Stable vector bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of by there is an exact sequencewhich represents a non-zero element since the trivial exact sequence representing the vector isIf we consider the family of vector bundles in the extension from for , there are short exact sequenceswhich have Chern classes generically, but have at the origin. This kind of jumping of numerical invariants does not happen in moduli spaces of stable vector bundles. Stable vector bundles over curves A slope of a