Twisted forms with real points of a real Grassmannian

Question

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 $X({\Bbb C})={\rm Gr}_{n,k,{\Bbb C}}$. We consider the twisted $\Bbb R$-forms $_c X$ of $X$ where $c\in {\rm Aut\,} X_{\Bbb C}$ is a $1$-cocycle. They have the following property: $({}_c X)\times_{\Bbb R}{\Bbb C}\simeq X\times_{\Bbb R}{\Bbb C}$.

Question 1. What are the twisted $\Bbb R$-forms $_c X$ of $X$ having ${\Bbb R}$-points?

Question 2. What are references for the version of Question 1 over an arbitrary field rather than over ${\Bbb R}$?

Concerning Question 1, I have found the following twisted forms $_c X$, for which I write the set of ${\Bbb R}$-points $({}_c X)({\Bbb R})$.

1. ${\rm Gr}_{n,k,{\Bbb R}}.$

2. For $n=2n',\ k=2k'$, the quaternionic Grassmannian ${\rm Gr}_{n',k',{\Bbb H}}$ of $k'$-dimensional subspaces in ${\Bbb H}^{n'}$, where ${\Bbb H}$ denotes the division algebra of Hamilton's quaternions.

3. For $n=2k$, the isotropic Hermitian Grassmannian whose ${\Bbb R}$-points are the $k$-dimensional subspaces $W\subset{\Bbb C}^n={\Bbb C}^{2k}$ that are totally isotropic with respect to the Hermitian form $\mathcal H$ with matrix ${\rm diag}(-1,\dots,-1,+1,\dots, +1)$ where $-1$ appears $k$ times and also $+1$ appears $k$ times. Here a subspace $W\subset {\Bbb C}^{2k}$ is called totally isotropic if the restriction of $\mathcal H$ to $W$ is zero.

Question 3. Do these exhaust all twisted real forms of $X$ having real points?

Answer 1

Updated

Let $G:={\rm Aut}(X)^0$. Then it is well-known that $$G_{\mathbb C}= ({\rm Aut}(X)^0)_{\mathbb C}= ({\rm Aut}(X)_{\mathbb C})^0= {\rm Aut}(X_{\mathbb C})^0\cong {\rm PGL}(n,\mathbb C).$$ So $G$ is a real form of ${\rm PGL}(n,\mathbb C)$ which means that $G$ is isomorphic to ${\rm PGL}(n,\mathbb R)$, ${\rm PGL}(n/2,\mathbb H)$, or ${\rm PU}(p,n-p)$. On the other hand, $X$ has a real point $x$ which means $X\cong G/P$ with $P=G_x$. It follows from $G_{\mathbb C}/P_{\mathbb C}\cong X_{\mathbb C}\cong {\rm Gr}_{n,k,\mathbb C}$ that $P_{\mathbb C}$ is a maximal parabolic which is defined over $\mathbb R$. Looking at the Satake diagram of $G$ it corresponds therefore to a non-compact self-conjugate simple root $\alpha_k$. This gives the following possibilities $$ G={\rm PGL}(n,\mathbb R), k=1,\ldots,n-1\quad\Rightarrow\quad\text{your first case} $$ $$ G={\rm PGL}(n/2,\mathbb H), k=2,4\ldots,n-2\quad\Rightarrow\quad\text{your second case} $$ $$ G={\rm PU}(p,n-p), k=p=n/2\quad\Rightarrow\quad\text{your third case} $$ This argument should generalize to arbitrary fields. The first two cases combine to $X=$ space of $k$-dimensional subspaces of $D^n$ where $D$ is a central simple algebra.

Answer 2

Let $X$ be a smooth projective variety over a field $K$ of characteristic zero such that $$ X_{\bar{K}} \cong \mathrm{Gr}(k,n)_{\bar{K}} $$ and $X(K) \ne \varnothing$.

First, consider the case $n \ne 2k$. Let $r = \gcd(k,n)$. Then there is an $r$-torsion Brauer class $\beta \in \mathrm{Br}(K)$ representable by a Severi--Brauer variety of dimension $k - 1$ and a Severi--Brauer variety $Y$ of dimension $n - 1$ with class $\beta$ such that $$ X \cong \mathrm{Hilb}_{\mathrm{P}^{k-1}}(Y) $$ is the component of the Hilbert scheme that parameterizes subschemes with the same Hilbert polynomial as $\mathrm{P}^{k-1}$. Let us call such $X$ a Severi--Brauer Grassmannian of class $\beta$.

Now consider the case $n = 2k$. Then either $X$ is a Severi--Brauer Grassmannian as before, or there is a quadratic extension $K'/K$ such that $X_{K'}$ is a Severi--Brauer Grassmannian.