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The log Grothendieck ring of varieties

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Pith's one-line read The log Grothendieck ring of varieties is $K_0(\mathrm{Var}_k)[P]/(P^2+P[G_m])$, and the log Euler polynomial is its motivic invariant.

desk verdict A useful and mostly convincing presentation of the log Grothendieck ring, with one genuine proof gap in Theorem 3.14 that is likely repairable; worth a serious referee. read the letter →

arxiv 2412.07715 v1 pith:VBE5XPDH submitted 2024-12-10 math.AG

Andreas Gross , Leo Herr , David Holmes , Pim Spelier , Jesse Vogel This is my paper · ORCID
classification math.AG MSC 14C3514M2514C30
keywords GrothendieckringofvarietieslogschemesblowuprelationsHodgenumbersEulerpolynomialtoricmotivicinvariantsstandardpoint
topics The Hodge Conjecture
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

This paper builds a Grothendieck ring for log schemes and computes it completely in terms of the classical Grothendieck ring of varieties. The result is that $K_0(\mathrm{LogSch}_k)$ is the polynomial ring $K_0(\mathrm{Var}_k)[P]$ with a single relation $P^2 + P[G_m] = 0$, where $P$ is the class of the standard log point. It then shows that the naive log Hodge numbers, computed from cohomology of the sheaves of log differentials, cannot be motivic invariants: no scissors-compatible map can agree with them on all smooth projective log schemes. The positive replacement is the log Euler polynomial $E_1^{\mathrm{log}}$, the generating function of the Euler characteristics of the wedge powers of log differentials, and the paper proves that $E_1^{\mathrm{log}}$ extends to a ring homomorphism $t_1$ from $K_0(\mathrm{LogSch}_{\mathbb{C}})$ to $\mathbb{Z}[u]$ that agrees with it on log smooth and on constant-free projective log schemes. The payoff is a computable invariant of log schemes that is unchanged by log modifications.

What carries the argument

The machinery is the presentation theorem and the two quotient maps it defines. The generator $P$ is the class of the standard log point, and the single relation $P^2 + P[G_m] = 0$ encodes all log blowup relations. From this presentation the paper defines the log Betti map $\tau(P)=0$ and the log Hodge map $\rho(P)=-[G_m]$; the invariant $t_1$ is the composite of $\rho$ with the usual $e$-polynomial followed by the substitution $v=-1$. The proof of the presentation reduces arbitrary log schemes to locally constant free log schemes by log blowups, uses the class computation for smooth toric varieties, and then shows that the kernel is generated by the one quadratic relation. The agreement theorem for $E_1^{\mathrm{log}}$ is carried by invariance under log modifications for log smooth projective pairs plus an inductive argument on the number of components of the boundary divisor, using an exact sequence for wedge powers of log differentials.

What would settle it

Compute $E_1^{\mathrm{log}}$ and $t_1$ for a log smooth projective log scheme whose only log blowups to a strict normal crossings pair are non-projective; if the two numbers differ, the equality asserted in Theorem 3.14 fails. A second check is to test the scissor relation for $E_1^{\mathrm{log}}$ on a strict closed embedding with log smooth projective pieces; any deviation would contradict the claim that $t_1$ is a ring homomorphism.

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Extended reading notes

Core claim

The central claim is a complete presentation of the log Grothendieck ring together with a motivic substitute for log Hodge numbers. The authors define $K_0(\mathrm{LogSch}_k)$ by strict scissor relations and log blowup relations, then prove Theorem 2.1: $K_0(\mathrm{LogSch}_k) \cong K_0(\mathrm{Var}_k)[P]/(P^2+P[G_m])$. The class $P$ is the standard log point, and the quadratic relation is forced by comparing the plane with toric log structure and its log blowup at the origin. On the Hodge side, they prove Proposition 3.4: no map from log schemes to $\mathbb{Z}[u,v]$ can satisfy the strict scissor relations and agree with the log Hodge polynomial $E^{\mathrm{log}}$ on smooth projective log schemes; the paper gives $\mathbb{P}^1$ with toric log structure and with trivial log structure as an explicit obstruction. The replacement $E_1^{\mathrm{log}}(X) = \sum_p \chi(\wedge^p \Omega_X^{\mathrm{log}}) u^p$ does descend: setting $\rho(P) = -[G_m]$, composing with the usual $e$-polynomial, and then setting $v = -1$ produces a ring homomorphism $t_1$ that agrees with $E_1^{\mathrm{log}}$ whenever $X$ is log smooth and projective or constant-free and projective.

