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....)?

  • $\begingroup$ What do you mean by fixed HN filtration? Do you fix the factors, or only their invariants? $\endgroup$
    – Sasha
    Jan 31, 2011 at 14:35
  • $\begingroup$ Only the invariants $\endgroup$
    – ginevra86
    Jan 31, 2011 at 15:11

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.