# Moscow Mathematical Journal

Volume 21, Issue 3, July–September 2021 pp. 453–466.

Some Automorphism Groups are Linear Algebraic

**Authors**:
Michel Brion (1)

**Author institution:**(1) Institut Fourier, University of Grenoble, 100 rue des Mathematiques, 38610 Gieres, France

**Summary: **

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.

2020 Math. Subj. Class. 14L30, 14M17, 20G15.

**Keywords:**Automorphism group, linear algebraic group.

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