Some automorphism groups are linear algebraic

Michel Brion

Institut Fourier, University of Grenoble, 100 rue des Mathematiques, 38610 Gieres, France

Abstract: Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of $\mathrm{Aut}(X)$, and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of $\mathrm{Aut}(X)$ that fixes $K$ pointwise is linear algebraic. If $K$ has transcendence degree $1$ over the base field $k$, then $\mathrm{Aut}(X)$ is an algebraic group.

Key words and phrases: automorphism group, linear algebraic group.

MSC: 14L30, 14M17, 20G15

Language: English

DOI: 10.17323/1609-4514-2021-21-3-453-466

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