Notes on acceptable bundles I

1. Problem Statement and Context

The paper addresses the foundational theory of acceptable bundles, a concept central to the Simpson–Mochizuki theory of harmonic bundles and non-abelian Hodge theory. While Takuro Mochizuki provided a comprehensive treatment of acceptable bundles in higher dimensions (specifically in [M4]), his exposition on the most basic case—acceptable bundles on a punctured disk ($\Delta^*$)—was relatively brief.

The authors aim to fill this gap by providing a rigorous, self-contained, and detailed study of acceptable bundles on a punctured disk. Their primary objectives are:

• To establish the existence and properties of prolongations of acceptable bundles by increasing orders.
• To introduce a new invariant, $\gamma(aE)$, to characterize the structure of these prolongations.
• To provide proofs and arguments that differ from or clarify those of Simpson and Mochizuki, making the theory more accessible.
• To lay the groundwork for subsequent work on acceptable bundles over partially punctured polydisks (Paper [FFO]).

2. Methodology

The paper employs a blend of complex differential geometry, analysis on Riemann surfaces, and sheaf theory. Key methodological components include:

• Definition of Acceptability: A holomorphic vector bundle $(E, h)$ on $\Delta^*$ is "acceptable" if the norm of its curvature form $\Theta_h(E)$ is bounded with respect to the Poincaré metric $\omega_P$.
  $$|\Theta_h(E)|_{h, \omega_P} \leq C$$
• Prolongation by Increasing Orders: For a real number $a$, the sheaf $^aE$ is defined as the subsheaf of sections $f$ of $E$ (on $\Delta^*$) satisfying a growth condition:
  $$|f|_h = O\left(\frac{1}{|z|^{a+\varepsilon}}\right) \quad \forall \varepsilon > 0$$
  The core challenge is proving that $^aE$ extends to a holomorphic vector bundle on the full disk $\Delta$.
• Analytic Techniques:
  • $L^2$-methods and $\bar{\partial}$-equations: Used to construct holomorphic sections with specific growth rates (Lemma 5.2).
  • Subharmonic Functions: The authors extensively utilize the properties of subharmonic functions (e.g., mean value inequalities, Lelong numbers) to control the growth of metrics and sections.
  • Simpson's Key Lemma: A crucial technical tool (Lemma 10.1) is adapted to derive estimates on the existence of sections with controlled norms.
  • Cyclic Covers: Used to reduce problems involving irrational weights to rational ones via pullback arguments (Section 11).
  • Diophantine Approximation: Employed in the proof of the main identity relating the invariant $\gamma$ to parabolic weights (Section 12).

3. Key Contributions and New Invariants

A. The Invariant $\gamma(aE)$

The authors introduce a new real-valued invariant $\gamma(aE)$ for an acceptable bundle $(E, h)$ and a real number $a$. It is defined via the asymptotic behavior of the determinant of the metric matrix $H(h, v)$ for a local frame $v$:
$$\gamma(aE) := -\frac{1}{2} \lim_{z \to 0} \frac{\log \det H(h, v)}{\log |z|}$$
The paper proves that this limit exists and is independent of the choice of frame.

B. Explicit Description of Prolongations

For line bundles, the authors provide an explicit description of the prolonged bundle $^aL$. They show that $^aL$ is generated by a section of the form $z^{-\lfloor a - \gamma \rfloor}$, where $\gamma$ is a limit related to the metric's growth. This explicit formulation is noted as a new contribution.

C. Fundamental Identity

The most significant technical result is the proof of the identity linking the invariant $\gamma(aE)$ to the parabolic weights of the bundle. If $\{b_1, \dots, b_r\}$ are the parabolic weights (elements of the set $\text{Par}_a(E, h)$), then:
$$\gamma(aE) = \sum_{i=1}^r b_i$$
This equality is established through a sophisticated argument involving cyclic covers and Simpson's key lemma, resolving the inequality $\gamma(aE) \leq \sum b_i$ (which is easier) and proving the reverse inequality.

4. Main Results

1. Coherence of Prolongations (Theorem 1.3):
   For any acceptable vector bundle $(E, h)$ on $\Delta^*$ and any $a \in \mathbb{R}$, the sheaf $^aE$ is a holomorphic vector bundle on $\Delta$. This confirms the coherence of the prolongation.

2. Structure of Line Bundles (Theorem 4.4):
   The paper gives a precise characterization of $^aL$ for line bundles, showing that sections belong to the stalk if and only if their order of pole is bounded by the invariant $\gamma$.

3. Determinant Bundles (Theorem 1.10 / 7.5):
   The determinant bundle $\det(^aE)$ is isomorphic to $^{\gamma(aE)}(\det E)$. This reduces the study of vector bundles to line bundles in many contexts.

4. Duality (Theorem 1.12 / 13.2):
   The authors establish the relationship between the prolongation of a bundle and its dual:
   $$(^aE)^\vee = {}^{-a+1-\varepsilon}(E^\vee)$$
   for sufficiently small $\varepsilon > 0$. They also describe how the parabolic weights transform under duality (mapping $b_i$ to $-b_i$).

5. Tensor Products and Hom Bundles (Theorem 1.14 / 16.2, 17.1):
   The paper proves that the prolongation of a tensor product (or Hom bundle) coincides with the tensor product (or Hom) of the prolonged bundles:
   $${}^a(E_1 \otimes E_2) = \sum_{a_1+a_2 \leq a} {}^{a_1}E_1 \otimes {}^{a_2}E_2$$
   This confirms that the category of acceptable bundles is closed under these operations and that the parabolic filtration behaves naturally.

6. Weak Norm Estimates (Theorem 1.13):
   The authors derive bounds on the metric matrix $H(h, v')$ (where $v'$ is a frame normalized by the parabolic weights), showing it grows at most polynomially in $(-\log |z|)$.

5. Significance

• Foundational Clarity: By focusing exclusively on the punctured disk, the authors provide a "clean" version of the theory, stripping away the complexities of higher dimensions to reveal the core analytic mechanisms. This serves as an essential pedagogical and reference tool for researchers entering the field.
• New Tools: The introduction of the invariant $\gamma(aE)$ and the explicit proofs of the identity $\gamma(aE) = \sum b_i$ offer new perspectives and potentially simpler routes to results that previously required more opaque arguments.
• Self-Contained Nature: The paper is designed to be self-contained, proving necessary lemmas on subharmonic functions, Lelong numbers, and cyclic covers, making it a standalone resource.
• Bridge to Higher Dimensions: The results on the punctured disk are explicitly stated as the building blocks for the subsequent paper [FFO], which tackles the more complex case of partially punctured polydisks.

In summary, this paper rigorously formalizes the local theory of acceptable bundles, introduces a powerful new invariant to classify their structure, and provides detailed proofs for the fundamental operations (duality, tensor products) that define the category of these bundles.