semistabile

Semistabile is the term used in mathematics, especially in algebraic geometry, to denote a weaker or more flexible notion of stability for objects such as vector bundles, sheaves, or curves. Semistability typically allows certain subobjects or degenerations that would be forbidden by strict stability, and it plays a central role in constructing moduli spaces and in geometric invariant theory.

In the context of vector bundles on a smooth projective variety, semistability is defined with respect to a polarization. For a vector bundle E, the slope μ(E) is the ratio of its degree to its rank. E is semistable if for every proper nonzero subbundle F, μ(F) ≤ μ(E). It is stable if the inequality is always strict, and polystable if it is a direct sum of stable bundles of the same slope. The Harder–Narasimhan filtration provides a canonical way to break any bundle into semistable pieces with strictly decreasing slopes.

Semistability extends beyond vector bundles to coherent sheaves, principal bundles, and Higgs bundles, with corresponding notions of slope or central charge. In geometric invariant theory (GIT) and moduli theory, semistability is a key condition that allows the construction of moduli spaces as quotients. Points representing semistable objects admit a well-behaved quotient, up to S-equivalence, where objects with the same associated graded object are identified.

For curves, a semistable curve is a connected projective curve with only nodes as singularities; typically, components like rational curves may meet the rest of the curve in at least two points, with a stronger “stability” condition requiring at least three points to ensure a finite automorphism group. Semistability concepts also appear in reduction theory and in the study of moduli of abelian varieties and other geometric objects.