semistable

Semistable is a term used in several branches of mathematics to describe objects that satisfy a weaker form of stability. In geometric invariant theory, a point of a projective variety with a group action is semistable with respect to a linearization if some invariant section of a positive power of the ample line bundle does not vanish at the point; equivalently, its orbit under the group action does not have zero in its closure after taking a suitable quotient. Semistable points, together with stable points, form a locus that admits a good quotient by the group, though semistable points may have nontrivial stabilizers and non-closed orbits.

In algebraic geometry over curves, the notion of semistability is central for vector bundles. A vector bundle E on a smooth projective curve is semistable if every nonzero proper subbundle F satisfies mu(F) <= mu(E), where mu is the slope deg/ rank. If the inequality is strict for all F, E is stable. Semistable bundles are parameterized in moduli spaces, and S-equivalence classes identify bundles with the same associated graded object of a Jordan–Hölder filtration.

The term is also used in other contexts to denote a relaxed stability condition, reflecting a common theme: semistability allows a controlled amount of instability while avoiding pathologies that would obstruct moduli constructions.