Polystability

Polystability is a concept in mathematics, particularly in the field of differential geometry and complex geometry, that describes the stability of certain geometric structures. It is a generalization of the notion of stability in the context of holomorphic vector bundles over complex manifolds. The term was introduced by Simon Donaldson in the 1980s as part of his work on the Yang-Mills equations and their applications to four-dimensional manifolds.

A holomorphic vector bundle over a complex manifold is said to be polystable if it can be decomposed into a direct sum of stable holomorphic vector bundles, each with the same slope. The slope of a holomorphic vector bundle is a measure of its stability, defined as the ratio of its first Chern class to its rank. A holomorphic vector bundle is stable if its slope is greater than the slope of any of its proper subbundles.

Polystability is a crucial concept in the study of moduli spaces of holomorphic vector bundles, which are spaces that parametrize equivalence classes of holomorphic vector bundles with fixed topological invariants. The moduli space of polystable holomorphic vector bundles is particularly well-behaved and has been extensively studied in the literature.

The concept of polystability has also been generalized to other contexts, such as the study of sheaves and coherent sheaves over complex manifolds, and has found applications in various areas of mathematics, including algebraic geometry, complex analysis, and mathematical physics.