04.12.15 · algebraic-geometry / tropical

Log Gromov-Witten Invariants (pointer)

shipped3 tiersLean: partial

Anchor (Master): Gross-Siebert 2013 *J. AMS* 26 *Logarithmic Gromov-Witten invariants*; Abramovich-Chen 2014 *Asian J. Math.* 18 *Stable logarithmic maps to Deligne-Faltings pairs II*; Chen 2014 *Annals of Math.* 180 *Stable logarithmic maps to Deligne-Faltings pairs I*; Abramovich-Chen-Marcus-Wise 2017 *Compos. Math.* 153 *Boundedness of the space of stable logarithmic maps* (equivalence of the two constructions); Kato 1989 *Logarithmic structures of Fontaine-Illusie* (foundations); Gross-Pandharipande-Siebert 2010 *Publ. IHÉS* 122 *The tropical vertex* (BPS invariants); Mandel-Ruddat 2020 *Algebr. Geom.* 7 (canonical scattering diagrams from log GW); Nishinou-Siebert 2006 *Duke* 135 (tropical-correspondence application); Behrend-Fantechi 1997 *Invent. Math.* 128 (virtual fundamental class); Mikhalkin 2005 *J. AMS* 18 (the $n = 2$ predecessor of the tropical correspondence)

Intuition [Beginner]

Counting curves on a variety is the oldest game in algebraic geometry. How many lines pass through two points? One. How many smooth conics through five generic points? One. How many rational cubics through eight generic points? Twelve. These integer answers are the Gromov-Witten invariants, and they were defined in the 1990s by counting holomorphic maps from a Riemann surface to a target variety, weighted by a delicate machinery that makes the count well-defined.

Log Gromov-Witten invariants extend the count to handle a refinement that ordinary Gromov-Witten theory cannot: curves that are tangent to a chosen boundary divisor with prescribed contact orders. If your target is a log Calabi-Yau like the plane $P^{2}$ together with a chosen line at infinity, and you want to count rational curves of degree $d$ that touch the line at infinity at a single point with full contact order $d$ , ordinary Gromov-Witten theory has no clean way to impose that. Log Gromov-Witten theory does: it remembers contact data along the boundary as part of the curve's structure.

The deeper reason this matters is that log Gromov-Witten counts work cleanly on degenerate targets — singular reducible varieties built from gluing simpler pieces along their boundary divisors. Ordinary Gromov-Witten theory breaks down on a singular target; log Gromov-Witten theory does not. This is the technical innovation that powers the Gross-Siebert mirror programme and the tropical-correspondence theorem of Nishinou-Siebert: the count on the smooth target equals the count on its singular degeneration, computed component-by-component, and the gluing of component-counts is governed by combinatorial tropical data.

Visual [Beginner]

A two-panel picture. Left panel: a smooth complex surface (drawn as a curved sheet) with a marked boundary curve along its edge, and a rational curve drawn on the sheet touching the boundary at a single labelled tangency point. Right panel: the same surface broken along its boundary into two pieces glued at an interior curve, with two halves of the rational curve drawn one on each piece, meeting along the interior curve with a labelled contact order matching the original tangency on the left.

The picture is the visual content of the degeneration formula for log Gromov-Witten invariants: the count of curves tangent to the boundary of the smooth surface equals a combinatorial sum of counts on the broken pieces, each piece tracking the tangency data at every internal interface.

Worked example [Beginner]

Take the projective line $P^{1}$ together with the divisor of two points $D = {0, \infty}$ , forming the simplest log Calabi-Yau pair $(P^{1}, D)$ . Count rational curves of degree $d$ from a $2$ -pointed source curve to $(P^{1}, D)$ with contact order $d$ at $0$ and contact order $d$ at $\infty$ . The answer is $1$ .

