arxiv: 2605.11156 · v1 · submitted 2026-05-11 · 🧮 math.AG

Recognition: no theorem link

Functoriality of logarithmic Hochschild homology of log smooth pairs

\'Ad\'am Gyenge, Leo Herr, M\'arton Hablicsek

Pith reviewed 2026-05-13 02:18 UTC · model grok-4.3

classification 🧮 math.AG

keywords logarithmic Hochschild homologylog smooth pairsFourier-Mukai transformsfunctorialitylogarithmic correspondencesblow-up compactificationsdg bicategorylogarithmic Chern characters

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The pith

Logarithmic Hochschild homology is functorial under strong log Fourier-Mukai transforms for smooth proper log pairs.

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

The paper shows that logarithmic Hochschild homology behaves functorially for smooth proper log pairs when one uses special kernels called strong log Fourier-Mukai kernels. These kernels live on canonical blow-up compactifications of the product rather than on the ordinary product. The construction supplies explicit unit and counit maps that restore the necessary adjunctions even though standard adjunction theory breaks down due to blow-up discrepancies. This yields a dg bicategory of logarithmic correspondences in which the homology and cohomology become invariants, and it produces compatible logarithmic Chern characters and Euler pairings.

Core claim

For smooth proper log pairs the authors construct strong log Fourier-Mukai kernels supported on canonical blow-up compactifications and prove that the induced transforms make logarithmic Hochschild homology functorial. They achieve this by building explicit unit- and counit-type morphisms that supply the required adjunction data directly, without relying on an ambient dg category of logarithmic sheaves.

What carries the argument

strong log Fourier-Mukai kernels supported on canonical blow-up compactifications, which carry the induced transforms and supply explicit adjunction morphisms

If this is right

Logarithmic Hochschild homology and cohomology become categorical invariants of the dg bicategory of logarithmic correspondences.
Logarithmic Chern characters can be defined so that they are compatible with the logarithmic Fourier-Mukai formalism.
A logarithmic Euler pairing exists that is compatible with the same formalism.
The approach supplies an alternative to a full dg category of logarithmic coherent sheaves by working directly with correspondences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same kernel construction might be tested on log pairs that are not proper, once suitable compactifications are chosen.
The bicategory of log correspondences could be compared with existing categories of log motives or log derived categories.
The explicit adjunction maps may serve as a model for restoring functoriality in other logarithmic invariants that currently lack ambient categories.

Load-bearing premise

Explicit unit- and counit-type morphisms can always be constructed to give the necessary adjunction data for these kernels.

What would settle it

A concrete smooth proper log pair together with two composable log correspondences for which the induced map on logarithmic Hochschild homology fails to equal the composition of the individual transforms, or for which the constructed unit or counit maps do not satisfy the triangle identities.

Figures

Figures reproduced from arXiv: 2605.11156 by \'Ad\'am Gyenge, Leo Herr, M\'arton Hablicsek.

**Figure 1.** Figure 1: The triple log product A 1 ×log A 1 ×log A 1 is the toric variety with fan the barycentric subdivision of the triangle, viewed as the height-one slice of the fan of A 3 . identified as a sequential blow-up along two divisors given by the strict transforms of DX × Y × DZ and X × DY × DZ. Example 2.7. If X, Y, Z are A 1 with boundary D = ⃗0, the log product A 1 ×log A 1×log A 1 is the blowup of A 3 at the or… view at source ↗

read the original abstract

The construction of a satisfactory dg category of logarithmic coherent sheaves remains a central open problem in logarithmic geometry. In this paper, we propose an alternative correspondence-theoretic approach based on logarithmic Fourier--Mukai transforms. For smooth proper log pairs, we introduce strong log Fourier--Mukai kernels supported on canonical blow-up compactifications and prove that logarithmic Hochschild homology is functorial with respect to the induced transforms. Unlike the classical setting, logarithmic correspondences do not naturally live on ordinary products, and the standard adjunction formalism fails because of blow-up discrepancies. We overcome these difficulties by constructing explicit unit- and counit-type morphisms that provide the necessary adjunction data without requiring an ambient dg category of logarithmic sheaves. As applications, we construct a dg bicategory of logarithmic correspondences in which logarithmic Hochschild homology and cohomology become categorical invariants. We also define logarithmic Chern characters and a logarithmic Euler pairing compatible with the logarithmic Fourier--Mukai formalism.

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.

