Let $V$ be a subvariety in the complex projective space $\mathbb{CP}^n$ (with finite number of connected components). Is it true that the group of projective automorphisms preserving $V$ is a linear algebraic group with finitely many connected components?
Thanks.
Best Answer
Yes. Suppose that $V$ is a closed subvariety of projective space, then it is the zero locus of finitely many polynomials $f_1, \cdots, f_m$. The set of $g \in \operatorname{PGL}_n$ that preserve the zero locus of these polynomials will be closed because it is cut out by certain equations for a representing matrix of $g$ (which can be turned into polynomial equations in the coordinate ring). In particular the stabiliser $G \subset \operatorname{PGL}_n$ of $V$ is a finite type closed subscheme, hence quasi-compact, hence has finitely many connected components.
Suppose now that $V$ is an arbitrary (locally closed) subvariety, so that $V \subset \overline{V}$ (Zariski closure) is an open subvariety. The complement $Z$ of $V$ in $\overline{V}$ is closed, and cut out by finitely many equations, so as above there is a finite type closed subgroup $H \subset G$ preserving $Z$, hence preserving $V$.