Step 1. The classical fact. A degree- $d$ rational map $f : P^{1} \to P^{1}$ with prescribed full tangency $d$ at both $0$ and $\infty$ is forced to be the map $z \mapsto z^{d}$ (up to torus rescaling of the source). The torus action by $z \mapsto λ z$ on the source is exactly the residual symmetry, and quotienting by it leaves a single map.

Step 2. The log GW count. The log Gromov-Witten invariant $⟨[pt]_{0}^{d} ∣ [pt]_{\infty}^{d} ⟩_{0, 2, d}^{l o g}$ is the integral of the empty constraint class against the virtual fundamental class of the moduli of stable log maps of genus $0$ , $2$ marked points with contact order $d$ at $0$ and $d$ at $\infty$ , in degree $d$ . The virtual dimension is $0$ .

Step 3. The integer. The moduli space contains the single map $z \mapsto z^{d}$ as its unique closed point, and its contribution to the virtual count is $1$ . The log GW invariant equals $1$ .

What this tells us: log Gromov-Witten theory cleanly recovers the count $1$ for the map $z \mapsto z^{d}$ , encoding the contact data $(d, d)$ at $(0, \infty)$ as part of the moduli structure. Ordinary Gromov-Witten theory of $P^{1}$ would count all degree- $d$ rational maps and give a different answer; the log refinement is the one suited to mirror symmetry and tropical counting.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

What distinguishes a log Gromov-Witten invariant from an ordinary Gromov-Witten invariant?

A. Log GW invariants are defined over $Q$ rather than $Z$ . B. Log GW invariants count curves on a smooth target $X$ together with prescribed tangency / contact orders to a fixed boundary divisor $D \subseteq X$ , while ordinary GW invariants ignore boundary contact data. C. Log GW invariants only work for curves of genus zero, while ordinary GW invariants work for all genera. D. Log GW invariants are computed by direct enumeration in a computer algebra system, while ordinary GW invariants are computed by integration.

Hint

The "log" prefix refers to logarithmic geometry, which is the formalism for tracking boundary divisors and their stratifications.

Answer

B. Log Gromov-Witten invariants are integer counts of stable log maps from a log smooth curve to a log smooth target, where the target carries a fixed log structure (typically the divisorial log structure of a chosen boundary divisor $D \subseteq X$ ) and the maps remember tangency orders at the boundary. Ordinary GW invariants count maps to a smooth target without boundary structure. Option A is wrong: both are integer-valued in good cases. Option C is wrong: log GW theory handles all genera. Option D is wrong: log GW counts are defined via the virtual fundamental class, exactly the same technique as ordinary GW.

Exercise (easy, true-false).

True or false: log Gromov-Witten theory was developed in part to give a well-defined count of curves on a singular reducible target obtained by degenerating a smooth one.

Hint

The degeneration formula is the central application motivating the Abramovich-Chen and Gross-Siebert constructions.

Answer

True. A primary motivation for log Gromov-Witten theory is the degeneration formula: a smooth target $X$ degenerating to a reducible log smooth central fibre $X_{0}$ in a one-parameter family $X \to A^{1}$ has ordinary GW invariants on $X$ equal to a combinatorial sum of log GW invariants on $X_{0}$ weighted by the contact-order distributions at the internal divisors. Ordinary GW theory has no clean meaning on $X_{0}$ ; log GW does. This is the technical input that the Nishinou-Siebert correspondence and the Gross-Siebert reconstruction programme both rely on.

Formal definition [Intermediate+]

Fix a base scheme $S$ over $C$ (typically $S = Spec C$ ). A log scheme $(X, M_{X})$ over $S$ is a scheme $X$ together with a fine saturated (fs) étale sheaf of monoids $M_{X}$ and a homomorphism $α_{X} : M_{X} \to (O_{X}, \cdot)$ such that $α_{X}^{- 1} (O_{X}^{\times}) ≅ O_{X}^{\times}$ via $α_{X}$ ^{[Kato 1989]}. The standard examples are: a smooth scheme with the identity log structure $M_{X} = O_{X}^{\times}$ ; a smooth scheme $X$ with a normal-crossings divisor $D \subseteq X$ and the divisorial log structure $M_{X} = (j_{*} O_{X ∖ D}^{\times}) \cap O_{X}$ ; a chart $Spec Z [P]$ for $P$ an fs monoid, with the canonical log structure $P ↪ Z [P]$ .

