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Good and geometric quotient on the stable locus

Statement

Assume AC inherited from the named suppliers. In the setting of The projective GIT quotient from the invariant section ring let $X^{s} (L) = {x \in X^{s s} (L) : G x is closed in X^{s s} (L), G_{x} is finite}$ , and let $π : X^{s s} (L) \to Y = Proj R (X, L)^{G}$ be the good quotient. Then:

(i) $X^{s} (L)$ is open and $G$ -stable in $X$ , $Y^{s} : = π (X^{s} (L))$ is open in $Y$ , and $X^{s} (L) = π^{- 1} (Y^{s})$ ;

(ii) $π : X^{s} (L) \to Y^{s}$ is a geometric quotient: its fibres are exactly the $G$ -orbits in $X^{s} (L)$ , and $O_{Y^{s}} ≅ (π_{*} O_{X^{s} (L)})^{G}$ ;

(iii) for every $m \geq 1$ the identification $X^{s s} (L) = X^{s s} (L^{\otimes m})$ carries $X^{s} (L)$ onto $X^{s} (L^{\otimes m})$ , and the two geometric quotients are identified by the Veronese isomorphism;

(iv) the stable locus admits the invariant-chart description: $x \in X^{s} (L)$ if and only if $G_{x}$ is finite and there exist $m \geq 1$ and $σ \in Γ (X, L^{\otimes m})^{G}$ with $σ (x) \neq 0$ and the action of $G$ on the affine chart $X_{σ}$ having all orbits closed;

(v) if $X^{s} (L) = X^{s s} (L)$ then $π$ itself is a geometric quotient of $X^{s s} (L)$ .

Facts & Assumptions

Given: The setting of the projective GIT theorem: a complex reductive affine algebraic group $G$ , a complex projective variety $X$ , an ample $G$ -linearized invertible sheaf $L$ , the good quotient $π : X^{s s} (L) \to Y = Proj R (X, L)^{G}$ , and the locally closed stable locus $X^{s} (L)$ .

[F1]

Linear case. For a linear action of $G$ on a $G$ -stable closed $X^{'} \subseteq P (V)$ : $R (X^{'})^{G}$ is a finitely generated graded $C$ -algebra, $X^{' s s}$ and $X^{' s}$ are open $G$ -stable subsets, the chart morphisms glue to a good quotient $π^{'} : X^{' s s} \to Y^{'} = Proj R (X^{'})^{G}$ , the set $Y^{' s} = π^{'} (X^{' s})$ is open with $X^{' s} = π^{' - 1} (Y^{' s})$ , the restriction $π^{'} : X^{' s} \to Y^{' s}$ is a geometric quotient with orbit fibres and $O_{Y^{' s}} ≅ (π_{*}^{'} O_{X^{' s}})^{G}$ , a point of $X^{' s s}$ is stable if and only if it has finite stabilizer and lies in a chart $X_{F}^{'}$ in which all $G$ -orbits are closed, equivalently if and only if it has finite stabilizer and closed orbit in $X^{' s s}$ , and $X^{' s} = X^{' s s}$ implies that $π^{'}$ is a geometric quotient of $X^{' s s}$ . (Projective GIT quotient for a linear action)

[F2]

Ample case and Veronese. Let $L$ be an ample $G$ -linearized invertible sheaf with $R (X, L)$ its section ring and $π : X^{s s} (L) \to Y = Proj R (X, L)^{G}$ the good quotient; for every $m \geq 1$ one has $X^{s s} (L) = X^{s s} (L^{\otimes m})$ and $X^{s} (L) = X^{s} (L^{\otimes m})$ , the Veronese isomorphism identifies the quotient data, and for a $G$ -equivariant closed immersion $i : X ↪ P (V)$ with $i^{*} O (1) ≅ L^{\otimes m}$ as $G$ -linearized invertible sheaves one has $X^{s s} (L) = i^{- 1} (X^{' s s})$ and $X^{s} (L) = i^{- 1} (X^{' s})$ for $X^{'} = i (X)$ with its linear action. (The projective GIT quotient from the invariant section ring, Semistable and stable points for a linearization, An ample linearization embeds equivariantly after a positive power, Proj is invariant under Veronese regrading)

