Notes on acceptable bundles II

1. Problem Statement and Context

The paper addresses the theory of acceptable bundles, a concept central to the Simpson–Mochizuki theory of harmonic bundles and non-abelian Hodge theory. While the foundational theory was established by Mochizuki (particularly in [M4]), the existing literature is often dense and relies on specific frameworks (e.g., Andreotti–Visentini) that can be difficult to access.

The primary goal of this paper is to:

1. Extend the theory of acceptable bundles from the one-dimensional case (punctured disks, $\Delta^*$) to higher-dimensional settings, specifically partially punctured polydisks $(\Delta^*)^l \times \Delta^{n-l}$.
2. Provide a more accessible, self-contained exposition of Mochizuki's theory, utilizing modern tools like the Ohsawa–Takegoshi $L^2$ extension theorem.
3. Prove that acceptable bundles admit a canonical prolongation to locally free sheaves on the compactification (the full polydisk $\Delta^n$) and possess a natural filtered bundle structure (parabolic structure).

2. Methodology

The authors employ a systematic reduction strategy, moving from curves to higher dimensions, while leveraging specific analytic estimates.

• Definitions and Metrics:

  • They define an acceptable bundle $(E, h)$ on a partially punctured polydisk as a holomorphic vector bundle with a smooth Hermitian metric $h$ such that its curvature $\sqrt{-1}\Theta_h(E)$ is bounded with respect to the Poincaré metric $\omega_P$ (which has logarithmic singularities at the punctures).
  • They utilize twisted metrics $h(a, N) = h \cdot e^{-\chi(a, N)}$ to adjust the growth rates of sections and ensure positivity properties (Nakano semipositivity/negativity).

• Analytic Tools:

  • $L^2$ Extension Theorem: Unlike Mochizuki's original approach, the authors treat the Ohsawa–Takegoshi $L^2$ extension theorem as a "black box" to extend sections from lower-dimensional subvarieties (curves) to the ambient space. This simplifies the technical machinery significantly.
  • Plurisubharmonic (psh) Functions: They extensively use properties of psh functions, including Lelong numbers and Siu's theorem, to analyze the growth of sections near the singular locus.
  • Reduction to Curves: A core methodological pillar is proving results for the case $l=1$ (punctured disk $\times$ polydisk) and then using cyclic coverings and descent arguments to generalize to $l \geq 2$.

• Algebraic Structures:

  • The paper rigorously defines prolongations $\mathbf{a}E$ (sheaves of sections with specific growth orders) and filtered bundles (families of these sheaves indexed by $\mathbb{R}^l$).
  • They establish the compatibility of parabolic filtrations along the boundary divisors.

3. Key Contributions and Results

A. Prolongation and Filtered Bundle Structure (The Main Theorem)

Theorem 1.1 & 1.3: The central result states that for an acceptable vector bundle $(E, h)$ on $(\Delta^*)^l \times \Delta^{n-l}$, the sheaf $\mathbf{a}E$ (defined by growth conditions of order $\mathbf{a} \in \mathbb{R}^l$) is a locally free sheaf on the full polydisk $\Delta^n$.

• Furthermore, the family $\{\mathbf{a}E\}_{\mathbf{a} \in \mathbb{R}^l}$ forms a filtered bundle in the sense of Mochizuki.
• Local Frame Description: Theorem 1.3 provides an explicit local description: there exists a local frame $\{v_j\}$ and weights $\mathbf{a}(v_j)$ such that $\mathbf{b}E$ is generated by sections of the form $z^{\lfloor \mathbf{b} - \mathbf{a}(v_j) \rfloor} v_j$. This confirms the existence of a parabolic structure.

B. Weak Norm Estimates

Theorem 1.4: The authors establish a weak norm estimate for the Hermitian matrix $H(h, v')$ associated with a frame $v'$ adapted to the parabolic filtration.
$$C^{-1} \left( -\sum \log |z_i| \right)^{-N} I_r \leq H(h, v') \leq C \left( -\sum \log |z_i| \right)^{N} I_r$$
This estimate is fundamental for controlling the behavior of the metric near the singularities and is crucial for proving the existence of the prolongation.

C. Stability under Operations

The paper proves that the category of acceptable bundles is closed under standard operations, extending known results from the curve case:

• Duals (Theorem 1.5): $(\mathbf{a}E)^\vee \cong -\mathbf{a} + \mathbf{1} - \varepsilon (E^\vee)$.
• Tensor Products (Theorem 1.6): $\mathbf{b}(E_1 \otimes E_2) = \sum_{\mathbf{a}_1 + \mathbf{a}_2 \leq \mathbf{b}} \mathbf{a}_1 E_1 \otimes \mathbf{a}_2 E_2$.
• Hom Bundles (Theorem 1.7): Characterization of $\mathbf{a}\text{Hom}(E_1, E_2)$ via filtration preservation.
• Parabolic Endomorphisms (Corollary 1.8): $\diamond \text{End}(E)$ is isomorphic to the sheaf of endomorphisms preserving the parabolic filtration.

D. Technical Innovations

• Cyclic Coverings and Descent: The authors refine the pull-back and descent arguments (Section 10) to handle the multi-variable case, showing how parabolic weights behave under cyclic coverings $\psi_c(z) = z^c$.
• Diophantine Approximation: In Lemma 13.4, they use Diophantine approximation to choose a cyclic covering degree $c$ that "flattens" the parabolic weights into a small interval, allowing the application of local freeness results.

4. Significance

• Accessibility: By replacing Mochizuki's original analytic framework with the Ohsawa–Takegoshi $L^2$ extension theorem, the authors make the theory of acceptable bundles more accessible to researchers in complex geometry and algebraic geometry who are familiar with $L^2$ methods but not necessarily with the specific machinery of Andreotti–Visentini.
• Higher-Dimensional Generalization: This work successfully bridges the gap between the well-understood 1-dimensional theory (Riemann surfaces) and higher-dimensional complex varieties. It provides the necessary technical foundation for studying harmonic bundles and non-abelian Hodge theory on higher-dimensional manifolds with normal crossing divisors.
• Foundation for Future Work: The explicit construction of the filtered bundle structure and the weak norm estimates serve as essential tools for further developments in the Simpson–Mochizuki correspondence in higher dimensions, potentially impacting the study of moduli spaces of Higgs bundles and representations of fundamental groups.

In summary, this paper is a rigorous, self-contained, and technically refined extension of the theory of acceptable bundles to higher dimensions, offering a clearer path to Mochizuki's deep results through modern $L^2$ techniques.