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This is an excerpt on a set of lecture notes I'm going over:

Classically, a lot of interest in stable vector bundles is due to the fact that stability allows the study of moduli of vector bundles via nicely behaved (finite type, separated) moduli spaces. In particular, the moduli of stable vector bundles of fixed Chern class is “bounded”, which is not true for the moduli of arbitrary vector bundles.

The notes do not give a reference for the result stating that the moduli space of stable bundles of a fixed Chern class is bounded. Does anyone here happen to know where I could find the details of this?

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    $\begingroup$ This is covered in great detail in Huybrechts-Lehn The Geometry of Moduli Spaces of Sheaves (see Theorem 3.3.7). $\endgroup$
    – abx
    Commented Oct 1 at 8:33
  • $\begingroup$ Thank you! @abx $\endgroup$
    – Johannes
    Commented Oct 1 at 8:51

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