[F3]

Transfer of coordinate charts. If $F \in R (X^{'})_{k}^{G}$ is a homogeneous invariant in the embedded homogeneous coordinate ring, its restriction is an invariant section $σ = i^{*} F \in Γ (X, L^{\otimes m k})^{G}$ and $i (X_{σ}) = X_{F}^{'}$ . Conversely every invariant homogeneous coordinate-ring element has an invariant polynomial lift by naturality of the Reynolds operator under the surjection from the polynomial ring. Since $i$ is a $G$ -equivariant isomorphism onto $X^{'}$ , corresponding points have the same stabilizer and orbit closedness on these matching charts agrees. This fact concerns sections coming from the embedded coordinate ring; arbitrary global sections need not arise this way. (Affine chart quotients for invariant sections of a linear action, The Reynolds operator and the ideal theory of the invariant subring)

[F4]

AC. The Axiom of Choice is inherited from the linear-action and quotient suppliers and is used only through them. (The Axiom of Choice)

[F5]

Saturation of a section chart. For a positive-degree invariant section $σ \in R_{n}^{G}$ and any positive-degree invariant section $f \in R_{d}^{G}$ defining an overlapping chart, the degree-zero function $σ^{d} / f^{n}$ on $X_{f}$ pulls back from the Proj chart. Therefore, on $X_{f}$ , its value is nonzero exactly where $σ$ is nonzero. Since the charts $X_{f}$ cover $X^{s s} (L)$ , the chart quotient has target $D_{+} (σ)$ and $X_{σ} = π^{- 1} (D_{+} (σ))$ . (The projective GIT quotient from the invariant section ring, Affine chart quotients for invariant sections of a linear action)

Proof

technique · direct

1.1F2

The equivariant embedding and the reduction. Choose $m \geq 1$ and a $G$ -equivariant closed immersion $i : X ↪ P (V)$ with $i^{*} O (1) ≅ L^{\otimes m}$ as $G$ -linearized invertible sheaves ([F2]); put $X^{'} = i (X)$ and $R^{'} = R (X^{'})$ . By [F2] the isomorphism $i$ identifies $X^{s s} (L)$ with $X^{' s s}$ and $X^{s} (L)$ with $X^{' s}$ ; the Veronese isomorphism identifies $Proj R (X, L)^{G}$ with $Proj R (X, L^{\otimes m})^{G}$ . The latter full section ring need not equal the homogeneous coordinate ring $R^{'}$ . Instead, the main theorem [F2] identifies their quotient targets canonically on each invariant coordinate chart: both localized degree-zero rings equal the invariant regular functions on that affine chart, these charts cover both targets, and their localization maps agree. The resulting canonical isomorphism $Y ≅ Y^{'}$ identifies $π$ with $π^{'}$ because both chart morphisms arise from the same inclusions of invariant functions.

2.1F1step 1.1

Transport of (i) and (ii). By [F1] applied to the linear action on $X^{'}$ the stable locus $X^{' s}$ is open and $G$ -stable, $Y^{' s} = π^{'} (X^{' s})$ is open in $Y^{'}$ , $X^{' s} = π^{' - 1} (Y^{' s})$ , and $π^{'} : X^{' s} \to Y^{' s}$ is a geometric quotient with orbit fibres and $O_{Y^{' s}} ≅ (π_{*}^{'} O_{X^{' s}})^{G}$ . Transporting along $i$ and the identification of step 1.1 gives that $X^{s} (L)$ is open and $G$ -stable, $Y^{s} = π (X^{s} (L))$ is open in $Y$ , $X^{s} (L) = π^{- 1} (Y^{s})$ , and $π : X^{s} (L) \to Y^{s}$ is a geometric quotient with orbit fibres and $O_{Y^{s}} ≅ (π_{*} O_{X^{s} (L)})^{G}$ . This proves (i) and (ii).

