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arxiv: 2606.30983 · v1 · pith:ZF2NQV4Qnew · submitted 2026-06-29 · 🧮 math.AG

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

Kenta Hashizume This is my paper

Pith reviewed 2026-07-01 00:50 UTC · model grok-4.3

classification 🧮 math.AG
keywords generalized canonical bundle formulalc-trivial fibrationscomplex analytic geometrygeneralized pairsbirational geometryalgebraic geometry
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The pith

The generalized canonical bundle formula holds for generalized lc-trivial fibrations without the nef part assumption in the complex analytic setting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the generalized canonical bundle formula applies to generalized lc-trivial fibrations in the complex analytic category even when the nef part is omitted. This removes a prior restriction that had limited the formula's scope. A parallel algebraic statement is recorded as well. A reader would care because the result widens the range of fibrations to which the formula can be applied without extra conditions on the base or fibers.

Core claim

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations without the assumption on the nef part in the complex analytic setting. We also record the corresponding algebraic statement.

What carries the argument

generalized canonical bundle formula applied to generalized lc-trivial fibrations

Load-bearing premise

The fibrations in question meet the definition of generalized lc-trivial fibrations in the complex analytic category.

What would settle it

A concrete generalized lc-trivial fibration in the complex analytic setting where the generalized canonical bundle formula fails in the absence of the nef part would disprove the result.

read the original abstract

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations without the assumption on the nef part in the complex analytic setting. We also record the corresponding algebraic statement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is an addendum establishing the generalized canonical bundle formula for generalized lc-trivial fibrations in the complex analytic setting after removing the nef-part hypothesis from prior results; it also records the corresponding algebraic statement.

Significance. If the derivation holds, the result extends the canonical bundle formula to a wider class of generalized lc-trivial fibrations without the nef restriction, strengthening tools for boundedness of complements in both analytic and algebraic categories. The explicit removal of the nef assumption is a concrete technical advance over the referenced prior work.

minor comments (2)
  1. The abstract and title refer to 'prior results on lc-trivial fibrations'; the introduction should explicitly cite the specific theorems (with equation or section numbers) from the base paper that are being extended.
  2. Clarify whether the analytic-category definitions of generalized lc-trivial fibrations introduce any new local analytic conditions not present in the algebraic case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were raised in the report, so we have no points to address point-by-point. The manuscript stands as submitted.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is presented as an addendum extending the generalized canonical bundle formula to generalized lc-trivial fibrations in the complex analytic setting by removing the nef-part hypothesis. The abstract and title indicate this is an incremental removal of an assumption from prior results rather than a derivation that reduces by construction to self-defined inputs, fitted parameters, or a load-bearing self-citation chain. No equations or explicit reductions are visible in the provided text that would match the enumerated circularity patterns, and the central claim retains independent content as an extension in the analytic category.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard domain assumptions from birational geometry rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Generalized lc-trivial fibrations are well-defined in the complex analytic setting
    Invoked as the object to which the formula applies.
  • ad hoc to paper Prior results on canonical bundle formulas hold when the nef assumption is dropped
    This is the load-bearing new assertion of the addendum.

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discussion (0)

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Reference graph

Works this paper leans on

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