Stability of fibrations

Abstract: The notion of K-stability of a polarised variety has been heavily studied in recent years, due to its link both with moduli theory (one should be able to form moduli spaces of K-stable varieties) and to Kahler geometry (K-stability should be equivalent to the existence of a constant scalar curvature Kahler metric on the variety). This story has been particularly successful for Fano varieties. I will describe a notion of stability for polarised fibrations, which generalises K-stability of polarised varieties when the base of the fibration is a point, and slope stability of a vector bundle when the variety is the projectivisation of a vector bundle. I will speculate that one should be able to form moduli spaces of stable fibrations, much as one can form moduli spaces of slope stable vector bundles over a fixed base. The main result, however, will be a description of the link with certain canonical metrics on fibrations. This is joint work with Lars Sektnan.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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