models of Fano fibrations. I will explain the joint work with Hamid Abban and Igor Krylov in which we show that the CM-minimizers for del Pezzo fibrations are Corti’s standard models and related work in progress on quartic threefold hypersurfaces.