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REVIEW 4 major objections 4 minor 29 references

The Log Product Formula

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves the Log Gromov-Witten Product Formula: the log virtual fundamental class of stable log maps to $V\times W$ equals the log Gysin pullback of the product of the factor classes.

desk verdict The log product formula is likely correct, but the proof rests on a normal-sheaf identification deferred to a companion paper—worth a serious referee, not a desk reject. read the letter →

arxiv 1908.04936 v1 pith:EOTJRQTC submitted 2019-08-14 math.AG

Leo Herr This is my paper · ORCID
classification math.AG MSC 14N3514C1714A2014C1514H10
keywords logGromov-WitteninvariantsnormalconevirtualfundamentalclassGysinmapproductformulasmoothschemesfsstructuresintersectiontheory
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 proves a product formula for logarithmic Gromov-Witten invariants, the curve counts attached to stable maps from marked curves into log smooth targets: when the target is a product $V\times W$ of log smooth quasiprojective schemes, the log virtual fundamental class of stable log maps to the product is determined by the corresponding classes for $V$ and $W$. The equality is stated as a log Gysin pullback along the diagonal after pushing forward from the space of stable log maps to $V\times W$ to the fiber product $Q$. Earlier work had the same formula but only when one factor carried the trivial log structure; the present paper removes that restriction. A sympathetic reader should care because the result makes curve counting on product targets reducible to curve counting on the factors, and because the proof constructs the log normal cone, log virtual fundamental class, and log Gysin map that logarithmic intersection theory needs. Those constructions are the paper's main new tools and are reusable beyond this particular formula.

What carries the argument

The central object is the log normal cone $C^\ell_{X/Y}=C_{X/LY}$, defined by sending a log map $X\to Y$ to the ordinary normal cone of the strict map $X\to LY$, where $LY$ is the universal stack of log structures over $Y$; for strict maps this cone agrees with the ordinary normal cone, and the definition is chosen to be invariant under log \'etale base change. Its companion is the log normal sheaf $N^\ell_{X/Y}$, whose points are square-zero deformations of log structures. A log perfect obstruction theory is an embedding $C^\ell_{X/Y}\subseteq E$ into a vector bundle stack, from which the log virtual fundamental class and the log Gysin map $f^!$ are extracted. The proof's load-bearing diagram is an fs pullback square involving the stack of partial stabilizations; the log Costello formula (Theorem 4.1) and the commutativity theorem for log Gysin maps (Theorem 3.12) are the two inputs that make the two computations agree.

What would settle it

Check Definition 1.7 directly on the log blowup example of Example 2.13: compute the square-zero deformations of log structures along a non-strict map such as the inclusion of a log point into the exceptional divisor, and compare the resulting sheaf with the ordinary normal sheaf of the map into the universal stack of log structures; a mismatch would invalidate the log virtual fundamental classes used in Theorem 5.6.

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Extended reading notes

Core claim

The paper's central claim is Theorem 5.6: for log smooth quasiprojective schemes $V$ and $W$, the identity $$h_*[M^\ell_{g,n}(V\times W), E(V\times W)]^{\ell\mathrm{vir}} = \$\Delta$^!([M^\ell_{g,n}(V), E(V)]^{\ell\mathrm{vir}}\times [M^\ell_{g,n}(W), E(W)]^{\ell\mathrm{vir}})$$ holds in the Chow group of $Q$. Here $E(\cdot)$ is the natural log perfect obstruction theory for stable log maps, $h$ is the comparison map to the fiber product $Q$, and $\Delta^!$ is the log Gysin pullback. The proof follows a classical two-way computation: it forms a fine-and-saturated (fs) pullback square involving the stack of partial stabilizations, and computes the log normal cone of the comparison map $Q\to Q'$ in two ways. One computation uses the log analogue of Costello's pushforward formula, the other uses commutativity of log Gysin maps; the two must agree, which is exactly the displayed formula.

Load-bearing premise

The load-bearing premise is that the log normal sheaf defined in this paper coincides with the ordinary normal sheaf of the map $X\to LY$ to the universal stack of log structures; the identification is asserted and deferred to another paper, yet Section 2's exact sequences, the obstruction-theory definitions, and the log Gysin machinery all depend on it.

