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On the $K$-theoretic logarithmic double ramification class

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Pith's one-line read A single explicit operator computes the K-theoretic log double ramification class and also yields a GL_r(Z)-invariant product formula.

desk verdict K-theoretic log DR class with a genuinely new stack-level Thom-Porteous formula; the formula and product are likely right, but the colimit logK ring rests on a sketched Artin-fan functoriality that needs a real proof. read the letter →

arxiv 2607.13376 v1 pith:LOQIUR2D submitted 2026-07-15 math.AG

Kamyar Amini , You-Cheng Chou , Leo Herr , David Holmes , Irit Huq-Kuruvilla , Yuan-Pin Lee This is my paper · ORCID
classification math.AG MSC 14H1014C1714C3514N3514D2319E08
keywords doubleramificationcyclelogarithmicgeometryalgebraicK-theorycolimitlogThom–PorteousformuladegeneracylocuscompactifiedJacobianmoduliofcurves
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs a K-theoretic analogue of the logarithmic double ramification cycle, the virtual class of curves on which a given line bundle becomes fiberwise trivial. It establishes two main results: the class lives in the colimit log K-theory ring and satisfies a product formula with GL_r(Z)-invariance—products for the columns of A equal those for the columns of AM and both equal the higher-rank log DR class—and, at positive genus, the class is explicitly computed by applying an operator eG, a Grothendieck-type polynomial built from exterior powers and determinants, to the derived pushforward of the universal line bundle over a compactified Jacobian. The operator is finite, thanks to a new K-theoretic Thom–Porteous formula for algebraic stacks that uses an explicit finite free resolution rather than divergent infinite series. Together these give both a computational formula for K-theoretic DR classes and the correct ring structure needed to multiply them.

What carries the argument

The central object is the operator eG on the K-theory of a stack, defined by eG(E−F) = [O_Y] − [det E]·(λ_t(F∨)/λ_{t^{-1}}(E))_{t^{r1}}, where λ_t is the exterior power series and the subscript means the coefficient of t^{r1}. This operator converts the class of a vector-bundle map to the pushforward of the structure sheaf of the locus where the map does not have maximal rank. Its stack-valid form comes from an explicit finite resolution (the Eagon–Northcott–Buchsbaum–Rim complex) of that structure sheaf, bypassing the infinite λ/divided-power expansions that truncate on schemes but can diverge on algebraic stacks. The surrounding structure is colimit log K-theory logK°(X) = colim over log a

What would settle it

Using the explicit eG operator, compute both sides of the product formula for g=1, n=2 with interaction matrix A=[[−1,3],[1,−3]] and M=[[−5,2],[−3,1]] (a worked example in the paper); the two products of log DR classes must coincide in logK°(M_{1,2}). Any difference in a computed class would disprove the product formula.

Watch

Extended reading notes

Core claim

At positive genus, the structure sheaf of the zero section e: M_{g,n}→J in a compactified Jacobian equals eG([Rπ'_*F]) in K°(J), where eG(E−F) = [O_Y] − [det E]·(λ_t(F∨)/λ_{t^{-1}}(E))_{t^{r1}}. Consequently, after pulling back along the Abel–Jacobi map, the K-theoretic log double ramification class satisfies [DR^log_{g,n,a}]^{ℓvir} = aj^!_a eG(Rπ'_*F). The proof identifies the zero section with the (r1−1)-degeneracy locus of a map of vector bundles whose ranks differ by 1−g, and applies a finite-resolution K-theoretic Thom–Porteous formula valid for algebraic stacks, bypassing the infinite λ/divided-power expansions that diverge there.

Load-bearing premise

The product formula holds only if the pullback maps between different log refinements of the moduli space are compatible enough to form a true ring; a single incompatible pair of refinements would destroy the product structure.

