A relative Yau-Tian-Donaldson conjecture and stability thresholds

Abstract: On a Fano variety, the Yau–Tian–Donaldson correspondence connects the existence of Kähler–Einstein metrics to an algebro-geometric notion called $K$-stability. In the last decade, the latter has proved to be very valuable in Algebraic Geometry: for instance, it is used for the construction of moduli spaces. In the first part of the talk, partly motivated by the study of Kähler–Einstein metrics with prescribed singularities, a new relative $K$-stability notion will be introduced for a fixed smooth Fano variety. A particular focus will be given to motivations and intuitions, making a comparison with the log $K$-stability/log Kähler–Einstein metrics. The relative $K$-stability and the Kähler–Einstein metrics with prescribed singularities will then be related to each other through a Yau–Tian–Donaldson correspondence, which will be the core of the talk. An important role will be played by algebro-geometric valuative criteria, which will be also used to link the relative $K$-stability to the genuine $K$-stability.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html