On the meagerness of the set of irregular bundles on Hopf surfaces

1. Problem Statement and Motivation

The paper addresses the structure of the moduli space of stable vector bundles on Hopf surfaces, which are non-Kähler complex surfaces diffeomorphic to $S^3 \times S^1$. Specifically, the authors investigate the properties of irregular bundles within these moduli spaces.

• Context: The research is motivated by open questions regarding the moduli spaces of instantons on $S^3 \times S^1$, as described by Edward Witten. A specific goal is to resolve the question of the connectedness of the moduli space of $SU(2)$ instantons.
• The Core Question: In algebraic geometry (projective surfaces), stable bundles often exhibit "good" behavior (e.g., being filtrable). However, on non-Kähler surfaces like Hopf surfaces, the behavior is less understood. The authors aim to determine whether "irregular" bundles (those with jumps or specific structural degeneracies) form a significant portion of the moduli space or if they are negligible.
• Definitions:
  • Hopf Surface ($X$): A classical Hopf surface elliptically fibered over $\mathbb{P}^1$ with fibers isomorphic to an elliptic curve $T$.
  • Moduli Space ($M_{r,c}$): The space of stable rank $r$ vector bundles with second Chern class $c > 0$ and trivial determinant.
  • Jump: A bundle $E$ has a jump if its restriction to a fiber $T_x$ is not semistable.
  • Irregular Bundle: A bundle $E$ is irregular if its restriction to some fiber $T_x$ is an irregular bundle on the elliptic curve (defined as having indecomposable factors that are not pairwise distinct in $\text{Pic}(T)$).
  • Meager Set: A set is meager if it is a countable union of nowhere dense subsets. In topology, a meager set is "small" (its complement is dense).

2. Methodology and Technical Approach

The authors employ a combination of deformation theory, cohomological analysis, and the classification of vector bundles on elliptic curves.

A. Contrast with the Algebraic Case

The paper begins by highlighting why standard algebraic arguments fail in this non-Kähler setting:

1. Filtrability: On projective surfaces, bundles are often filtrable (extensions of line bundles). On Hopf surfaces, filtrable bundles must have jumps. Consequently, the dense open set of stable bundles without jumps consists entirely of non-filtrable bundles.
2. Unstable Components: In general type varieties, moduli spaces can have large components consisting solely of unstable bundles. The authors investigate whether similar "large" components of irregular bundles exist on Hopf surfaces and prove they do not.

B. Cohomological Analysis (Lemma 3)

The core technical engine of the proof is Lemma 3, which analyzes the endomorphism space of vector bundles on an elliptic curve $T$.

• Premise: Let $F$ be a rank $r \ge 2$ semistable bundle on $T$ with $\det(F) \cong \mathcal{O}_T$.
• Result:
  • If $F$ is regular (indecomposable factors are distinct), $\dim H^0(T, \text{End}(F)) = r$.
  • If $F$ is irregular (repeated factors in the decomposition), $\dim H^0(T, \text{End}(F)) \ge r + 2$.
• Significance: This jump in the dimension of the endomorphism space (from $r$ to at least $r+2$) implies that irregular bundles have a higher degree of symmetry (automorphisms), which geometrically corresponds to a "thinner" locus in the moduli space.

C. Deformation Theory and Surjectivity (Theorem 4)

The authors use the exact sequence of sheaves to relate the deformation space of the bundle on the surface to the deformation space on the fiber:
$$0 \to \text{End}(E)(-T_x) \to \text{End}(E) \to \text{End}(F) \to 0$$

• Surjectivity: By analyzing the cohomology groups (specifically $H^1$), they show that the restriction map $\rho: H^1(X, \text{End}(E)) \to H^1(T_x, \text{End}(F))$ is surjective.
• Codimension Argument: Since the space of irregular bundles on the fiber $T_x$ has a higher codimension (due to the increased dimension of $H^0(\text{End}(F))$), and the map $\rho$ is surjective, the locus of bundles $E$ in the moduli space $M_{r,c}$ that restrict to irregular bundles on $T_x$ must have codimension at least 2 in the moduli space.
• Meagerness: Since the base of the fibration is $\mathbb{P}^1$ (dimension 1), the union of these loci over all $x$ remains meager (a countable union of codimension $\ge 2$ sets).

3. Key Contributions and Results

Main Theorem

Theorem 4: Let $U$ be a connected component of the moduli space $M_{r,c}$. The set of all bundles $E \in U$ that are not regular (i.e., those with jumps or those that are irregular on fibers) is meager in $U$.

• Implication: The set of regular bundles is dense in every open subset of the moduli space. Regular bundles are the "generic" case.

Corollary: Connectedness of Moduli Spaces

Corollary 5: For rank $r=2$, the complex manifold $M_{2,c}$ is connected.

• Proof Logic:
  1. By the Main Theorem, it suffices to prove the connectedness of the set of regular bundles.
  2. Regular bundles correspond to spectral curves $C_E$ which are smooth, connected curves of genus $2n-1$.
  3. The set of graphs associated with these bundles forms an irreducible Zariski open subset.
  4. The fibers of the map from bundles to spectral curves are Jacobians of these smooth curves, which are connected.
  5. Therefore, the entire moduli space is connected.

4. Significance and Impact

1. Resolution of Witten's Question: The paper provides an affirmative answer to the question posed by Edward Witten regarding the connectedness of the moduli space of $SU(2)$ instantons on $S^3 \times S^1$.
2. Non-Kähler Geometry: It establishes that despite the lack of Kähler metrics and the failure of standard algebraic properties (like filtrability), the moduli spaces of stable bundles on Hopf surfaces still possess a "generic" structure dominated by regular bundles.
3. Technical Novelty: The paper demonstrates that "bad" behavior (irregularity) in non-Kähler settings does not necessarily lead to large, disconnected components of the moduli space, contrary to what might be expected from analogies with general type varieties.
4. Foundational Definitions: It rigorously defines and distinguishes between "jumps" and "irregularity" in the context of Hopf surfaces, clarifying the geometric landscape for future research in non-Kähler gauge theory.

In summary, the paper proves that irregular bundles are topologically negligible (meager) within the moduli spaces of Hopf surfaces, thereby establishing the density of regular bundles and confirming the connectedness of the rank 2 moduli space.