Editorial extensions

If this is right

  • For g>0, the K-theoretic log double ramification class can be computed by evaluating the finite operator eG on Rπ'_*F, without virtual localization or infinite series; the formula is valid on algebraic stacks and in mixed characteristic.
  • The GL_r(Z)-invariance of products means the class attached to a matrix A is unchanged, after base change, under the action of M; this gives a universal identity in logK°(M_{g,n}) that reduces products of DR classes to a single higher-rank class.
  • Because the classes lie in colimit log K-theory, which is a ring, intersections of log DR classes with one another and with other K-theoretic classes are well-defined—something that fails for limit log K-theory—so this ring is the natural home for double DR intersections and quantum K-theoretic integration.
  • The K-theoretic Thom–Porteous formula extends degeneracy-locus computations from schemes to stacks and is valid in mixed characteristic, making the same finite-resolution tool applicable to other moduli stacks where classes of virtual rank zero are not nilpotent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The finite form of eG suggests that all K-theoretic degeneracy loci on algebraic stacks admit finite alternating-sum formulas, so the apparent need for infinite Grothendieck polynomials is a scheme-specific artifact of nilpotence; analogous finite formulas should hold for rank-drop loci beyond the first.
  • Because the r-th root variant pushes down to the same eG formula, one can conjecture a full K-theoretic Pixton formula—[DR^{log}]·ψ^u = r^{u+1} ϵ_* c_{g+u}(−Rπ_* L^{1/r})—with explicit r-dependence for each u>0, parallel to the Chow precursor; this is testable by the same degeneracy-locus method.
  • The reliance on compactified Jacobians is likely temporary; once the logarithmic Picard stack LogPic has sufficiently developed Brill–Noether theory, the same formula should be provable directly on LogPic, eliminating the quasistable-model and admissibility detour.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper constructs a K-theoretic logarithmic double ramification (log DR) class and studies its formal properties. It defines log perfect obstruction theories for the log DR space and the Abel--Jacobi pullback description, proves their equivalence, introduces a colimit log K-theory ring, and establishes a log product formula together with GL_r(Z)-invariance. It also proves a finite K-theoretic Thom--Porteous formula for degeneracy loci on algebraic stacks, using an Eagon--Northcott resolution, and applies it to compactified Jacobians to express the structure sheaf of the zero section as eG([Rπ'_*F]) and the log DR class as an explicit Gysin pullback of this operator. The main theorems are Theorem 1.8, Theorem 4.19, and Theorem 6.9/Proposition 6.14.

Significance. If correct, the paper is a substantial contribution: it gives a K-theoretic refinement of logarithmic double ramification cycles, a product formula in a genuinely ring-valued log K-theory, and an explicit finite Thom--Porteous formula that remains valid on algebraic stacks where the usual infinite Grothendieck-polynomial series can fail to converge. The degeneracy-locus argument in §5--§6 is independent of the target statement and is not circular; in particular, the Eagon--Northcott resolution is a real methodological strength. However, the framework of colimit log K-theory, and hence Theorems 1.3, 1.4 and 4.19, rests on the Artin-fan functoriality claims in Appendix B, whose proofs are currently not complete. The central formula in Theorem 1.8 also requires clarification of the K-theory group in which the class [Rπ'_*F] is interpreted. For these reasons the paper needs substantial revision before I can recommend acceptance.

