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$\newcommand{\sss}{{\rm ss}} \newcommand{\ssc}{{\rm sc}} \newcommand{\tor}{{\rm tor}} \newcommand{\X}{{\sf X}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\qed}{{$\blacksquare$}} $Let $G$ be a (connected …

You can find answers to your questions in: Deligne, Pierre: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L …

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\im{im}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\SL{SL}$I add details to Friedrich's comment: The question easily reduces to the case of a sem …

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ …

$\renewcommand{\O}{{\rm O}} \newcommand{\GO}{{\rm GO}} \newcommand{\PO}{{\rm PO}} \newcommand{\PGO}{{\rm PGO}} \newcommand{\GL}{{\rm GL}} \newcommand{\PGL}{{\rm PGL}} \newcommand{\SL}{{\rm SL}} \new …

Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$. We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{ …

Let $X$ be a smooth projective variety over a field $k$ of characteristic 0, and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$ where $I$ is an ideal such that $I^2=0$. Let $\frak X$ be a …

$\newcommand{\ssc}{{\rm sc}} \newcommand{\sss}{{\rm ss}} \newcommand{\ad}{{\rm ad}} \newcommand{\wh}{\widehat} \newcommand{\wt}{\widetilde} \newcommand{\pitil}{\tilde\pi} \newcommand{\rhotil}{\tilde\r …

$\newcommand{\ssc}{{\rm sc}} \newcommand{\ad}{{\rm ad}} \newcommand{\Fbar}{{\overline F}} $ Let $F$ be a field and $\Fbar$ be a fixed algebraic closure of $F$. Let $G$ be a (connected) reductive group …

$ \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 $U$ …

The answer is Yes. Let $G\hookrightarrow G'$ be a smooth regular embedding. We write $Z'=Z(G')$ for the center of $G'$, which is an $F$-torus where $F={\Bbb F}_q$. We construct a regular embedding $G' …

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0 (a reduced separated scheme of finite type over $k$). Let $G$ be a connected linear algebraic group over $k$ …

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 …

1. Let $X$ be a smooth irreducible $\Bbb C$-variety, on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$ (the additive group) acts freely on the right: $$ X\times _{\Bbb C} G\to X,\quad (x, …

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 …