# Stable pairs

## 5 pm, 31 Jul 2024

In this seminar, Fields Medalist Prof. Caucher Birkar talks about sheaf stable pairs in algebraic geometry, especially for Fano varieties.

Birational geometry studies when two algebraic varieties are isomorphic outside lower-dimensional subsets. Quot schemes were constructed by Grothendieck to parameterise families of sheaves with fixed numerical invariants. Stability is a key property of a sheaf over an algebraic variety, characterising, for examples, the existence of flat Hermitian metrics.

In this seminar, Prof. Birkar will explore the moduli of sheaf stable pairs, examining their relationships with Quot schemes and birational geometry. The focus will be on Fano varieties and their singularities. Drawing from the recent paper "Sheaf Stable Pairs, Quot-Schemes, and Birational Geometry" by Birkar, Jia, and Sheshmani, this presentation connects moduli theory of sheaf stable pairs with birational geometry. It offers in-depth analysis of moduli on smooth projective curves and explores calculations in low degrees. This seminar aims to bridge these concepts and give their implications for algebraic geometry.

## Event info

The event is on Wednesday 31 July at the London Institute, which is on the second floor of the Royal Institution. The seminar starts at 5pm and is followed by a Q&A. There will be drinks afterwards.

## Speakers

Caucher Birkar, Distinguished Professor at Tsinghua University, received the 2018 Fields Medal for his groundbreaking work in algebraic geometry. He did his PhD in mathematics at the University of Nottingham. He works on algebraic geometry, particularly birational geometry.