Editorial extensions

If this is right

  • The log Gromov-Witten invariants of $V\times W$ are determined by those of $V$ and $W$, so curve counting on product targets reduces to smaller problems.
  • The log Gysin map and log virtual fundamental class introduced here become available for any DM-type log map equipped with a log perfect obstruction theory, not only for stable log maps.
  • The log Costello formula gives a valid pushforward formula along log blowups of log smooth targets, even though pushforward compatibility with log Gysin maps fails in general.
  • The earlier product formula, proved only when one factor has trivial log structure, is included as the special case of the new theorem.

Reading between the lines

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

  • Beyond the paper: the same two-way log normal cone computation should prove product formulas for other log moduli problems, such as stable log maps with contact-order conditions or relative expansions, since the machinery is not tied to the specific stack of stable maps.
  • Beyond the paper: the paper's counterexamples show ordinary pushforward loses logarithmic information, suggesting that a logarithmic Chow group is the right home for these classes; with such a theory, the equality $p_*[\hat X]^{\ell\mathrm{vir}}=[X]^{\ell\mathrm{vir}}$ may hold without the log smoothness hypothesis.
  • Beyond the paper: because the log normal cone can differ from the ordinary normal cone even when the underlying schemes agree, the formula implies that ordinary product-formula computations can miss logarithmic contributions in mixed-degree cases; comparing the two outputs on a product of nontrivial log points would make this visible.
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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 / 4 minor

Summary. The paper develops logarithmic analogues of the intrinsic normal cone, perfect obstruction theory, Gysin pullback, and Costello's pushforward formula, and uses them to prove a product formula for logarithmic Gromov-Witten invariants of V × W in terms of those of V and W for log smooth quasiprojective schemes. The main theorem, Theorem 5.6, identifies the pushforward of the log virtual fundamental class of the moduli stack of stable log maps to V × W with the log Gysin pullback of the product of the two factor classes. The proof follows Behrend's strategy via a cartesian diagram of moduli stacks, replacing the ordinary normal cone and Gysin machinery with the newly introduced log versions. The manuscript is explicit about known failures of naive pushforward-Gysin compatibility (Remarks 2.14, 3.8, 3.9) and acknowledges several dependencies on forthcoming papers [WH] and [AHW].

Significance. If the technical foundations hold, the paper proves a natural and expected generalization of the product formula of Lee–Qu and Behrend to the logarithmic setting, removing the earlier restriction that one factor have trivial log structure. The paper also contributes a framework of log normal cones, log perfect obstruction theories, and log Gysin maps that is likely to be useful beyond the product formula. The author deserves credit for stating counterexamples to naive log analogues (Remark 2.14, Remarks 3.8 and 3.9) and for clearly separating what is proved here from what is deferred to [WH] and [AHW]. The central mathematical idea is coherent and the strategy of reducing log statements to strict, ordinary cases is attractive.