2.2F2step 1.1

Veronese compatibility (iii). The locus equalities $X^{s s} (L) = X^{s s} (L^{\otimes m})$ and $X^{s} (L) = X^{s} (L^{\otimes m})$ and the identification of the quotient data by the Veronese isomorphism are the corresponding clauses of [F2]; closedness of an orbit in the semistable locus and finiteness of a stabilizer are read in the same identified locus, so the geometric restrictions to the stable loci are identified as well. This proves (iii).

3.1F1F2F3F5step 2.1

Chart description (iv). Let $x \in X^{s} (L)$ , so $x^{'} = i (x) \in X^{' s}$ by step 2.1. By [F1] the point $x^{'}$ lies in a chart $X_{F}^{'}$ , $F \in R_{> 0}^{' G}$ , in which all $G$ -orbits are closed, and $G_{x^{'}}$ is finite; by [F3] the form $F$ restricts to an invariant section $σ \in Γ (X, L^{\otimes m \deg F})^{G}$ with $i (X_{σ}) = X_{F}^{'}$ , and $i$ identifies the stabilizers and the closedness of orbits, so all orbits in $X_{σ}$ are closed and $G_{x}$ is finite. Conversely let $σ \in Γ (X, L^{\otimes n})^{G}$ and $x \in X_{σ}$ with all orbits in $X_{σ}$ closed and $G_{x}$ finite. By [F5], $X_{σ} = π^{- 1} (D_{+} (σ))$ is saturated. The quotient map is constant on $G x$ and hence on its closure in $X^{s s} (L)$ , since the fibre over $π (x)$ is closed. That fibre lies in $X_{σ}$ , because $π (x) \in D_{+} (σ)$ . Therefore any point of $\overline{G x} \cap X^{s s} (L)$ lies in $X_{σ}$ ; as the orbit is closed there by assumption, it has no boundary point in $X^{s s} (L)$ and is closed in the semistable locus. Thus $x \in X^{s} (L)$ . This proves (iv).

3.2F1step 1.1

The case $X^{s} (L) = X^{s s} (L)$ (v). If $X^{s} (L) = X^{s s} (L)$ , then $X^{' s} = X^{' s s}$ by step 2.1, so $π^{'}$ is a geometric quotient of $X^{' s s}$ by [F1]; transporting along the identifications of step 1.1 gives that $π$ is a geometric quotient of $X^{s s} (L)$ . This proves (v).

4.1F4step 2.1step 2.2step 3.1step 3.2∎

Assertions (i)-(v) are established: (i) and (ii) in step 2.1, (iii) in step 2.2, (iv) in step 3.1 and (v) in step 3.2. No new selection is made; the Axiom of Choice is inherited from the named suppliers [F4].

Remarks

Both descriptions of stability. The invariant-chart description (iv) is the form in which stability is checked on the companion page; it is transported from the linear-action theorem through the equivariant embedding in steps 1.1 and 3.1, so the ample case is reduced to the linear one rather than re-proved chart by chart.
No numerical criterion. As in the linear-action theorem, no Hilbert--Mumford criterion is involved; all statements are about orbits and invariant sections.

Depends on

Used by

GIT quotients of the projective line for different linearizations Example

Dependency tree · two levels

76 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. E. Newstead, Geometric Invariant Theory, lecture notes, CIMAT Guanajuato 2006 (CEL/HAL; archived copy) (standard reference, not scraped)
Michel Brion, Introduction to actions of algebraic groups, Les cours du CIRM 1 (2010), no. 1, 1-22 (standard reference, not scraped)
Victoria Hoskins, Moduli Problems and Geometric Invariant Theory, FU Berlin lecture notes (2015/16) (standard reference, not scraped)
I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series 296, Cambridge University Press, 2003 (standard reference, not scraped)