Algebraic Geometry Seminar: From stability conditions to generalized Bogomolov-Gieseker inequalities

The classical Bogomolov-Gieseker inequality provides a bound on the second Chern character of semistable sheaves, which is an essential ingredient in the construction of stability conditions on surfaces. Generalizing this, Bayer-Macrì-Toda proposed to use a conjectural ch_3 inequality for tilt stable objects to construct stability conditions on threefolds. This inequality found many applications ranging from counting invariants to linear series, but people were only able to prove it in special cases.

The recent breakthrough of Chunyi Li provides a different route to construct stability conditions over any projective variety. In my talk, I will recall this beautiful idea, and explain that, via an enhancement of Li’s construction together with some reverse-engineering, we can produce a series of inequalities on any Chern characters for suitable objects on any variety, with Bogomolov-Gieseker and Bayer-Macrì-Toda as the first two instances. This is based on joint work in progress with Chunyi Li, Zhiyu Liu and Benjamin Sung.

If time permits, I will also discuss a strengthen of this result for ch_3 inequality using Frobenius method, based on joint work with Chunyi Li, Zhiyu Liu and Alex Perry. Parts of these projects are AI-assisted, which will be explained in the talk.