Higher rank bundles on Hopf surfaces

Technical Summary: Higher Rank Bundles on Hopf Surfaces

Problem Statement
The paper investigates the geometry and stability properties of higher rank ($r \ge 2$) vector bundles and torsion-free sheaves on classical Hopf surfaces. A classical Hopf surface $X$ is a non-Kähler complex surface, diffeomorphic to $S^1 \times S^3$, which admits an elliptic fibration $\pi: X \to \mathbb{P}^1$. The authors address several interconnected problems:

1. The relationship between filtrability (the existence of a filtration by subbundles) and jumping behavior (instability of the bundle's restriction to fibers of the elliptic fibration).
2. The existence of stable and irreducible bundles with specific topological invariants, specifically trivial determinant and prescribed second Chern class ($c_2$).
3. The structure of the moduli spaces $M_{r,n}$ of stable rank $r$ bundles with $c_2=n$, particularly their boundaries and the nature of limits of families of bundles (torsion-free sheaves).
4. The introduction and characterization of very irreducible bundles, defined as bundles whose symmetric powers $S_n(E)$ remain irreducible for all $n > 0$.

Methodology
The authors employ a combination of algebraic geometry techniques adapted to the non-Kähler setting of Hopf surfaces:

• Stability and Metrics: Stability is defined via slope stability with respect to a Gauduchon metric (a $(1,1)$-form $\omega$ satisfying $\partial\bar{\partial}\omega = 0$). The degree of a bundle is defined via integration against this metric.
• Filtration Analysis: The authors utilize the concept of filtrable sheaves, defined by a filtration $0 = E_0 \subset E_1 \subset \dots \subset E_r = E$ where quotients are torsion-free. They analyze the Chern classes of such bundles, noting that on Hopf surfaces, the intersection form is trivial ($H^2(X, \mathbb{Z}) \times H^2(X, \mathbb{Z}) \to \mathbb{Z}$ is zero), which simplifies the calculation of $c_2$ for iterated extensions.
• Elementary Transformations: To study the boundary of moduli spaces, the authors perform elementary operations in codimension 2. Starting with a stable bundle $E \in M_{r,n}$ and a surjection $v: E \to \mathbb{C}_p$ (where $\mathbb{C}_p$ is a skyscraper sheaf at $p \in X$), they construct the kernel $E'$. This $E'$ is a torsion-free sheaf with $c_2(E') = c_2(E) + 1$.
• Deformation Theory: The authors analyze the local deformation spaces of these sheaves using Ext groups. They prove that for stable bundles, $\text{Ext}^2(E, E) = 0$, and extend this analysis to the constructed torsion-free sheaves to show they lie in the closure of the moduli space of bundles with higher $c_2$.
• Graph of Bundles: For rank 2 bundles, the authors utilize the "graph" of the bundle, a divisor in $\mathbb{P}^1 \times \mathbb{P}^1$ derived from the first derived image of the bundle twisted by the universal Poincaré line bundle. The geometry of this graph (specifically the presence of a component corresponding to a surjective morphism $\mathbb{P}^1 \to \mathbb{P}^1$) is used to deduce irreducibility properties.

Key Contributions and Results

1. Filtrability and Jumps:

   • Theorem 3.9: The authors prove that any filtrable vector bundle on a classical Hopf surface with $c_2 > 0$ must possess jumps (i.e., there exists a fiber where the restriction is not semistable).
   • Theorem 3.19: Conversely, they prove the existence of stable, filtrable rank $r$ vector bundles with trivial determinant and any prescribed $c_2 = c > 0$.

2. Bundles without Jumps:

   • Theorem 4.12: In a contrasting result, the authors prove the existence of irreducible (and thus stable) rank $r$ vector bundles with trivial determinant and $c_2 = 1$ that have no jumps (their restriction to every fiber is semistable). This highlights a dichotomy between filtrable and irreducible bundles regarding jumping behavior.

3. Boundary of Moduli Spaces:

   • Theorem 5.6: The paper establishes that the torsion-free sheaves obtained via elementary transformations (kernels of surjections $E \to \mathbb{C}_p$) from stable bundles $E \in M_{r,n}$ lie in the closure of the moduli space $M_{r,n+1}$.
   • Proposition 5.5: They show that for a simple bundle $E$, the kernels of surjections to skyscraper sheaves at a point $p$ form a family of dimension $2r-1$ of pairwise non-isomorphic torsion-free sheaves.
   • Interpretation: These results describe the "bubbling" phenomenon in the algebro-geometric context, where a sequence of stable bundles degenerates into a torsion-free sheaf with a point singularity.

4. Very Irreducible Bundles:

   • Definition 6.1: The authors introduce "very irreducible" bundles, where all symmetric powers $S_n(E)$ are irreducible.
   • Theorem 6.3: They demonstrate that any rank 2 bundle whose graph contains a component corresponding to a surjective morphism $\mathbb{P}^1 \to \mathbb{P}^1$ is very irreducible.
   • Theorem 6.4: This result is generalized to representations of the symmetric group, showing that $S_\rho(E)$ is irreducible (or zero) under similar conditions.

Significance and Claims
The paper claims to provide a comprehensive picture of the landscape of higher rank bundles on Hopf surfaces. Specifically:

• It clarifies the constraints on filtrable bundles, showing they are intrinsically linked to jumping phenomena when $c_2 > 0$.
• It demonstrates that the moduli spaces $M_{r,n}$ are rich enough to contain both filtrable bundles (with jumps) and very irreducible bundles (without jumps).
• It provides a concrete algebro-geometric description of the boundary of these moduli spaces, identifying specific torsion-free sheaves as limits of stable bundles. This connects the geometry of Hopf surfaces to the differential geometric concept of instanton bubbling.
• The introduction of "very irreducible" bundles offers a new class of rigid bundles with strong stability properties under symmetric powers, extending previous work on strong irreducibility.

The authors conclude that the moduli of stable higher rank bundles on Hopf surfaces are complex structures containing diverse types of bundles, ranging from those that are easily decomposable (filtrable) to those that are highly rigid and irreducible in all symmetric powers.