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.

💡 Research Summary

The paper investigates holomorphic vector bundles of rank $r\ge2$ on a classical Hopf surface $X$, which is the quotient of $\mathbb C^2\setminus{0}$ by a contraction $\mu$ and carries an elliptic fibration $\pi\colon X\to\mathbb P^1$ with fibre $T\simeq\mathbb C^*/\mu$. Because $H^2(X,\mathbb Z)=0$, every line bundle has trivial first Chern class and the only non‑trivial topological invariant of a bundle $E$ is its second Chern class $c_2(E)$. The authors work with a fixed Gauduchon metric, define degree and slope in the usual way, and adopt the standard notion of (Mumford‑Takemoto) stability.

A central notion is that of a “jump”. For a fibre $J$ of $\pi$, the restriction $E|J$ is a degree‑zero vector bundle on an elliptic curve; it is called a jump if $E|J$ is not semistable. This generalises the classical notion of jumping fibres for rank‑2 bundles. The authors also associate to each bundle $E$ a divisor $G(E)\subset\mathbb P^1\times\mathbb P^1$, called the graph of $E$, obtained as the support of $R^1\pi*(E\otimes V)$ where $V$ is the universal Poincaré line bundle. When $c_2(E)=n$, $G(E)$ lies in the linear system $|\mathcal O{\mathbb P^1\times\mathbb P^1}(n,1)|$. The presence of a non‑constant rational map component in $G(E)$ guarantees stability, and in particular any bundle with $c_2>0$ and no jumps is automatically stable.

Theorem 3.9 proves that any filtrable bundle $E$ with $c_2(E)>0$ must have a jump. The proof proceeds by taking a filtration $0=E_0\subset E_1\subset\cdots\subset E_r=E$, saturating it, and observing that the quotient $E/E_{r-1}$ is a rank‑1 torsion‑free sheaf of the form $I_Z\otimes L$, where $Z$ is a zero‑dimensional subscheme. If $Z\neq\emptyset$, one can choose a fibre $J$ passing through a point of $Z$; the restriction $E|_J$ then acquires a subsheaf of positive degree, violating semistability. Hence $Z$ must be empty for a jump‑free filtrable bundle, which forces $c_2(E)=0$, contradicting the hypothesis.

Theorem 3.19 shows that for any integers $r\ge2$ and $c>0$ there exist filtrable, stable bundles $E$ with $\det E\simeq\mathcal O_X$ and $c_2(E)=c$. The construction uses an extension \