Semistable and stable points for a linearization

Let $G$ be a complex reductive affine algebraic group (Reductive and linearly reductive complex algebraic groups) acting algebraically on a complex projective variety $X$ (projective variety classical, Classical complex affine algebraic actions and rational modules), and let $L$ be an ample $G$-linearized invertible sheaf (G-linearizations of invertible sheaves on a complex G-variety).

A point $x\in X$ is semistable with respect to $L$ if there exist $n\ge 1$ and $\sigma \in \mathrm{\Gamma }(X,L^{\otimes n}{)}^G$ with $\sigma (x)\mathrm{\ne }0$. The set of such points is written $X^{ss}(L)$, and its complement $X^{us}(L)=X\setminus X^{ss}(L)$ is the unstable locus.

A point $x\in X^{ss}(L)$ is stable with respect to $L$ if its orbit $Gx$ is closed in $X^{ss}(L)$ and its stabilizer $G_x$ is finite. The set of stable points is written $X^s(L)$.

These definitions agree with the embedded definitions for a $G$-equivariant closed immersion $X\hookrightarrow \mathbf{P}(V)$ with $L^{\otimes m}\cong \mathcal{O}(1){\mathrm{\mid }}_X$ as $G$-linearized invertible sheaves: $X^{ss}(L)=X\cap \mathbf{P}(V{)}^{ss}$ and $X^s(L)=X\cap \mathbf{P}(V{)}^s$, where semistability in $\mathbf{P}(V)$ is the nonvanishing of a positive-degree invariant homogeneous form (homogeneous coordinate ring, affine cone projective set) and stability adds closedness of the orbit in the semistable locus and finiteness of the stabilizer; this equivalence is asserted here and proved in the two main theorems of this page.

By construction $X^{ss}(L)$ and $X^s(L)$ are $G$-stable subsets of $X$. The definition itself claims no openness, nonemptiness or finiteness of either locus.

• The linearization is part of the data. The two loci depend on the linearization and not only on $(X,G,L)$; the companion page demonstrates this in The semistable locus depends on the linearization, not only on the sheaf ↗ and GIT quotients of the projective line for different linearizations ↗.
• The invariant-section formulation. The definition uses invariant sections of positive tensor powers of $L$, not only of $L$ itself; this is why a suitable common multiple of the degrees of a finite generating set, and not a single power of $L$, is needed in the projectivity arguments of this page (The invariant section ring and the projective GIT quotient).
• Embedded comparison. The comparison requires relating the full section ring to the embedding’s homogeneous coordinate ring and lifting invariants under the polynomial-ring surjection. These arguments, in addition to the positive-power comparison, are proved in Projective GIT quotient for a linear action and The projective GIT quotient from the invariant section ring.
• Finiteness of the stabilizer. For a complex affine algebraic group $G$, a closed subgroup $G_x$ is finite if and only if $\dim G_x=0$ (Global and local dimension of classical varieties).