Bridgeland Stability Condition

A Bridgeland stability condition on a triangulated category consists of a central charge homomorphism from its Grothendieck group to and a locally finite slicing indexed by real phases. A nonzero object has with . The slicing satisfies , and morphisms from higher to lower phases vanish. Every nonzero object has a finite Harder-Narasimhan filtration by distinguished triangles. Local finiteness requires the extension categories of sufficiently short phase intervals to be quasi-abelian of finite length.

For a smooth projective variety, a numerical condition factors through the numerical Grothendieck group , the quotient by the radical of the Euler pairing. The full numerical support property requires a norm and a constant such that

for every nonzero semistable object. This controls its entire numerical class, including directions invisible to selected intersection degrees. The heart is the extension category . A geometric condition has every point sheaf stable of phase 1.