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Let $A$ be a (semi-)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a non-trivial finite stabilizer ? By stabilizer, I mean the closed subgroup $S\subset A$ such that $X$ is stable by translation by any point in $S$. This question is equivalent to the one in the title since you can always take the quotient by the neutral component of $S$ to reduce to a finite group.

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    $\begingroup$ No. Let $f:A \to B$ be a separable isogeny and $D$ a smooth ample divisor in $B$. Then $f^{−1}(D)$ is irreducible if $\dim(X)>1$ (since it is smooth and connected) and is stabilized by the kernel of $f$. $\endgroup$
    – naf
    Commented May 15, 2013 at 13:12

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No. Take a curve in its Jacobian and pull it back by multiplication by some n. The resulting pullback is a curve invariant by the n torsion.

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