major comments (4)
  1. [Appendix B, Proposition B.7; Definition 4.3; Theorem 4.19] The definition of colimit log K-theory and the ring structure on logK^0(X) depend on the existence, uniqueness and compatibility of maps between Artin fans: these are the maps φ: Θ_{X1}→Θ_{X2} used to define the Gysin pullbacks φ^! in Definition 4.3 and in the proof of Theorem 4.4. Proposition B.7 is therefore load-bearing. Its proof is a sketch: Lemma B.6 asserts a uniqueness/extension statement for sheaves on Spec P, but the proof claims that global sections are determined by the stalk at the deepest closed stratum, which is false for general constructible sheaves (e.g. a skyscraper supported on a higher stratum has zero stalk at the deepest stratum but nonzero global sections). The statement of Lemma B.6 also does not assume the constructibility or constancy hypotheses used in the proof. Lemma B.5 similarly asserts an induced étale representable map without a complete verification of
  2. [§6, proof of Theorem 6.9 and Proposition 6.14] The identification of the degeneracy locus with the zero section E^{1/r}→J^{1/r} is imported: the proof says 'which [CH25, Lemma 4.4] then identifies with the zero locus', and Proposition 6.14 uses '[CH25, Lemma 4.5]' to identify a gerbe. These identifications are central to Theorem 1.8. The paper should either state these lemmas with their precise hypotheses and prove them in the required generality, or give a complete reference and explain why the cited statement applies to the present compactified-Jacobian setting. As written, a key step of the main formula is outsourced.
  3. [Definitions 1.5 and Theorem 6.9; Eq. (35)] The operator eG is defined on differences E−F of vector bundles. In Theorem 6.9 it is applied to [Rπ'_*F]=[π'_*F]−[R^1π'_*F]. The proof derives equality of this class with [π'_*F(D)]−[π'_*F_D(D)], where the latter two are vector bundles, but the equality is obtained in G-theory. Unless the cohomology sheaves are locally free, or K^0 is interpreted as K-theory of perfect complexes and eG is extended to that group, the formula eG([Rπ'_*F]) is not literally well-defined as stated. Please clarify which K-group is used throughout §6 and prove that eG extends to the relevant perfect-complex class.
  4. [Theorem 4.19 and §4.3] The proof of the product formula invokes a pullback square involving the simultaneous triviality locus and then applies '[CHL23, Remark 1.8]' to identify ∆^†_B(⊠_i [DR_{P_i,a_i}]^{ℓvir}) with [DR_{P,A}]^{ℓvir}. This is a substantial compatibility statement between the log Gysin map and the log perfect obstruction theories. The paper should justify this square and the obstruction-theory identification in detail, or state the exact theorem from [CHL23] being used.
minor comments (5)
  1. [§1, Theorem 1.8] The notation \(\tilde M_{g,n}\) appears in the statement without definition; it is later described as a log alteration. Please define it in the glossary or in the theorem statement.
  2. [§5, Lemma 5.2] The proof cites '[hg]' (a MathOverflow answer) with only an initial; please give the full author name and a stable reference, or replace by a direct proof.
  3. [§1.6] The paragraph on motivations from quantum K-theory and physics is long and disconnected from the mathematical content. It could be condensed to a few sentences or moved to a remarks subsection.
  4. [§4.3, proof of Theorem 4.19] The statement 'By Remarks 2.18, 2.19 and their analogue in higher rank, we have a pullback square' would be easier to check if the square were written explicitly with arrows and the maps identified.
  5. [§5, Corollary 5.5] The corollary states that it suffices that the grade of the ideal sheaf I_{X/Y} is r_2−r_1+1, but the preceding proof assumes a regular immersion. Please include a short explanation of the grade condition and its compatibility with the determinantal complex.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central formula [e]=eG(Rπ'_∗F) is derived from an independent Eagon–Northcott resolution and degeneracy-locus identification, not from the target class.

full rationale

The paper's main derivation is not a restatement of its inputs. Theorem 1.8 is obtained by writing the universal line bundle on a compactified Jacobian, forming the exact sequence (34)–(35), observing via Lemma 6.11 that the ranks satisfy r1−r2 = 1−g, and then applying the finite Eagon–Northcott/Buchsbaum–Rim resolution (Theorem 5.1, Corollary 5.5) to identify the K-theory class of the zero section. The identification of the degeneracy locus with the zero locus is an external geometric statement cited from [CH25], not from the class being computed. The log product formula (Theorem 4.19) is derived from the pullback square and the log Gysin map, with compatibility facts cited from [CHL23]/[Her23]; these are published prior theorems with stated assumptions, so under the rules they count as independent support rather than circularity. The appendix's Artin-fan functoriality lemmas (B.5–B.6) are the weakest point and were revised after an AI-assisted error correction, but even if those proofs are incomplete, the failure would be a gap in the construction of the ring, not a reduction of the conclusion to its inputs. No parameter is fitted and no 'prediction' is statistically forced by construction. Hence no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical free parameters. The paper introduces no new physical entities; the operator eG is a notation, not an entity. All axioms are imported from (or stated variants of) established log-geometry theorems.

