Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected

Question

Let $G$ be an affine algebraic group over $\mathbb{C}$. It is well known that when working with principal $G$ bundles it is too restrictive to require bundles to be locally trivial in the Zariski topology. Instead $G$-bundles are defined to be locally trivial in the etale topology. The finite map $z \to z^n$ from $\mathbb{C}^\times$ to itself is a $\mu_n$ (= $n$th roots of unity) bundle that is not Zariski locally trivial. Also setting $B = Spec \mathbb{C}[s^\pm,t^\pm]$ then $\{x^2 + s y^2 + t z^2 = 0\} \subset \mathbb{P}^2_B$ is a $\mathbb{P}^1$-bundle that gives rise to a $PGL_2$-bundle that is not Zariski locally trivial.

However for some groups (dubbed special groups) being locally trivial in the etale topology implies locally trivial in the Zariski topology. If $G$ is semisimple then Grothendieck proved the only such special $G$ are products of $SL_n$ and $Sp_{2m}$.

QUESTION

I would like an example of a non Zariski locally trivial principal $G$-bundle for a simple, simply connected group $G$, e.g. $Spin(n)$ or $G_2$.

My motivation is largely out curiosity. For $G$ as in the question, I study the moduli space of $G$-bundles on a curve (and although for a curve over $\mathbb{C}$ a bundle will be Zariski locally trivial it wont be in families) and I would like to have a non Zariski locally trivial bundle in my back pocket.

A vague idea would be to mimic the $PGL_2$ example above by looking at some varying family of $G/P$. You could try this with $G = E_8$ because it is simply connected and adjoint and if you're lucky you might have $Aut(G/P) = G$. I have no idea if this works but even if it does it would be nice if there was a simpler example.

Answer 1

I think a construction like the one for ${\rm PGL}_2$ in the question should work. The bundle in the question corresponds to the quaternion algebra over $\mathbb{C}[s^{\pm},t^{\pm}]$ whose norm form is $T^2-sX^2-tY^2+stZ^2$, i.e., the quaternion algebra where $i$ and $j$ are non-commuting square roots of $s$ and $t$, respectively.

Now we do the same thing for octonion algebras. The underlying ring is going to be $A=\mathbb{C}[s^{\pm},t^{\pm},r^{\pm}]$. Apply the Cayley--Dickson doubling construction to the above quaternion algebra, with parameter $r$. (This works not just over fields, definitions or locally ringed spaces can be found in papers of H.P. Petersson.) This produces an octonion algebra over $A$ whose norm form is the 3-fold Pfister form $$ \langle 1,-s\rangle\otimes\langle 1,-t\rangle\otimes\langle 1,-r\rangle= $$ $$=(X^2-sY^2-tZ^2+stT^2)-r(U^2-sV^2-tW^2+stS^2). $$ This is a non-trivial 3-fold Pfister form over the function field $\mathbb{C}(s,t,r)$ and so gives a non-trivial decomposable element in Galois cohomology ${\rm H}^3(\mathbb{C}(s,t,r),\mathbb{Z}/2)$. (You can think of this topologically, it's the top cohomology of the 3-torus ${\rm Spec}A$.) In particular, the corresponding octonion algebra is going to be nonsplit over the function field. Then we can take the corresponding $G_2$-torsor (of local automorphisms of the octonion algebra); by the above this torsor cannot be Zariski-locally trivial.

An example of a torsor for ${\rm Spin}(7)$ or ${\rm Spin}(8)$ can be obtained by the natural change-of-structure-group, alternatively, these can be explicitly described in terms of the 3-fold Pfister form above. I guess the 3-fold Pfister form gives a family of smooth affine quadrics whose generic fiber is anisotropic, another example of a torsor which is not Zariski-locally trivial.

Edit: If I interpret the description of the projective bundle in the question correctly, the $\mathbb{P}^1$-bundle is exactly the variety cut out by the vanishing of the norm on the projectivization of the quaternions of trace zero. The same can be done for the octonion algebra above: the equation $(sY^2-tZ^2+stT^2)-r(U^2-sV^2-tW^2+stS^2)=0$ should define a hypersurface in $\mathbb{P}^6_A$, defined by vanishing of the norm on the projectivization of the octonions of trace zero. If I'm not messing up something, this should be a homogeneous space bundle with fibers $G_2/P_1$ which is not locally trivial in the Zariski topology. I think this is the appropriate $G_2$-analogue of the $\mathbb{P}^1$-bundle in the question.

Answer 2

This may not be what you are looking for, but the first example would be over the spectrum of a commutative field $K$. Then a principal $G$-bundle is a $G$-torsor over $K$; any nontrivial such torsor will answer the question. For instance, a $G_2$-torsor is given by a $K$-form of the standard octonion algebra; there is a classical such form over $\mathbb{R}$, called the split-octonion algebra, see here.