Geometric Invariant Theory - Stability

Stability

If a reductive group G acts linearly on a vector space V, then a non-zero point of V is called

• unstable if 0 is in the closure of its orbit,
• semi-stable if 0 is not in the closure of its orbit,
• stable if its orbit is closed, and its stabilizer is finite.

There are equivalent ways to state these:

• A non-zero point x is unstable if and only if there is a 1-parameter subgroup of G all of whose weights with respect to x are positive.
• A non-zero point x is unstable if and only if every invariant polynomial has the same value on 0 and x.
• A non-zero point x is semistable if and only if there is no 1-parameter subgroup of G all of whose weights with respect to x are positive.
• A non-zero point x is semistable if and only if some invariant polynomial has different values on 0 and x.
• A non-zero point x is stable if and only if every 1-parameter subgroup of G has positive (and negative) weights with respect to x.
• A non-zero point x is stable if and only if for every y not in the orbit of x there is some invariant polynomial that has different values on y and x, and the ring of invariant polynomials has transcendence degree dim(V)−dim(G).

A point of the corresponding projective space of V is called unstable, semi-stable, or stable if it is the image of a point in V with the same property. "Unstable" is the opposite of "semistable" (not "stable"). The unstable points form a Zariski closed set of projective space, while the semistable and stable points both form Zariski open sets (possibly empty). These definitions are from (Mumford 1977) and are not equivalent to the ones in the first edition of Mumford's book.

Many moduli spaces can be constructed as the quotients of the space of stable points of some subset of projective space by some group action. These spaces can often by compactified by adding certain equivalence classes of semistable points. Different stable orbits correspond to different points in the quotient, but two different semistable orbits may correspond to the same point in the quotient if their closures intersect.

Example: (Deligne & Mumford 1969) A stable curve is a reduced connected curve of genus ≥2 such that its only singularities are ordinary double points and every non-singular rational component meets the other components in at least 3 points. The moduli space of stable curves of genus g is the quotient of a subset of the Hilbert scheme of curves in P5g-6 with Hilbert polynomial (6n−1)(g−1) by the group PGL_5g−5.

Example: A vector bundle W over an algebraic curve (or over a Riemann surface) is a stable vector bundle if and only if

for all proper non-zero subbundles V of W and is semistable if this condition holds with < replaced by ≤.