Load-bearing premise

The most delicate step is the assumption that every smooth projective log scheme can be turned, by a log blowup, into a strict normal crossings pair with the blowup still smooth and projective; for the constant-free case this is not proved in the paper, though a separate theorem handles that case.

Editorial extensions

If this is right

  • $E_1^{\mathrm{log}}$ is invariant under log modifications for log smooth and for constant-free projective log schemes, so the log Euler polynomial is a well-defined invariant of the log scheme class in those cases.
  • The class of any toric variety $X$ with fan $\Sigma$ is $[G_m]^n + (1-\chi_c(\Sigma))P[G_m]^{n-1}$, giving $E_1^{\mathrm{log}}(X)=\chi_c(\Sigma)(-u-1)^n$ and log Euler characteristic $\chi^{\mathrm{log}}(X)=0$.
  • The compactly supported Euler characteristic extends uniquely to $\chi^{\mathrm{log}}: K_0(\mathrm{LogSch}_{\mathbb{C}}) \to \mathbb{Z}$, and it equals the Euler characteristic of the locus where the log structure is trivial, equivalently of the Kato-Nakayama space.
  • The duality involution on $K_0(\mathrm{Var}_{\mathbb{C}})$ extends to the log Grothendieck ring in two ways, $i_1$ and $i_2$, and yields a restricted log Serre duality relating $\chi(\wedge^{k+n-i}\Omega_X^{\mathrm{log}})$ and $\chi(\wedge^i \Omega_X^{\mathrm{log}})$ for constant-free projective log schemes.
  • Naive log Hodge numbers are not motivic, but their alternating sums over $q$, the holomorphic Euler characteristics of the wedge powers of log differentials, are.

Reading between the lines

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

  • If the presentation is correct, every ring-valued motivic invariant of log schemes is fixed by its values on ordinary varieties and its value on the standard log point; constructing new log invariants reduces to choosing one number satisfying the quadratic relation.
  • The failure of the full two-variable log Hodge polynomial suggests that a future logarithmic mixed Hodge theory will not have scissor-compatible Hodge numbers; the $E_1^{\mathrm{log}}$ specialization, or a refinement carrying more topology, is the level at which motivic behavior can be expected.
  • For toric varieties the class depends only on the compactly supported Euler characteristic of the fan, so toric examples can serve as a testing ground for any proposed log motivic invariant.
  • The unproved reduction of log smooth projective log schemes to s.n.c. pairs with projective blowups is the main point to check; a counterexample would shrink the domain of Theorem 3.14, while the presentation and the constant-free case would stand.
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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

2 major / 5 minor

Summary. The paper defines a Grothendieck ring K0(LogSch_k) for fine and saturated log schemes over k, imposing strict scissor relations and log blowup relations. The main algebraic result is a presentation K0(LogSch_k) = K0(Var_k)[P]/(P^2 + P[G_m]), where P is the class of the standard log point and the relation is derived from the blowup of A^2. The paper then studies motivic log Hodge invariants over C. It shows that the naive log e-polynomial cannot satisfy scissor relations, and constructs two maps t1, t2 out of K0(LogSch_C); the map t1 is intended to compute the alternating Euler characteristics chi(Omega^p_log) for log smooth projective log schemes and constant-free log schemes. An appendix proves a cohomological vanishing statement for fibers of toric blowups.

