Borel fixed-point theorem fails for non-split linear algebraic groups

Statement

Short version

The analogue of the Borel fixed-point theorem for the case of an infinite non-algebraically closed field where the group is not a split algebraic group is not true.

Long version

For any infinite non-algebraically closed field $K$, there exists a solvable connected linear algebraic group $G$ over $K$ that is not a split algebraic group, such that there exists a projective variety $V$ over $K$ and an algebraic group action of $G$ on $V$ with no fixed point.

Proof

Example of the multiplicative group of a field extension

Since $K$ is not algebraically closed, it has finite-dimensional field extensions of degree strictly greater than one.

We consider the case that $G$ is the multiplicative group of a finite-dimensional field extension $L$ of $K$ of degree $d>1$. Consider $L$ as a $d$-dimensional vector space over $K$. The action of $G$ on $L$ by multiplication defines an embedding of $G$ in $GL(d,K)$. This induces an action of $G$ on projective space $V=\mathbb {P} ^{d-1}(K)$. We verify all the things:

• $G$ is solvable: In fact, it is abelian
• $G$ is connected: See algebraic torus is connected
• $G$ is a linear algebraic group: This follows from its embedding in $GL(d,K)$
• $G$ is not split: $G=L^{*}$ has a normal subgroup $K^{*}$ but the quotient group does not contain any copies of the additive or multiplicative group of $K$.
• $V$ is a projective variety: By definition, projective spaces are projective varieties.
• $G$ acts on $V$ algebraically (also called regularly): The action is a restriction of the natural action of $GL(d,K)$, which is algebraic.
• The action has no fixed point: In fact, the action is transitive, and, if we think of $G$ as $L^{*}$, then $V$ is isomorphic to the quotient variety $L^{\ast }/K^{\ast }$.