Jenia Tevelev (UMass): On the existence of Bridgeland stability conditions (after Chunyi Li) - Part I.

Abstract. The notion of a stability condition on the derived category of a smooth projective variety was introduced by Tom Bridgeland around 2000, motivated by the study of D-branes in string theory, and especially by Michael R. Douglas’s work on Π-stability. The case of original interest - and of particular importance - is when this variety is a Calabi-Yau manifold.

A key result, Bridgeland deformation theorem, states that stability conditions (when they exist) can be varied, and their variations form a complex manifold. Furthermore, moduli spaces of Bridgeland-semistable objects exist and change in a controlled way as the stability condition varies, leading to many striking applications.

Despite enormous progress, the existence of Bridgeland stability conditions has proven to be a very hard problem in dimensions higher than 2. It was only known for a few Calabi-Yau threefolds, including the quintic threefold, but no examples for higher-dimensional Calabi-Yau varieties were known. It therefore came as a surprise when Chunyi Li posted a paper on the arXiv a few weeks ago claiming the existence of Bridgeland stability conditions on all smooth projective varieties. 

We will try to understand what a Bridgeland stability condition is, why it is hard to construct, and how the Chunyi Li's proof works.