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Let $X$ be a non-singular complex algebraic variety (quasi-projective if necessary) and $K$ a connected compact Lie group acting on $X$ real algebraically, i.e. the action map $K \times X \to X$ is real algebraic and for each $k \in K$ the map $k : X \to X$ is complex algebraic.

Question. Is there a cover of $X$ by $K$-invariant affine open sets?

The reason I'm asking is that I'm thinking about whether the $K$-action extends to a complex algebraic action of the complexification $K_{\mathbb{C}}$. If $X$ is affine, the action does extend, so a positive answer to the above question would settle the general case.

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    $\begingroup$ No, that is not true. Let $X$ be a complex Abelian variety of complex dimension $g>0$, and let $K$ be the underlying real Lie group of $X$ considered as a product of $2g$ copies of the real Lie group $U(1)$. The only $K$-invariant open subsets are the empty set and all of $X$. $\endgroup$
    – Jason Starr
    Commented Sep 1, 2020 at 16:48
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    $\begingroup$ ...and Jason's example also gives a counterexample to the OP's more general question: this action of K does not extend to an algebraic action of K_C, at least if one complexifies to a split algebraic torus. $\endgroup$
    – Dave Anderson
    Commented Sep 1, 2020 at 17:12
  • $\begingroup$ Homogenous varieties under a connected reductive group are another class of examples. $\endgroup$
    – Angelo
    Commented Sep 3, 2020 at 18:59

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