Modern Mathematics Lecture Series | Wall-crossing for Calabi-Yau fourfolds and applications

Abstract

The theory for counting sheaves on Calabi-Yau fourfolds was developped by Borisov-Joyce and Oh-Thomas, while my work focuses on proving wall-crossing in this setting. It is desirable that the end result can have many concrete applications to existing conjectures. For this purpose, I introduce a new structure into the picture - formal families of vertex algebras. Apart from being a natural extension of the vertex algebras introduced by Joyce, they allow us to wall-cross with insertions instead of the full virtual fundamental classes. Many fundamental hurdles needed to be overcome to prove wall-crossing in this setting. They included constructing Calabi-Yau four obstruction theories on (enhanced) master spaces and showing that the invariants counting semistable torsion-free sheaves are well-defined. Towards the end of the talk, I will use the complete package to address existing conjectures with applications to 3-fold DT/PT correspondences.

About the speaker

Arkadij Bojko

Academia Sinica

Currently, I am an Institute Research Scholar at Academia Sinica. Previously, I worked in the group of Rahul Pandharipande at ETH Zurich, and I did my PhD with Dominic Joyce at Oxford.

My research field is enumerative algebraic geometry where I have mainly focused on sheaf-counting theories thus far. Through my work, I help to advance our understanding of invariants and structures of moduli spaces of sheaves by using tools from representation theory, gauge theory, topology, derived algebraic geometry, and combinatorics. The techniques I often apply rely on the wall-crossing behaviour of sheaves under changing stability conditions and on equivariant localization. Much of my work has focused on sheaf-counting on Calabi-Yau fourfolds and Virasoro constraints for abelian categories where I used vertex algebras and their formal families combined with wall-crossing to prove multiple existing conjectures.