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Post Reopened by Daniel Moskovich, S. Carnahan
added 460 characters in body
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Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class of vector bundles is then equivalent to a stable orbit under the action of the gauge group on say, the space of all connections.

Edit:

To be more precise, The notion of a stable orbit of a group action requires that, say, the stabilizer of an element be discrete. Now, viewing vector bundles as isomorphism classes under the action of the gauge group, one says that a vector bundle is stable if it satisfies the slope condition (all sub-bundles have strictly less slope). The question is, how is this slope inequality equivalent to the definition of stable orbit for group actions.

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class of vector bundles is then equivalent to a stable orbit under the action of the gauge group on say, the space of all connections.

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class of vector bundles is then equivalent to a stable orbit under the action of the gauge group on say, the space of all connections.

Edit:

To be more precise, The notion of a stable orbit of a group action requires that, say, the stabilizer of an element be discrete. Now, viewing vector bundles as isomorphism classes under the action of the gauge group, one says that a vector bundle is stable if it satisfies the slope condition (all sub-bundles have strictly less slope). The question is, how is this slope inequality equivalent to the definition of stable orbit for group actions.

Post Closed as "Needs details or clarity" by Ben Webster
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Mumford's vector bundle stability equivalent the notion orbit stability for a G-space?

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class of vector bundles is then equivalent to a stable orbit under the action of the gauge group on say, the space of all connections.