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Addendum: On generalized canonical bundle formula and boundedness of complements in complex analytic setting

This paper establishes the generalized canonical bundle formula for generalized lc-trivial fibrations in the complex analytic setting without requiring assumptions on the nef part, while also recording the corresponding algebraic result.

Original authors: Kenta Hashizume

Published 2026-07-01

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CC BY 4.0

Original authors: Kenta Hashizume

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand a massive, complex building (let's call it Building X). This building has a unique property: it is built over a foundation that stretches out into a landscape (let's call this Landscape Z).

In the world of mathematics, specifically a field called "Complex Geometry," mathematicians try to describe the "shape" and "structure" of these buildings using special blueprints called Canonical Bundle Formulas. These formulas act like a translation guide, telling you how the complex details of the big building (X) relate to the simpler, underlying landscape (Z).

For a long time, there was a strict rule for using this translation guide. The rule said: "You can only use this formula if the building's structure is built from a specific, finite set of pre-fabricated, sturdy beams." (In math terms, this was the assumption that the "nef part" $M$ had to be a finite combination of specific divisors).

The Problem:
This rule was too limiting. It meant that if a building was made of a slightly different, more fluid mix of materials, the translation guide didn't work. You couldn't predict the shape of the landscape just by looking at the building.

The Breakthrough (This Paper):
Kenta Hashizume, the author of this paper, says: "We don't need those specific pre-fabricated beams anymore."

He has proven that the translation guide (the Generalized Canonical Bundle Formula) works even if the building is made of a more flexible, infinite variety of materials. He removed the strict "finite beam" requirement.

Here is how he did it, broken down into simple concepts:

1. The "Stein Compact" Safety Net

Imagine you are looking at a specific, cozy neighborhood within the Landscape (called a Stein compact subset). The paper says that if you zoom in on this specific neighborhood, you can always rearrange the building slightly (like shrinking the map) so that the rules hold true. It's like saying, "If you look at this specific corner of the city, the blueprint works perfectly, no matter how weird the materials are."

2. The "Shadow" and the "Reflection"

The paper deals with two main things that come out of the formula:

The Discriminant (The Shadow): This describes the "rough spots" or singularities on the landscape where the building gets weird.
The Moduli Part (The Reflection): This describes the "smooth" or flexible parts of the landscape.

Hashizume proves that even without the strict "finite beam" rule, the Reflection (the Moduli part) is still well-behaved. It remains "nef" (a mathematical way of saying it's stable and points in a good direction). This means the blueprint for the landscape is still valid and predictable.

3. Two Worlds, One Solution

The paper provides this new rule for two different "universes":

The Algebraic World: This is like a world made of perfect, rigid Lego bricks (algebraic geometry).
The Complex Analytic World: This is like a world made of flowing water or smooth clay (complex analytic geometry).

Hashizume shows that his new, more flexible rule works in both worlds. He first proves it for the rigid Lego world, and then uses a clever trick to show it works for the flowing clay world too.

The "Why It Matters" (Without the Jargon)

Before this paper, mathematicians had to check if a building was made of "standard beams" before they could use the blueprint. If it wasn't, they were stuck.

Now, Hashizume has shown that the blueprint works for almost any building, as long as the building isn't completely broken (it must be "effective" in certain areas). This removes a huge barrier, allowing mathematicians to study a much wider variety of complex shapes and structures without getting stuck on technical restrictions.

A Note on the "Supplement"

At the end of the paper, the author adds a small appendix. Think of this as a "How-To" guide for drawing the blueprints in the first place. He explains exactly how to define the "Canonical Divisor" (the core measurement of the building's shape) in a way that is consistent, even if you look at the building from different angles or through different lenses. He proves that no matter how you zoom in or out, the core measurement remains consistent and well-defined.

In Summary:
Kenta Hashizume removed a major "speed bump" in the road of complex geometry. He proved that the fundamental formulas used to understand the relationship between complex shapes and their underlying landscapes work much more broadly than previously thought, without needing strict, artificial constraints on the materials used to build them.

Technical Summary: Addendum on Generalized Canonical Bundle Formula and Boundedness of Complements in Complex Analytic Setting

Problem Statement
The paper addresses a gap in the theory of generalized pairs within the complex analytic setting. Specifically, it targets the generalized canonical bundle formula for generalized lc-trivial fibrations. In the author's previous work ([Has26]), the validity of this formula relied on a restrictive assumption: the nef part $M$ of the generalized pair $(X, B+M)$ had to be a finite $\mathbb{R}_{>0}$ -linear combination of $b$ -nef/ $S$ $\mathbb{Q}$ - $b$ -Cartier $\mathbb{Q}$ - $b$ -divisors. The current work aims to remove this assumption, establishing the formula for general generalized sub-pairs where $M$ is not necessarily of this specific linear combination form. This is crucial for extending the boundedness of complements and the structural theory of generalized pairs to the broader complex analytic context without artificial constraints on the nef part.

