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On the log-local principle for the toric boundary

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Pith's one-line read For toric varieties with nef boundary, log and local Gromov-Witten counts match for every effective degree.

desk verdict A serious extension of the log-local principle with closed forms for all degrees, but the log-side proof as written has a false avoidance claim and a wrong vanishing argument that need repair. read the letter →

arxiv 1908.04371 v3 pith:V4MIFFHZ submitted 2019-08-12 math.AG hep-thmath-phmath.MP

Pierrick Bousseau , Andrea Brini , Michel van Garrel This is my paper · ORCID
classification math.AGhep-thmath-phmath.MP MSC 14N3514M2514T05
keywords logGromov-Witteninvariantslocaltoricboundarymaximaltangencyfakeweightedprojectivespacestropicalcorrespondenceprinciplemirrorsymmetryforstacksnefpairs
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 establishes the log-local principle for toric pairs $(X,D)$ in which $X$ is a possibly singular $\mathbb{Q}$-factorial projective toric variety and every component of the toric boundary $D$ is nef, meaning numerically nonnegative on curves. Such pairs are forced to be products of fake weighted projective spaces, and the paper computes both sides of the correspondence in closed form: the genus-zero log Gromov-Witten invariants of maximal tangency (each boundary divisor met at one point with maximal contact) and the genus-zero local invariants of the total space of $\bigoplus_j \mathcal{O}(-D_j)$. The result is that, after multiplying the local invariants by the explicit factor $N_X^d=\prod_j (-1)^{d\cdot D_j+1}d\cdot D_j$, the two sides agree for every effective curve class $d$, with up to two point insertions and one descendant $\psi$-insertion. This extends the known smooth normal-crossing case to singularities and to boundaries that are not normal crossing.

What carries the argument

The paper's computations are carried by the tropical correspondence principle and the equivariant mirror theorem for toric stacks. Tropical correspondence identifies the log invariants with weighted counts of maximally tangent rational tropical curves, and the multiplicity algorithm computes each curve's weight by iterated exterior contractions in the dual lattice. The mirror theorem identifies the small $J$-function of the local geometry with the stacky $I$-function, a hypergeometric series whose coefficients are products over the intersection numbers $e^X_j(d)$; comparing its $z$-expansion isolates the descendant invariants. The link between the two sides is the normalization factor $N_X^d$, and the classification of nef toric pairs as products of fake weighted projective spaces (quotients of weighted projective spaces by finite abelian groups) reduces every computation to the weights $(w^{(i)})_j$ and group orders $|G_i|$.

What would settle it

Take $X=\mathbb{P}(1,1,2)$ and degree $d=1$. The theorem predicts exactly one maximally tangent tropical curve through two general points, of multiplicity $2$, so the two-point log invariant equals $2$. An independent computation, for instance by degeneration to the toric boundary or by direct tropical enumeration that allows contact with the $\mathbb{Z}/2$ orbifold point, that yields any other value would falsify the claimed extension to singular toric pairs.

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Extended reading notes

Core claim

The central claim is Theorem 3.3: for every nef toric pair and every effective curve class $d$, the identities $N_X^d\, p^X_d = Rp^X_d$ and $N_X^d\, q^X_d = Rq^X_d$ hold. The paper evaluates both sides. The one-point log invariant $Rp^X_d$ is $1$; the two-point log invariant $Rq^X_d$ is $\prod_i |G_i|\prod_{i,j}(w^{(i)})_j\, d^{n_X}$. The local invariants are $p^X_d = (-1)^{e^X(d)-n_X-r_X}/\prod_j^\circ e^X_j(d)$ and $q^X_d = \prod_i |G_i|\prod_{i,j}(w^{(i)})_j\, d^{n_X} p^X_d$, where $e^X_j(d)=d\cdot D_j$ and $e^X(d)=-d\cdot K_X$. The theorem needs no positivity assumption on $d\cdot D_j$: when some intersection number vanishes, both log and local invariants vanish and the identities still hold.

Load-bearing premise

The proof assumes that a tropical curve-counting correspondence stated for smooth varieties remains valid for the singular fake weighted projective spaces that occur here, because the relevant curves avoid the deeper toric strata; the paper flags this in Remark 4.1 but does not prove it.

Editorial extensions

If this is right

  • For every effective curve class $d$ on a nef toric pair, the genus-zero log invariant with one maximal-tangency point and $\psi^{n_X+r_X-2}$ is exactly $1$.
  • The two-point log invariant equals $(\prod_i |G_i|)(\prod_{i,j} (w^{(i)})_j)\, d^{n_X}$, so it depends only on weights, group orders, and the degree, not on the finer structure of the fan.
  • The local invariants are given in closed rational form by $p^X_d = (-1)^{e^X(d)-n_X-r_X}/\prod_j^\circ e^X_j(d)$, and the universal multiplicative relation with the log invariants holds in all degrees.
  • The correspondence holds without assuming $d\cdot D_j>0$: when some intersection number vanishes, the virtual dimension drops and both log and local invariants vanish separately.
  • This extends the smooth normal-crossing case of the log-local principle to singular $\mathbb{Q}$-factorial toric varieties with non-normal-crossing toric boundary.

