Modern Theory of Moduli Spaces, Stability and Its related topics

We will first introduce the theory of good moduli space of Prof. Jarod Alper and his cooperators, more precisely, the first 6 sections in their paper¹ (the more earier version since the results in the paper are too general which we will not use).

Then we will introduce a very general moduli space in the 7th section in paper¹, the good moduli space of objects in abelian categories which will give our a very general idea to construct the moduli space of semistable sheaves and Bridgeland stable complexes.

Next we will see the detailed theory of stability and moduli theory of stable sheaves.

Fianlly We will introduce the general theory of Bridgeland stablility on the triangulated category and construct the good moduli space of Bridgeland semistable objects in heart. We will also prove the Wall-crossing structure and give some examples. Moreover we will construct the Bridgeland stablility on the surface in detail and give the second tilt about threefolds with some conjectures.

Here is my notes about this part: ModernModuliTheory, finished at 2023-10-14.