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Let $G$ be an algebraic group (not necessarily linear) defined over an algebraically closed field $k$, acting on a smooth integral $k$-variety $X$. Let $x_0\in X(k)$ and let $\pi_1(X,x_0)$ denote the …

Let $m>0$ be a natural number. Consider the following semisimple algebraic groups over ${\mathbb{R}}$: $$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$ We embed $H$ into …

Let $X$ and $Y$ be smooth varieties over the field of complex numbers $\bf C$ (smooth integral separated schemes of finite type over $\bf C$). Let $$f\colon X\to Y$$ be a surjective morphism such tha …

Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a spherical homogeneous space, then $X$ defines a spherical datum (Luna datum) $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\varsig …

A Cayley map is a $G$-equivariant birational isomorphism $\lambda: G\to \mathfrak{g}$ (which does not have to exist). A connected linear algebraic group $G$ over $\mathbb{C}$ is called a Cayley grou …

I am asking for a reference for the following lemma (for which I know a proof). Lemma. Let $f\colon X\to Y$ be a surjective morphism of irreducible smooth complex algebraic varieties (separated, red …

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason). Theorem. Let $X$ be a homogeneous space of a connected linear algebra …

What are the automorphisms of $SL_n$ as an algebraic variety? In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL …

I am looking for exact references for the comparison theorem for the étale fundamental group. I mean the following result: Theorem (Grothendieck). For a pointed algebraic variety $(X,x)$ over $\m …

$ \renewcommand{\C}{{\mathbb C}} \renewcommand{\R}{{\mathbb R}} $ In the preprint Taking quotient by a unipotent group induces a homotopy equivalence we proved the following result: Theorem. Let $U$ …

I translate into English Lemma 6.5 from Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. 327 (1981), 12-80. Let $X$ be …

Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety. By Losev's theorem, the spherical $G$-variety $X$ is uniquely determined by its spherical datum, see …

This question is almost a duplicate of that question, which has a good answer. The difference is that I ask for references rather than proofs. By a reference I mean a reference to a book, or to a pap …

Find it here. (Edit: Link removed). I hope you can read Russian. Enjoy! Edit: Find here another (better?) scan, also in Russian.

Colliot-Thélène's paper Résolutions flasques des groupes linéaires connexes, J. für die reine und angewandte Mathematik (Crelle) 618 (2008) 77--133, http://www.math.u-psud.fr/~colliot/resolflsq_211107 …