Higher rank bundles on Hopf surfaces

We show that all filtrable bundles on a Hopf surface $X$ must have jumps and we prove the existence of filtrable stable bundles on $X$ with any value of $c_2>0$. On a somewhat opposite direction, for each integer $r\ge 2$ we prove the existence of irreducible rank $r$ vector bundles on $X$ with trivial determinant, $c_2=1$, and no jumps. We then apply elementary operations in codimension $2$ to points of the moduli space $\mathcal M_{r,n}$ of rank $r$ stable vector bundles on $X$ with $c_2=n$ to obtain torsion free sheaves with $c_2=n+1$. Namely, starting with a surjection $v\colon E \rightarrow \mathbb C_p$ from a vector bundle $E \in \mathcal M_{r,n}$ to a skyscraper sheaf supported at a point $p\in X$, we prove that if $E'$ is any torsion free sheaf fitting into a short exact sequence of the form $0 \longrightarrow E'\longrightarrow E\stackrel{v}{\longrightarrow}\mathbb C_p \longrightarrow 0,$ then $E'$ is in the closure of $\mathcal M_{r,n+1}$. We discuss various properties of vector bundles and torsion free sheaves and introduce the concept of very irreducible bundles to describe bundles whose symmetric powers $S^n(E)$ are irreducible for all $n> 0$. We then show that any rank $2$ bundle on $X$ whose graph contains a component corresponding to a surjective morphism $\mathbb P^1\to \mathbb P^1$ is very irreducible.