Given a smooth projective surface, I consider vector bundles of rank $r$ with fixed HarderNarasimhan filtration. Does there exist a moduli space of such bundles? If yes, how is it constructed and which are its properties (dimension, smoothness....)?
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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 HNfactors themselves: Let $E,F$ be sheaves with Ext^1(E,F) onedimensional. 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 isotrivial family of nontrivial extensions. Therefore the modulispace cannot be separated. Nevertheless, you can construct the moduli space of sheaves with HNfactors as an ArtinStack: As pointed out by Arend Bayer, Bridgeland's Introduction to Hallalgebras (arXiv:1002.4372) is a good reference. 