major comments (4)
  1. [Section 1, Definition 1.7 (also Definition 2.2)] The equality N^ℓ_{X/Y} = N_{X/LY} is asserted with proof deferred to [WH], a forthcoming paper. This identification is load-bearing: Definition 2.2 defines the log normal cone as C_{X/LY}, Definition 3.1 requires the inclusion C^ℓ_{X/Y} ⊆ N^ℓ_{X/Y}, and the log virtual fundamental classes in Theorem 5.6 are constructed from this package. Without a proof or a published reference, the main theorem is conditional on an unverified premise. The paper should either prove this identification or replace the reference to [WH] with a published and accessible argument.
  2. [Section 1, Definitions 1.5 and 1.7] The notion of a square-zero closed immersion of log structures is not defined in the manuscript (the text says 'See [WH] for details'). Since the functor of points of N^ℓ is defined in terms of this notion, and Definition 1.7 claims agreement with N_{X/LY}, the reader cannot verify the central object of the paper. A precise definition and a proof of the claimed agreement are needed before the log normal sheaf can be used.
  3. [Section 4, Theorem 4.1 (and Section 3, Theorem 3.10)] The proof of Theorem 4.1 asserts that after the reductions of Construction 1.1, 'the proof of Costello's Formula' applies to the strict closed immersions X ⊆ X^θ and X' ⊆ X'^θ, and that the exact sequences of Proposition 2.5 then show the cone map s is pure degree d. The first assertion is plausible only because the cones in question are then ordinary normal cones, but this is not stated, and the second assertion about preservation of pure degree under quotient by the tangent bundle stack is not justified. Since Theorem 4.1 is the mechanism producing the pushforward identity used in Theorem 5.6, this step should be written out in detail.
  4. [Section 5, proof of Theorem 5.6] The displayed chain of equalities in the proof applies φ^! to M_{g,n} × M_{g,n}, but Definition 3.3 permits φ^! to be applied only to a log-smooth equidimensional stack over Q' (the target of φ), and M_{g,n} × M_{g,n} is not such an input. The intended use of Corollary 3.15 must be written out with the correct Gysin pullbacks (the bottom horizontal map of the square, rather than φ, in the middle term). As printed, this is a gap in the proof of the main theorem.
minor comments (4)
  1. [Definition 3.1] The parenthetical '(equiv. N^ℓ_{X/Y} ⊆ E)' is not precise: N^ℓ is a sheaf, and the equivalence with a closed immersion of the cone C^ℓ_{X/Y} into a vector bundle stack requires explanation.
  2. [Theorem 3.12, Eq. (5)] The expression C^ℓ_{C^ℓ_g|X / C^ℓ_g} uses a cone stack as the base of another cone stack, which is not defined in the text; please introduce notation for this construction.
  3. [References and external dependencies] The paper relies on the forthcoming works [WH] and [AHW] for several supporting statements; for a journal submission these dependencies should either be removed or the statements proved in the paper.
  4. [Throughout] There are frequent notational collisions between the prestable and stable curve stacks (both often rendered as 'M_{g,n}' in the text); the overline notation should be restored consistently to avoid ambiguity in the diagrams and the proof of Theorem 5.6.

Circularity Check

1 steps flagged · score 4.0 of 10

The Product Formula is not assumed, but the identification N^ℓ_{X/Y}=N_{X/LY} is deferred to the authors' forthcoming [WH] and is load-bearing for the log virtual class machinery.

  1. self citation load bearing [Definition 1.7; Definition 2.2]
    "We show that this definition agrees with Definition 1.5 in [WH]: N ℓ X/Y = NX/LY . ... We define the Log (Intrinsic) Normal Cone C ℓ X/Y := CX/LY ⊆ N ℓ X/Y after [GS11]."

    The intrinsic log normal sheaf of Definition 1.7 is identified with the ordinary normal sheaf of X→LY only by the sentence 'We show ... in [WH]', a forthcoming paper by Wise and Herr that includes the present author. Definition 2.2 then builds the log normal cone as C_{X/LY} and asserts its inclusion in N^ℓ, so the Log POTs (Definition 3.1), Log Gysin maps (Definition 3.3), and log virtual classes entering Theorem 5.6 all depend on this unproved equality. This is a load-bearing self-citation: if the [WH] identification failed, the classes compared in Theorem 5.6 would not be grounded by the stated obstruction theory. The Product Formula itself is not assumed, so the circularity is partial.

full rationale

No fitted parameters or statistically forced predictions occur in this paper. Theorem 5.6 is proved by comparing two computations of [Q, E(V)⊞E(W)]^{ℓvir}, using the Log Costello Formula (Theorem 4.1) and commutativity of Log Gysin maps (Theorem 3.12); neither equality is an input to the construction of the log normal cone or the log virtual fundamental class. The proof of Theorem 4.1 is given in the paper, and the compatibility of the obstruction theories is proved in Lemma 5.10. The only circularity-adjacent step is the deferred identification N^ℓ_{X/Y}=N_{X/LY} from [WH], a forthcoming self-citation that is load-bearing for the log intersection-theoretic formalism but does not encode the target product formula. Accordingly the central claim has independent content, and the score reflects that partial self-citation dependence rather than a reduction of the theorem to its own statement.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim is a theorem in pure mathematics. It introduces no fitted parameters or empirical inputs. It rests on the fs-log-geometric framework, on the unpublished identification of the log normal sheaf with an ordinary normal sheaf, on standard virtual pullback machinery, and on a log extension of Costello's formula. The last two are flagged because some proofs are deferred to forthcoming work [WH], [AHW] and to [Cos06].