Significance. If the main theorems are fully proved, the paper gives a remarkably simple presentation of the log Grothendieck ring and a well-defined motivic log Euler polynomial, with explicit computations for toric varieties. The paper contains no fitted parameters: the relation P(P+[G_m])=0 is computed from a concrete blowup rather than imposed, and the later invariances are stated as falsifiable equalities. The appendix, due to Mike Roth, is a useful standalone contribution on the structure sheaf of fibers of toric blowups. The significance is real but conditional: the central claims currently rest on two proof gaps that need to be repaired.

major comments (2)
  1. [§2, Proposition 2.9] The kernel computation in Proposition 2.9 is load-bearing for Theorem 2.1, but its proof is only sketched. The reduction to smooth toric blowups is asserted via an undefined diagram: the object T0 is not defined in the text, and the sentence "The ideal I is then generated by relations of the form [T0]lcf = P^r" is not derived. In particular, the argument that every log blowup of a constant-free log scheme is dominated by a pullback of a smooth toric blowup, and that this suffices to generate the kernel I, needs to be written out. This is a proof gap rather than a demonstrated error, but it blocks the main presentation theorem as written.
  2. [§3, Theorem 3.14] The proof of Theorem 3.14 says, after a suitable log blowup, to assume X=(X,D) is an s.n.c. pair, but no justification is given that such a log blowup exists with both underlying schemes smooth and projective. This matters because Corollary 3.6, used immediately before, requires both underlying schemes to be projective. For a log smooth projective scheme with a nontrivial constant part, for example a product of an s.n.c. pair with the standard log point, a log blowup acting on the constant factor need not preserve projectivity of the underlying scheme. The purely constant-free case is treated separately in Theorem 3.13, but Theorem 3.14 does not make a case distinction, so the equality E1^log = t1 is not established for log smooth projective schemes with mixed constant and divisorial log structure. This appears repairable, for instance by combining Theorem 3.13 with the s.n.c. case, but as written it is a genuine gap in a central claim.
minor comments (5)
  1. [§3, Proposition 3.4] In the proof of Proposition 3.4, the displayed difference should be 2+u-uv if the equation is phi(P1)-2phi(P)=phi(P1^o)-2phi(pt); as printed the sign before the constant term is different. The divisibility conclusion is unchanged, but the sign should be corrected.
  2. [§1.2 and §3] The notation P1^o is used for the scheme P1 with the trivial log structure, but the superscript o can suggest the open torus. Please state this convention explicitly at first use.
  3. [§3.2, Definition 3.11] The map t is overloaded: t, t, t1, and t2 are all introduced in a short space. Using distinct symbols, for example T for the ring homomorphism and T1,T2 for its two components, would improve readability.
  4. [§3, Corollary 3.6] The proof relies on [CHL20, Lemma 2.1], an unpublished preprint by one of the authors. Since this lemma is used to justify invariance under log modifications, either a proof should be included or the dependence should be replaced by the simpler invariance argument available for s.n.c. pairs via Remark 3.10.
  5. [§2, Proposition 2.11] The toric class formula is stated for all toric varieties after the smooth case, but the reduction from the singular case to the smooth case via log blowups deserves a sentence explaining why the class [X] in K0(LogSch_k) is invariant under the relevant toric log blowups; this is a small clarity issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ring presentation is derived from computed blowup strata and the agreement of t1 with E1^log is proved by independent Hodge-theoretic calculations, not by definition.