Reading between the lines

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

  • The paper proves the two-point identity only with the descendant $\psi^{r_X-1}$ placed at one marked point; the same closed-form structure suggests the equality persists for arbitrary distributions of $\psi$-classes among the two points, which the paper itself leaves as an exercise from the symplectic S-matrix.
  • If the extended tropical correspondence is valid, the log invariants are determined entirely by the intersection numbers $e^X_j(d)=d\cdot D_j$; one could test this by enumerating tropical curves for a toric surface with the same boundary intersection pattern but a different fan.
  • The rational closed forms for the local invariants give concrete base cases for the refined/BPS formulation of the log-local principle, since expanding them in the degree produces the integrality statements one would expect from the log side.
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Editorial analysis

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

Referee Report

3 major / 4 minor

Summary. The paper proves an extension of the log-local principle of van Garrel--Graber--Ruddat to toric pairs (X,D) where X is a Q-factorial projective toric variety and D is the toric boundary. Under the assumption that every component of D is nef, Proposition 2.1 reduces X to a product of fake weighted projective spaces. The main result, Theorem 3.3, asserts that for every effective curve class d, the genus-zero maximally tangent log Gromov--Witten invariants with one or two point insertions and descendant insertions at one point satisfy the log-local relation N_d p_d = Rp_d and N_d q_d = Rq_d. The log side is computed in Theorem 3.1 using tropical correspondence and the multiplicity algorithm of [18,19], yielding Rp_d=1 and Rq_d equal to a product of group orders, weights, and d^{n_X}. The local side is computed in Theorem 3.2 via the mirror theorem for toric stacks, yielding closed hypergeometric formulas. The two sides are computed independently and then compared. The paper explicitly claims the correspondence holds for all degrees, including when some d.D_j=0, without invoking the original smoothness assumptions of Conjecture 1.1.

Significance. If the proof is completed, the paper gives a substantial generalization of the log-local principle: it replaces smooth X and normal-crossing D by possibly singular Q-factorial toric varieties with nef toric boundary, and it removes the positivity assumption d.D_j>0. The main formulas are explicit, closed-form, and parameter-free, which is a notable strength. The local computations are grounded in the established mirror theorem of Coates--Corti--Iritani--Tseng, and the log and local sides are genuinely computed by independent methods rather than fitted to each other. The paper also clearly situates itself relative to the parallel work of Nabijou--Ranganathan. However, the proof as written contains two load-bearing gaps: the extension of the tropical correspondence to singular targets is not justified, and the vanishing statement for d.D_j=0 is supported by a false dimension claim. These issues currently prevent the paper from fully establishing its central theorem.

major comments (3)
  1. [Section 4.1, Remark 4.1] The proof of the log-side formulas in Theorem 3.1 relies on applying [18, Theorem 1.1] and [19, Theorem 1.2] to Q-factorial fake weighted projective spaces, but [18] is stated for smooth varieties. Remark 4.1 asserts that in the cases of interest the curves never meet the deeper toric strata, so the arguments carry through. This assertion is false as stated: for X = P(1,1,2) and degree d = 1, a degree-one curve is given by an equation ax + by = 0, and every such curve passes through the singular point [0:0:1], which is the intersection of two toric divisors and hence a deeper toric stratum. Since all log invariants in Theorem 3.1 are obtained through this correspondence, the authors need either a proof of the correspondence for these singular toric varieties or a separate argument showing that the tropical counts compute the log invariants without the stated avoidance condition.
  2. [Section 5, first paragraph] The vanishing claim for d with some d.D_j = 0 is justified by saying that 'the virtual dimension of the moduli problem is negative in that case.' This contradicts the paper's own formula in Section 2.2, vdim = n_X + m + l_D - 3, which is independent of d and positive for the moduli spaces considered with m=1 or m=2. The invariant may indeed vanish, but the provided reason is internally inconsistent. A correct argument is needed, for instance by analyzing the contact-order zero conditions or by a deformation argument, because this vanishing is part of the statement of Theorem 3.1 and is used in Theorem 3.3 for the case d.D_j=0.
  3. [Sections 2.3 and 3.2] The local invariants p_d and q_d are defined in Section 2.3 only under the hypothesis d.D_j > 0 for all j, since R^1 pi_* f^* O(-D_j) is then a vector bundle and the virtual class (2.7) is defined via its top Chern class. However, Theorem 3.2 states formulas for all effective curve classes d, including those with d.D_j = 0, and Theorem 3.3 explicitly claims the log-local relation with no assumptions on d.D_j. The manuscript does not define the local invariants in the case d.D_j = 0, nor does it explain how the hypergeometric formulas in Theorem 3.2 are to be interpreted geometrically in that case. This is a load-bearing point for the 'no assumptions on d.D_j' part of the main theorem and must be addressed.
minor comments (4)
  1. [Abstract and Theorem 3.3] There are small typos: 'hold s' in the abstract and 'assumtions' in Theorem 3.3 should be 'holds' and 'assumptions'.
  2. [Section 2.1] The notation for the monomial product after defining x = alpha_{j_1}^{1} ... is confusing: the displayed expression writes x^n but the variable v in N^m is not used consistently. Clarifying the notation would improve readability.
  3. [Section 5, Propositions 5.1-5.4] The tropical curve uniqueness arguments are terse; for instance, Proposition 5.1 asserts the unique element without discussing why the fixed general point condition is compatible with the star-shaped tropical curve. Adding a sentence explaining the general position argument would help.
  4. [Section 6, Remark 6.5] Remark 6.5 leaves the computation of two-point descendant invariants with distributed psi-powers to the reader. Since the paper's main theorem is about descendent insertions at one point only, this is acceptable, but a brief statement of the expected result would make the scope clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the log and local invariants are computed independently and compared after the fact.