assumptions (4)
  • domain assumption All log structures are fine and saturated (fs), and the base field is C (Section 0.4).
    The moduli stacks M^ℓ_{g,n} and Olsson's stacks L□/V□ are used in the fs category; the paper assumes fs log structures throughout, and the fsiification pullbacks are the relevant ones.
  • ad hoc to paper The identification N^ℓ_{X/Y}=N_{X/LY} holds; proof deferred to [WH].
    Definition 1.7 defines the log normal sheaf and states the identification with the ordinary normal sheaf of X→LY is shown in the forthcoming [WH]. This is load-bearing for Section 2's functoriality and for Definition 3.1's 'equiv. N^ℓ⊆E' phrasing.
  • domain assumption The ordinary intrinsic normal cone and virtual pullback formalism of [BF96], [Man08], and [Wis11] carries over to the log setting by reduction to strict maps.
    The paper repeatedly reduces statements about log normal cones to the strict case (Remarks 2.3, 2.7, proof of 2.11, Theorem 3.12) and relies on Manolache's Theorem 4.8, Kresch's cycle classes, and Wis11 obstruction theories without reproving them.
  • ad hoc to paper Costello's pure degree theorem [Cos06, Thm 5.0.1] remains valid for the log cone maps after the reductions in Section 4.
    Theorem 4.1 (Log Costello Formula) is used in the proof of Theorem 5.6; its proof invokes '[Cos06, Theorem 5.0.1]' in the reduced log setting rather than supplying the log analogue, and the upcoming [AHW] is cited as the place for more.

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Pith. "Pith review of The Log Product Formula." pith.science (2026). https://pith.science/paper/EOTJRQTC

@misc{pith2026190804936,
  author       = {Pith},
  title        = {Pith review of: The Log Product Formula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOTJRQTC}},
  note         = {Machine review of arXiv:1908.04936}
}
abstract

We prove a formula expressing the Log Gromov-Witten Invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. This extends results of F. Qu and Y.P. Lee, who introduced this formula analogously to K. Behrend. The proof requires notions of "log normal cone" and "log virtual fundamental class," as well as modified versions of standard intersection-theoretic machinery adapted to Log Geometry.

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Works this paper leans on

29 extracted references · 23 canonical work pages

  1. [1]

    Boundedness of the space of stable logarithmic maps

    Dan Abramovich, Qile Chen, Jonathan Wise, and Steffen Marcus. Boundedness of the space of stable logarithmic maps. J. Eur. Math. Soc. , 19:2783--2809, 2017

    work page 2017

  2. [2]

    Costello's pushforward formula

    Dan Abramovich, Leo Herr, and Jonathan Wise. Costello's pushforward formula. forthcoming

    work page

  3. [3]

    Birational invariance in logarithmic gromov–witten theory

    Dan Abramovich and Jonathan Wise. Birational invariance in logarithmic gromov–witten theory. Compositio Mathematica , 154(3):595–620, 2018

    work page 2018

  4. [4]

    L. J. Barrott . Logarithmic Chow theory . ArXiv e-prints , October 2018

    work page 2018

  5. [5]

    K. Behrend . The product formula for Gromov-Witten invariants . October 1997

    work page 1997

  6. [6]

    Behrend and B

    K. Behrend and B. Fantechi . The Intrinsic Normal Cone . In eprint arXiv:alg-geom/9601010 , January 1996

    arXiv 1996

  7. [7]

    A moduli stack of tropical curves

    Renzo Cavalieri , Melody Chan , Martin Ulirsch , and Jonathan Wise . A moduli stack of tropical curves . arXiv e-prints , page arXiv:1704.03806, Apr 2017

    arXiv 2017

  8. [8]

    Costello

    K. Costello. Higher genus gromov-witten invariants as genus zero invariants of symmetric products. Ann. of Math. , 164(2):561–601, 2006

    work page 2006

Show all 29 references
  1. [9]

