Universal vector bundle over the reals

TL;DR

This paper characterizes when universal real or quaternionic vector bundles exist over certain algebraic curves and their moduli spaces, based on parity conditions of degrees and Euler characteristics.

Contribution

It provides necessary and sufficient conditions for the existence of universal bundles over real algebraic curves and their moduli spaces, extending previous results to new cases.

Findings

Universal real line bundle exists iff hi(L) is odd.

Universal quaternionic line bundle exists iff degree d is odd.

Universal real vector bundle exists iff hi(E) is odd.

Abstract

Let X_R be a geometrically irreducible smooth projective curve, defined over R, such that X_R does not have any real points. Let X= X_R\times_R C be the complex curve. We show that there is a universal real algebraic line bundle over X_R x Pic^d(X_R)$ if and only if $\chi(L)$ is odd for L in Pic^d(X_R)$. There is a universal quaternionic algebraic line bundle over X x Pic^d(X) if and only if the degree d is odd. Take integers r and d such that r > 1, and d is coprime to r. Let M_{X_R}(r,d) (respectively, M_X(r,d)$) be the moduli space of stable vector bundles over X_R (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over X_R x M_{X_R}(r,d) if and only if \chi(E) is odd for E in M_{X_R}(r,d). There is a universal quaternionic vector bundle over X x M_X(r,d) if and only if the degree d is odd. The cases where X_R is geometrically reducible or X_R has real points are also investigated.