Methodology
The author employs a two-pronged approach, treating the algebraic and complex analytic settings sequentially:

Algebraic Setting (Theorem 1.3):
- The proof leverages Proposition 2.1, which establishes that for standard lc-trivial fibrations (where the nef part is zero), the moduli $R$ - $b$ -divisor is a finite $\mathbb{R}_{>0}$ -linear combination of specific divisors. This result is cited as a special case of [FuHa23] or a minor variation of [Has26].
- Building on this, the author applies arguments from [Fi20]. Crucially, the methodology utilizes weak semistable reductions ([Kar99]), generalized dlt blow-ups, and the Minimal Model Program (MMP) for $\mathbb{Q}$ -factorial generalized dlt pairs ([BiZh16]).
- The key technical insight is that these tools remain effective even when the nef parts of the generalized pairs are not represented by finite linear combinations of $b$ -nef/ $S$ $\mathbb{Q}$ - $b$ -Cartier $\mathbb{Q}$ - $b$ -divisors.
Complex Analytic Setting (Theorem 1.2):
- The proof relies on Lemma 2.16 from [Fu22] to construct a Stein compact subset $W' \subset S$ containing the original $W$ , satisfying property (P) regarding the morphism $\pi$ . This allows the reduction of the problem to a local neighborhood where standard analytic techniques apply.
- The argument mirrors the algebraic case but integrates specific analytic tools: the case of $M=0$ from Theorem 1.1 (the author's prior work), results from EH25, and Fu22.
- Remark 2.2 clarifies that the proof necessitates the full machinery of [Has26, Section 3]. The author notes that variations of mixed Hodge structures are not used; instead, the proof proceeds by first establishing the formula for generalized klt-trivial fibrations and then extending it to the lc case.
Supplementary Construction (Theorem 2.3):
- To ensure the rigorous definition of the canonical $b$ -divisor in the analytic setting, the author provides a construction for the global definition of the canonical $b$ -divisor $K$ on a normal analytic variety $X$ over a Stein compact set $W$ .
- This involves shrinking $S$ to ensure projectivity of a resolution $Y \to X$ and the existence of a very ample divisor. Using cohomology and base change theorems, along with Cartan's Theorem A, the author constructs global sections to define Weil divisors that are compatible under bimeromorphic contractions, thereby forming a well-defined $b$ -divisor.

Key Contributions and Results
The paper presents three main theorems:

Theorem 1.2 (Main Result, Complex Analytic Setting):
- Removes the assumption on the nef part $M$ for generalized lc-trivial fibrations over a Stein compact set $W$ .
- Establishes that the moduli $R$ - $b$ -divisor $N$ associated with the fibration is a $b$ -nef/ $S$ $R$ - $b$ -Cartier $R$ - $b$ -divisor.
- Confirms that $(Z, G_Z + N_Z) \to S$ forms a generalized sub-pair with nef part $N$ , preserving the generalized lc (or klt) property of the source.
- Proves a locality property: for any open subset $\tilde{S} \subset S$ , the moduli divisor $\tilde{N}$ is the restriction of $N$ .
- Adds a conditional statement: If the original $M$ is a finite $\mathbb{R}_{>0}$ -linear combination of $b$ -nef/ $S$ $\mathbb{Q}$ - $b$ -Cartier $\mathbb{Q}$ - $b$ -divisors, then $N$ inherits this property.
Theorem 1.3 (Main Result, Algebraic Setting):
- Provides the corresponding algebraic statement over a quasi-projective scheme $S$ .
- Asserts that $N$ is a $b$ -nef/ $S$ $R$ - $b$ -Cartier $R$ - $b$ -divisor and that the resulting pair on $Z$ maintains the generalized lc/klt properties.
- The author explicitly states this result is new and serves as a generalization of [FiSv23, Theorem 2.20].
Theorem 2.3:
- Rigorously defines the canonical $b$ -divisor and moduli $R$ - $b$ -divisors in the context of generalized lc-trivial fibrations for normal analytic varieties, ensuring their existence and compatibility under bimeromorphic morphisms after shrinking the base.

Significance and Claims
The paper claims to be a necessary refinement of the generalized canonical bundle formula theory. By removing the restrictive assumption on the nef part $M$ , the results broaden the applicability of the formula in the complex analytic setting, which is essential for the study of boundedness of complements in this context.

The author emphasizes that the proof is not merely a trivial extension but relies on the interplay of several deep results (including [Has26], [Fi20], [Kar99], and [BiZh16]). The work is presented as a foundational step ("addendum") that solidifies the theoretical framework for generalized pairs without artificial constraints on the nef part, thereby supporting further developments in the Minimal Model Program for complex analytic spaces. The paper does not propose new experimental applications but rather seeks to correct and generalize existing theoretical statements.