    Fantechi, L

    B. Fantechi, L. G \"o ttsche, American Mathematical Society, L. Illusie, S.L. Kleiman, N. Nitsure, and A. Vistoli. Fundamental Algebraic Geometry: Grothendieck's FGA Explained . Mathematical surveys and monographs. American Mathematical Society, 2005

    2005

  2. [10]

    Groupes de monodromie en g\'eom\'etrie alg\'ebrique

    Alexander Grothendieck, Michel Raynaud, and Dock Sang Rim. Groupes de monodromie en g\'eom\'etrie alg\'ebrique. I . Lecture Notes in Mathematics, Vol. 288. Springer-Verlag, 1972. S \'e minaire de G \'e om \'e trie Alg \'e brique du Bois-Marie 1967--1969 (SGA 7 I)

    1972

  3. [11]

    Gross and B

    M. Gross and B. Siebert . Logarithmic Gromov-Witten invariants . ArXiv e-prints , February 2011

    2011

  4. [12]

    T. Ito , K. Kato , C. Nakayama , and S. Usui . On log motives . ArXiv e-prints , December 2017

    2017

  5. [13]

    G\'eometrie logarithmique

    Luc Illusie. G\'eometrie logarithmique. Notes of a course at IHP , 1997

    1997

  6. [14]

    On log flat descent

    Luc Illusie, Chikara Nakayama, and Takeshi Tsuji. On log flat descent. Proceedings of the Japan Academy, Series A , 89(1), 2013

    2013

  7. [15]

    Log smooth deformation and moduli of log smooth curves

    Fumiharu Kato. Log smooth deformation and moduli of log smooth curves. 11, 11 1999

    1999

  8. [16]

    Cycle groups for A rtin stacks

    Andrew Kresch. Cycle groups for A rtin stacks. Invent. Math. , 138(3):495--536, 1999

    1999

  9. [17]

    Laumon and L

    G. Laumon and L. Moret-Bailly. Champs alg \'e briques . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer Berlin Heidelberg, 1999

    1999

  10. [18]

    Lee and F

    Y.-P. Lee and F. Qu. A Product Formula for Log Gromov-Witten Invariants . J. Math. Soc. Japan , 70(1):229--242, 01 2018

    2018

  11. [19]

    Manolache

    C. Manolache . Virtual pull-backs . ArXiv e-prints , May 2008

    2008

  12. [20]

    Monomorphisms in categories of log schemes

    Shinichi Mochizuki. Monomorphisms in categories of log schemes. Kodai Math. J. , 38(2):365--429, 06 2015

    2015

  13. [21]

    Toric singularities: Log-blow-ups and global resolutions

    Wies awa Nizio . Toric singularities: Log-blow-ups and global resolutions. J. Algebraic Geom. , 15(1):1--29, 2006

    2006

  14. [22]

    Lectures on Logarithmic Algebraic Geometry

    Arthur Ogus. Lectures on Logarithmic Algebraic Geometry . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2018

    2018

  15. [23]

    Logarithmic geometry and algebraic stacks

    Martin Olsson. Logarithmic geometry and algebraic stacks. Scientific Annals of the Ecole Normale Sup\'erieure , 36, 2003

    2003

  16. [24]

    The logarithmic cotangent complex

    Martin Olsson. The logarithmic cotangent complex. Mathematische Annalen , 2005

    2005

  17. [25]

    Sheaves on artin stacks

    Martin Olsson. Sheaves on artin stacks. Journal Fur Die Reine Und Angewandte Mathematik - J REINE ANGEW MATH , 2007:55--112, 01 2007

    2007

  18. [26]

    Logarithmic Gromov-Witten theory with expansions

    Dhruv Ranganathan . Logarithmic Gromov-Witten theory with expansions . arXiv e-prints , page arXiv:1903.09006, Mar 2019

    1903 arXiv

  19. [27]

    Stacks Project

    The Stacks Project Authors . Stacks Project. http://stacks.math.columbia.edu, 2018

    2018

  20. [28]

    Wise and L

    J. Wise and L. Herr . Log normal sheaves. forthcoming

  21. [29]

    J. Wise . Obstruction theories and virtual fundamental classes . ArXiv e-prints , November 2011

    2011

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