Given a smooth projective surface, I consider vector bundles of rank $r$ with fixed Harder-Narasimhan filtration. Does there exist a moduli space of such bundles? If yes, how is it constructed and which are its properties (dimension, smoothness....)?

## 1 Answer

Dear ginevra,

your question is related to my earlier question Moduli of Extensions.

There are some obvious problems which occure already if you fix the HN-factors themselves: Let $E,F$ be sheaves with Ext^1(E,F) one-dimensional. Then there are two isomorphism classes of extensions $0 \rightarrow F \rightarrow G \rightarrow E \rightarrow 0$. One is trivial, the other one is not. Moreover the trivial extension is the limit of a iso-trivial family of non-trivial extensions. Therefore the moduli-space cannot be separated.

Nevertheless, you can construct the moduli space of sheaves with HN-factors as an Artin-Stack: As pointed out by Arend Bayer, Bridgeland's Introduction to Hall-algebras (arXiv:1002.4372) is a good reference.