full rationale

The central derivation is self-contained rather than circular. The relation P(P+[G_m])=0 is obtained in Example 2.2 by stratifying A^2 and its blowup at the origin into constant-log strata and using the log blowup relation [X]=[~X]; it is computed, not imposed. Surjectivity of K0(Var_k)[P] -> K0(LogSch_k) is shown via log blowups to locally constant free log schemes, and the kernel computation reduces to classes of smooth toric varieties via Proposition 2.10, which uses only the already-derived relation. The map t1 is defined from the presentation, but the content of Theorems 3.13 and 3.14 is that t1 agrees with the independently defined E1^log; Theorem 3.13 computes E_log(X) from log Kähler differentials of constant free log schemes, and Theorem 3.14 uses an induction with the Esnault--Viehweg exact sequence for s.n.c. pairs. The cited results by overlapping authors, [CHL20, Lemma 2.1] in Corollary 3.6 and [Gro17, Proposition 3.1] in Remark 3.10, are standalone lemmas whose stated assumptions do not include the target identity; they are not being used as a hidden reformulation of the conclusion. There is a genuine proof gap in Theorem 3.14: the claim 'after applying a suitable log blowup, assume X=(X,D) is an s.n.c. pair' is not justified while preserving smoothness and projectivity of the underlying scheme for constant-free or mixed log structures, and the paper does not state a case distinction there. This is a correctness gap, not circularity, and the purely constant-free case is covered separately in Theorem 3.13, so the claim is likely repairable.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameters are fitted. The central theorem depends on cited standard results in log geometry and toric geometry, not on adjustable constants. The only introduced formal object is the class P, whose relation is computed, not postulated.

assumptions (5)
  • standard math Every fs log scheme admits a locally constant free log blowup, used in Proposition 2.6 and relying on [Niz06] and [KKMSD06].
    Used to prove surjectivity of the map from K0(Var_k)[P] to K0(LogSch_k) in Theorem 2.1; cited but not proved in the paper.
  • standard math For a proper representable morphism of Artin fans B -> C, Rτ_*O_B = O_C, as stated in [CHL20, Lemma 2.1].
    Used in Corollary 3.6 to equate Elog-polynomials under log modifications. This is a self-citation by one of the authors and is not independently verified here.
  • standard math For an s.n.c. pair (X,D) with component F, the residue sequence 0 -> ∧^p Ω_{X'}^{log} -> ∧^p Ω_X^{log} -> ∧^{p-1} Ω_{\hat F}^{log} -> 0 holds, citing [EV92, Property 2.3b].
    This sequence is the basis of the induction in the proof of Theorem 3.14.
  • standard math The Hodge-Deligne e-polynomial is a ring homomorphism on K0(Var_C) and factors through the ordinary Grothendieck ring.
    Used to define t1 and t2 as composites of ring homomorphisms; this is standard mixed Hodge theory.
  • standard math For constant free log schemes, the log Kähler differentials decompose as Ω_{X}^{log} = Ω_{X°} ⊕ O_X^r, citing [Ogu18, Proposition IV.1.2.15].
    Used in Theorem 3.13 to compute Elog for constant free log schemes and in Example 3.20.
invented entities (1)
  • P, the class of the standard log point in K0(LogSch_k)
    purpose: Formal generator encoding a rank-one log structure; used to present the ring as K0(Var_k)[P]/(P^2 + P[G_m]).
    P is a formal algebraic class, not an empirical entity. Its defining relation is derived from the blowup computation in Example 2.2, and the standard log point is a specific geometric object, so this is a normal construction rather than a pulled-from-thin-air addition.

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Pith. "Pith review of The log Grothendieck ring of varieties." pith.science (2026). https://pith.science/paper/VBE5XPDH

@misc{pith2026241207715,
  author       = {Pith},
  title        = {Pith review of: The log Grothendieck ring of varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBE5XPDH}},
  note         = {Machine review of arXiv:2412.07715}
}
abstract

We define a Grothendieck ring of varieties for log schemes. It is generated by one additional class ``$P$'' over the usual Grothendieck ring. We show the na\"ive definition of log Hodge numbers does not make sense for all log schemes. We offer an alternative that does.

Figures

Figures reproduced from arXiv: 2412.07715 by the authors.

Figure 1
Figure 1. Decompose the plane A 2 and its blowup Bl~0A 2 at the origin into locally closed strata on which the log structure is constant. Obtain [A 2 ] = [Gm] 2 + 2[Gm] · P + P 2 and [Bl~0A 2 ] = [Gm] 2 + 3[Gm] · P + 2P 2 . 2. The log Grothendieck ring and its presentation In this section we will prove the following simple presentation of the log Grothendieck ring. Theorem 2.1. The log Grothendieck ring is generated by P with… view at source ↗

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