full rationale

The paper's central claim, Theorem 3.3, verifies the log-local principle after computing each side independently. The log invariants in Theorem 3.1 are obtained from explicit tropical curve counts: Propositions 5.1, 5.2, 5.3, and 5.4 identify unique tropical curves and compute their multiplicities using the algorithm of [19], with the correspondence results of [18] invoked only after the tropical count is performed. The local invariants in Theorem 3.2 are obtained from the equivariant mirror theorem of [10], applied to the stacky I-function of the local toric orbifold; Lemma 6.1 shows the mirror maps are trivial by direct calculation, and the invariants are extracted as explicit Laurent coefficients. No parameter is fitted to the target invariants, and the log-local principle of [12] is not an input to either computation: it is stated as motivation and then checked as the final comparison. The factors N_d^X in Theorem 3.3 are simply the prefactors mandated by Conjecture 1.1, and the equalities N_d^X p_d^X = Rp_d^X and N_d^X q_d^X = Rq_d^X reduce to the independently derived closed forms. The only concerns in the paper, such as Remark 4.1's unproved extension of [18] from smooth varieties to singular fake weighted projective spaces and the questionable negative-virtual-dimension vanishing statement in Section 5, are potential mathematical gaps affecting correctness, not circularity: they do not make the conclusion equivalent to an input by construction. Self-citations to the authors' own work appear only as motivation or as statements of the principle being proved, and they are not load-bearing in the derivations.

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

The central claim rests on two external computational engines, tropical correspondence and toric mirror symmetry, plus an unproved extension of the former to singular toric targets. No numerical parameters are fitted and no new entities are introduced.

assumptions (5)
  • domain assumption Tropical correspondence [18, Theorem 1.1] extends to Q-factorial fake weighted projective toric varieties when curves avoid deeper strata.
    Remark 4.1 asserts the arguments of [18] carry through, but no proof or reference is given for non-smooth targets.
  • standard math Mirror theorem for toric DM stacks [10, Theorem 31 and Corollary 32] applies to X and X^loc_D and identifies the small J-functions with the I-functions in (4.5) to (4.8).
    External theorem from cited literature, used as a black box for the local invariant computation.
  • standard math If all effective toric divisors are nef, X is a product of fake weighted projective spaces (Proposition 2.1).
    Quoted from Fujino-Sato [11] and used to reduce the geometry to a product form.
  • domain assumption Log Gromov-Witten invariants of the toric boundary (X,D) are well-defined and satisfy the standard virtual dimension formula.
    Standard log smooth geometry; used throughout Section 2.2 to set up the invariants Rp_d and Rq_d.
  • ad hoc to paper Local Gromov-Witten invariants are defined for all effective curve classes, including d with d·Dj=0.
    Section 2.3 defines the local virtual class via a vector bundle Ob_D only when d·Dj>0, but Theorem 3.2 states formulas for all d.

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Pith. "Pith review of On the log-local principle for the toric boundary." pith.science (2026). https://pith.science/paper/V4MIFFHZ

@misc{pith2026190804371,
  author       = {Pith},
  title        = {Pith review of: On the log-local principle for the toric boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4MIFFHZ}},
  note         = {Machine review of arXiv:1908.04371}
}
abstract

Let $X$ be a smooth projective complex variety and let $D=D_1+\cdots+D_l$ be a reduced normal crossing divisor on $X$ with each component $D_j$ smooth, irreducible, and nef. The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of $(X,D)$ is equivalent to the genus 0 local Gromov-Witten theory of $X$ twisted by $\bigoplus_{j=1}^l\mathcal{O}(-D_j)$. We prove that an extension of the log-local principle holds for $X$ a (not necessarily smooth) $\mathbb{Q}$-factorial projective toric variety, $D$ the toric boundary, and descendent point insertions.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sheaves of maximal intersection and multiplicities of stable log maps

    math.AG 2019-08 accept novelty 8.0 of 10

    The paper proves explicit multiplicity formulas for non-rigid A1-curves and for unions of two rigid A1-curves in maximal-tangency genus 0 log Gromov-Witten invariants on surfaces.

  2. Gromov-Witten theory with maximal contacts

    math.AG 2019-08 conditional novelty 8.0 of 10

    For simple normal crossings divisors, logarithmic and local/naive Gromov-Witten invariants with maximal contacts differ, and this paper gives the first counterexamples plus a blowup formula measuring the difference.

Reference graph

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