Desk Editor's Note private letter to a colleague

They build explicit unit and counit maps for strong log Fourier-Mukai kernels on blow-up compactifications to get functoriality of log Hochschild homology without an ambient dg category.

read the letter

The paper's core move is to replace the missing dg category of log coherent sheaves with a correspondence setup. For smooth proper log pairs they define strong log kernels supported on canonical blow-ups and construct unit and counit morphisms by hand so that the induced transforms preserve logarithmic Hochschild homology. This is new: ordinary Fourier-Mukai adjunction breaks on the blow-up discrepancies, and the authors give concrete replacements rather than appealing to a nonexistent ambient category. They then assemble these into a dg bicategory where log HH and HC become invariants, and they define compatible log Chern characters and an Euler pairing. That package is useful for anyone tracking degeneration or mirror symmetry questions in the log setting. The constructions look explicit enough that a referee can check the naturality and triangle identities directly against the log structures. The main soft spot is exactly the one the stress-test flags: whether the hand-built unit and counit really compose to the required identities once the blow-up discrepancies are accounted for. The abstract claims they do, but that step carries the whole functoriality result, so it needs line-by-line verification. No circularity or hidden fitting appears in the outline. This is for algebraic geometers already comfortable with log schemes, dg-categories, and Hochschild invariants; a general reader will find the motivation but not the details. It is worth sending to peer review because the problem is central and the approach is concrete rather than hand-wavy.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes an alternative correspondence-theoretic approach to the open problem of constructing a dg category of logarithmic coherent sheaves. For smooth proper log pairs, it introduces strong log Fourier-Mukai kernels supported on canonical blow-up compactifications and proves that logarithmic Hochschild homology is functorial with respect to the induced transforms. Standard adjunction fails due to blow-up discrepancies, so the authors construct explicit unit- and counit-type morphisms to supply the necessary adjunction data without an ambient dg category of logarithmic sheaves. Applications include a dg bicategory of logarithmic correspondences in which log HH and HC become categorical invariants, plus definitions of logarithmic Chern characters and a compatible logarithmic Euler pairing.

Significance. If the explicit constructions and functoriality hold, the work would be a foundational contribution to logarithmic algebraic geometry and homological algebra. It circumvents the missing ambient category by providing concrete adjunction data for log FM transforms, enabling categorical invariants and log versions of classical invariants like Chern characters. The approach of using blow-up compactifications and explicit morphisms could serve as a template for other settings where standard formalism breaks.

major comments (1)

[Construction of unit and counit morphisms for strong log Fourier-Mukai kernels] The central claim rests on the explicit unit- and counit-type morphisms (described in the abstract and the section on adjunction data for strong log FM kernels) satisfying the triangle identities and naturality conditions with respect to the log structures. The manuscript notes that standard adjunction fails due to blow-up discrepancies but does not provide a detailed verification that counit ∘ unit equals the identity on the relevant Hom spaces in the log setting; this verification is load-bearing for the functoriality of logarithmic Hochschild homology.

minor comments (1)

The abstract and introduction could include a brief diagram or table contrasting the classical FM adjunction with the logarithmic version to clarify the role of the blow-up compactifications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness in the verification of the adjunction data. We address the major comment below and will incorporate the suggested expansion in the revised version.

read point-by-point responses

Referee: [Construction of unit and counit morphisms for strong log Fourier-Mukai kernels] The central claim rests on the explicit unit- and counit-type morphisms (described in the abstract and the section on adjunction data for strong log FM kernels) satisfying the triangle identities and naturality conditions with respect to the log structures. The manuscript notes that standard adjunction fails due to blow-up discrepancies but does not provide a detailed verification that counit ∘ unit equals the identity on the relevant Hom spaces in the log setting; this verification is load-bearing for the functoriality of logarithmic Hochschild homology.

Authors: We agree that a more detailed verification is required. The manuscript constructs the explicit unit- and counit-type morphisms in the section on adjunction data for strong log FM kernels and uses them to establish functoriality of logarithmic Hochschild homology, noting the failure of standard adjunction due to blow-up discrepancies. However, the step-by-step check that counit ∘ unit equals the identity on the relevant Hom spaces, together with naturality under the log structures, is presented at a level of detail that could be expanded. In the revised manuscript we will add an expanded subsection containing explicit computations of the compositions on Hom spaces, including the necessary diagrams and checks that account for the logarithmic structures and the discrepancies arising from the canonical blow-up compactifications. revision: yes

Circularity Check

0 steps flagged

No circularity: functoriality derived from explicit new morphisms on blow-up compactifications

full rationale

The paper's central derivation introduces strong log Fourier-Mukai kernels supported on canonical blow-up compactifications for smooth proper log pairs and constructs explicit unit- and counit-type morphisms to supply adjunction data, bypassing the absent ambient dg category of logarithmic sheaves. This construction is used to prove that logarithmic Hochschild homology is functorial with respect to the induced transforms, yielding a dg bicategory of logarithmic correspondences. No quoted step reduces the functoriality claim to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the explicit morphisms are presented as novel and independent of the target result. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions from logarithmic geometry and the existence of canonical blow-up compactifications; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence of canonical blow-up compactifications for smooth proper log pairs that support the strong log kernels
Invoked to place the kernels and overcome blow-up discrepancies.
standard math Standard properties of Hochschild homology and adjunctions in dg categories extend to the logarithmic setting once unit and counit maps are supplied
Background used to conclude functoriality once the explicit morphisms are constructed.

pith-pipeline@v0.9.0 · 5462 in / 1345 out tokens · 45347 ms · 2026-05-13T02:18:35.834892+00:00 · methodology

discussion (0)

Reference graph

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