Universal vector bundle over the reals

Abstract:

Let $X_{\mathbb R}$ be a geometrically irreducible smooth projective curve, defined over $\mathbb R$, such that $X_{\mathbb R}$ does not have any real points. Let $X = X_{\mathbb R}\times _{\mathbb R} \mathbb C$ be the complex curve. We show that there is a universal real algebraic line bundle over $X_{\mathbb R}\times \text {Pic}^d(X_{\mathbb R})$ if and only if the Euler characteristic $\chi (L)$ is odd for $L \in \text {Pic}^d(X_{\mathbb R})$. There is a universal quaternionic algebraic line bundle over $X\times \text {Pic}^d(X)$ if and only if the degree $d$ is odd. (Quaternionic algebraic vector bundles are defined only on a complexification.)

Take integers $r$ and $d$ such that $r \geq 2$, and $d$ is coprime to $r$. Let ${\mathcal M}_{X_{\mathbb R}}(r,d)$ (respectively, ${\mathcal M}_{X}(r,d)$) be the moduli space of stable vector bundles over $X_{\mathbb R}$ (respectively, $X$) of rank $r$ and degree $d$. We prove that there is a universal real algebraic vector bundle over $X_{\mathbb R}\times {\mathcal M}_{X_{\mathbb R}}(r,d)$ if and only if $\chi (E)$ is odd for $E \in {\mathcal M}_{X_{\mathbb R}}(r,d)$. There is a universal quaternionic vector bundle over $X\times {\mathcal M}_X(r,d)$ if and only if the degree $d$ is odd.

The cases where $X_{\mathbb R}$ is geometrically reducible or $X_{\mathbb R}$ has real points are also investigated.