Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme.



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> I am looking for a reference for the statement that the stablizer of $X$ in $A$ is a $K$-abelian subvariety of $A$.
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I hope that the reference will also include a discussion on the notion of stabilizers. I have two naive options in mind: The first is the group stabilizer of $X(\bar{K})$ in $A(\bar{K})$ together with the Galois action of $G\_K$; maybe taking Zariski closure is needed? The second is just $\{a\in A: a+X \subseteq X\}$.



Stabilizers in abelian varieties are also abelian? reference request
This is not true: it does not need to be connected even if $X$ is smooth connected over an algebraically closed field $K$ of characteristic zero. Indeed, if there is an isogeny $\pi \colon A \to B$ and a (necessarily smooth and connected) subscheme $Y \subseteq B$ such that $X = \pi^\*Y$ (the scheme-theoretic inverse image), then $a \in A(K)$ fixes $X$ if and only if $\pi(a)$ fixes $Y$, so we get a short exact sequence
$$0 \to \ker \pi \to \operatorname{Stab}\_A(X) \to \operatorname{Stab}\_B(Y) \to 0.$$
(Exactness on the right follows from surjectivity of $\pi$.) If moreover $\operatorname{Stab}\_B(Y)$ is finite, then we see that $\operatorname{Stab}\_A(X)$ is a nonzero finite étale algebraic group.


But example of such $X$ are aplenty; for instance we can start with a nontrivial isogeny $\pi \colon A \to B$ and take $Y \subseteq B$ a smooth complete intersection of dimension $1$ of very ample divisors. Then $X$ is a complete intersection of ample divisors [Hart, Exc. III.5.7], so $X$ is connected [Hart, Cor. III.5.7]. Since $Y$ is smooth and $X \to Y$ is finite étale, $X$ is smooth as well (see for instance Tag [01VA](https://stacks.math.columbia.edu/tag/01VA)). But $Y$ is a curve generating $B$, so it cannot contain a positive-dimensional translate of an abelian subvariety, so $\operatorname{Stab}\_B(Y)$ is finite.




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**Remark.** However, it is true that the stabiliser is a smooth closed subgroup of $A$ that is defined over $K$, so its identity component is an abelian variety. Here's how to set this up scheme-theoretically:


We should first say what is being stabilised. I think the natural candidate is the point $X$ in the Hilbert scheme $\mathbf{Hilb}\_A$ of closed subschemes of $A$ (see for instance [FGAE, Ch. 5]). The multiplication action $A \times A \to A$ of $A$ on itself naturally defines an action of $A$ on $\mathbf{Hilb}\_A$ using the functor of points, and one considers the stabiliser of $X \in \mathbf{Hilb}\_A(K)$, i.e. the pullback
$$\begin{array}{ccc}\operatorname{Stab}\_A(X) & \to & \mathbf{Hilb}\_A \\ \downarrow & & \downarrow \\ A & \to & \mathbf{Hilb}\_A \times \mathbf{Hilb}\_A,\!\end{array}$$
where the bottom arrow is given on functor of points by $a \mapsto (X,aX)$ and the right vertical map is the diagonal $\Delta\_{\mathbf{Hilb}\_A}$. (This formula is taken from Milne's notes on algebraic groups [MilneNotes, §9c].)


Then the functor of points point of view immediately shows that $\operatorname{Stab}\_A(X)$ is a subgroup functor of $A$, which is a closed subgroup scheme since $\Delta\_{\mathbf{Hilb}\_A}$ is a closed immersion (see also the remark below). It is smooth since every algebraic group in characteristic $0$ is smooth [MilneNotes, Cor. 10.36] (but a better proof is given in Tag [047N](https://stacks.math.columbia.edu/tag/047N)), hence its identity component is an abelian subvariety of $A$ [MilneNotes, Def. 10.13].


Since formation of Hilbert schemes and the pullback square above commute with base change along $\operatorname{Spec} \bar K \to \operatorname{Spec} K$ (see [FGAE, 5.1.5(5)] for the statement about Hilbert schemes), we get
$$\operatorname{Stab}\_{A\_{\bar K}}(X\_{\bar K}) = \operatorname{Stab}\_A(X) \underset{\operatorname{Spec} K}\times \operatorname{Spec} \bar K.$$
The Nullstellensatz says that reduced subschemes of $A\_{\bar K}$ are uniquely determined by their $\bar K$-points, so $\operatorname{Stab}\_{A\_{\bar K}}(X\_{\bar K})$ could be defined simply as the (set-theoretic) stabiliser of $X\_{\bar K}$ in $A\_{\bar K}$.


**Remark.** It's a little dissatisfying that we need properness of $A$ to get representability of the Hilbert scheme, so the same argument doesn't work for affine algebraic groups. But in fact we don't really use representability of $\mathbf{Hilb}\_A$, only that the diagonal $\Delta\_{\mathbf{Hilb}\_A}$ is representable by closed immersions (in the sense of Tags [0023](https://stacks.math.columbia.edu/tag/0023) and [025V](https://stacks.math.columbia.edu/tag/025V)). It's possible that this holds for a larger class of algebraic groups, but I don't immediately see whether or not this is true.




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**References.**


[FGAE] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, and A. Vistoli, [*Fundamental algebraic geometry: Grothendieck’s FGA explained*](https://doi.org/http://dx.doi.org/10.1090/surv/123). Mathematical Surveys and Monographs 123. American Mathematical Society (AMS), Providence, RI (2005).


[Hart] R. Hartshorne, [*Algebraic geometry*](https://doi.org/10.1007/978-1-4757-3849-0). Graduate Texts in Mathematics 52. Springer-Verlag, New York-Heidelberg-Berlin (1983).


[Milne] J. S. Milne, [*Algebraic groups. The theory of group schemes of finite type over a field*](http://dx.doi.org/10.1017/9781316711736). Cambridge Studies in Advanced Mathematics 170. Cambridge University Press (2017).


A variant is available online as lecture notes (which I cite because I don't have the book available to me at the moment):


[MilneNotes] J. S. Milne, [*Algebraic groups*](https://www.jmilne.org/math/CourseNotes/iAG200.pdf). Lecture notes.