assumptions (5)
  • domain assumption Log perfect obstruction theory compatibility under log alterations (Theorem B.1, [Her23 Thm 3.10], [CHL23 Prop 2.5])
    Imported theorem underpinning the definition and pushforward-compatibility of the log virtual classes in §4.1; without it, [Q]^{ℓvir} is not well-defined as a compatible system.
  • domain assumption Bounded functoriality of Artin fans: subdivision theorem [ACWM17 Thm 4.6.2] and existence of Artin-fan maps Θ_X→Θ_Y (Prop B.7)
    Needed to define the colimit logK ring and Gysin pullbacks between log alterations; proof in Appendix B relies on Lemmas B.5-B.6, which the acknowledgments say were corrected after an AI-assisted check.
  • domain assumption Log Picard stack properties from [MW22]: exact sequence 1→Pic[0]→LogPic→TroPic→0, representability of diagonal of LP^{ps}_{g,n}
    Used in §2 to define the Abel-Jacobi map and the DR fiber product; not re-proved.
  • domain assumption Existence of compactified Jacobians and universal admissible line bundle (Definition 2.24, [HKP18], [HMP+25])
    The explicit formula in §6 proceeds via J rather than LogPic, because the paper notes LogPic lacks developed Brill-Noether theory (§1.5, §6).
  • domain assumption Identification of the r-th root zero locus with the degeneracy locus of π'_*F^{1/r}(D) (Lemma 4.4 of [CH25])
    Makes the square (33) cartesian; if this identification fails outside Pic^[0], Theorem 6.9 would not compute [e^{1/r}].

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Pith. "Pith review of On the $K$-theoretic logarithmic double ramification class." pith.science (2026). https://pith.science/paper/LOQIUR2D

@misc{pith2026260713376,
  author       = {Pith},
  title        = {Pith review of: On the $K$-theoretic logarithmic double ramification class},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOQIUR2D}},
  note         = {Machine review of arXiv:2607.13376}
}
abstract

The logarithmic double ramification cycle is the virtual fundamental class of the locus where a line bundle on a family of curves is fiberwise trivial. We construct a K-theoretic logarithmic double ramification class and prove a product formula and a \(\mathrm{GL}_r(\mathbb Z)\)-invariance property. We also give an explicit formula for this class in terms of a Grothendieck polynomial via a novel $K$-theoretic Thom--Porteous formula for vector bundles on algebraic stacks.

Figures

Figures reproduced from arXiv: 2607.13376 by the authors.

Figure 1
Figure 1. The blowup Be → B = M1,2 through which DRlog OC (2p−2q) factors as a strict closed immersion. e1 t e2 e e1 1 t e2 e1 t e2 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. The Artin fans associated to the log blowup Be → B. The Artin fan of B consists of two cones joined along a ray, depicted in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The tropical abelian variety from Example 2.13 is the “blow down” in log geometry of a nodal cubic or the curve C ′ 0 from [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: A family of divisors Ds on a curve C/S. These smooth points Ds ∈ Cs correspond to a line bundle Ls on Cs. Taking D0 ∈ C0 to be the limit of the points Ds, we see it is not a smooth point and does not correspond to a line bundle on C0. Blow up the point D0 ∈ C to obtain…
Figure 5
Figure 5. Figure 5: A commutative, noncartesian diagram of the zero sections of J and J 1/r . Stability and marked points are irrelevant, so it suffices to show the same claim for prestable curves Mg,0(r) → Mg,0 with no marked points. Etale localize to replace ´ Mg,0 by S := A k , with co…

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