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Let $k$ be either the field $\Bbb C$ of complex numbers or the field $\Bbb R$ of real numbers. Let $X$ be an algebraic variety over $k$, say, quasi-projective and smooth (but not necessarily projectiv …

Let $k$ be an algebraically closed field of characteristic 0. Let $G$ be a connected reductive group over $k$. The notion of the algebraic fundamental group of $\pi_1(G)$ was introduced in my memoir …

The result in my preprint mentioned in the question was erroneous (the mistake was noticed by a referee). It is possible to construct a quotient $X=G/H$ and and automorphism $\tau$ of $\mathbb{C}$ su …

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$. The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence $$ \p …

I answer the question in the comment of Tom Goodwillie: What is known when $H=1$? Theorem. Let $G$ be a connected linear algebraic group over ${\mathbb{C}}$. Let $\tau$ be an automorphism of ${\mathb …

Let $X$ be a smooth irreducible algebraic variety over the field of complex numbers ${\mathbb{C}}$. Let $x\in X({\mathbb{C}})$. Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily continu …