Definition (log smooth morphism). A morphism of fs log schemes $f : (X, M_{X}) \to (Y, M_{Y})$ is log smooth if it is étale-locally on the source isomorphic to a chart morphism $Spec Z [P] \to Spec Z [Q]$ induced by an injective monoid homomorphism $Q ↪ P$ with the cokernel of $Q^{gp} \to P^{gp}$ a finite group (torsion-free in the appropriate setting). Log smoothness allows the source to be singular as an ordinary scheme (e.g., a normal-crossings variety) while remaining smooth in the logarithmic category.

Definition (log smooth curve). A log smooth curve over a base $(S, M_{S})$ is a flat proper log morphism $(C, M_{C}) \to (S, M_{S})$ whose underlying scheme is a (possibly nodal) curve, with the standard log structure recording the nodes as "smoothing data" via the chart $Spec Z [N^{2} \oplus_{N} N^{2}]$ at each nodal point. The log structure encodes both the marked points of an ordinary stable curve and the nodal points as boundary loci.

Definition (stable log map). A stable log map of genus $g$ , $m$ marked points, and class $β \in H_{2} (X, Z)$ to a log smooth target $(X, M_{X})$ over $S$ is a commutative square of fs log schemes $$ \begin{array}{ccc} (C, M_C) & \xrightarrow{f} & (X, M_X) \ \downarrow & & \downarrow \ (S, M_S) & \xrightarrow{} & (\mathrm{Spec},\mathbb{C}, \mathrm{triv}) \end{array} $$ with $(C, M_{C}) \to (S, M_{S})$ a log smooth curve of genus $g$ with $m$ marked points, $f_{*} [C] = β$ on underlying schemes, $f$ a log morphism (i.e., the pullback $f^{♯} : f^{- 1} M_{X} \to M_{C}$ is well-defined), and stability holding in the analogue of the ordinary Deligne-Mumford sense: the underlying map of schemes has finite automorphism group after fixing the marked points and log structure.

Definition (log Gromov-Witten invariant). The moduli stack $\overline{M}_{g, m}^{l o g} (X / S, β, c)$ of stable log maps with contact-order data $c = (c_{1}, \dots, c_{m})$ at the marked points is a proper Deligne-Mumford stack with a perfect obstruction theory ^{[Behrend-Fantechi 1997]}, giving rise to a virtual fundamental class $[\overline{M}_{g, m}^{l o g} (X / S, β, c)]^{vir}$ of expected dimension. The log Gromov-Witten invariant is the integral $$ \langle \tau_{a_1}(\gamma_1) \cdots \tau_{a_m}(\gamma_m) \rangle^{\log}{g, \beta, \mathbf{c}} ;:=; \int{[\overline{\mathcal{M}}{g, m}^{\log}(X / S, \beta, \mathbf{c})]^{\mathrm{vir}}} \prod{i = 1}^m \psi_i^{a_i} \cdot \mathrm{ev}_i^(\gamma_i), $$ with $ψ_{i}$ the $i$ -th cotangent line class, $\gamma_i \in H^(X, \mathbb{Q}) $co h o m o l o g y c l a sses, an d$ \mathrm{ev}i : \overline{\mathcal{M}}{g, m}^{\log} \to X$ the evaluation map.

Counterexamples to common slips

A log smooth scheme is not the same as a smooth scheme with the identity log structure. A normal-crossings divisor $X = {x y = 0} \subseteq A^{2}$ is singular as a scheme but log smooth with the divisorial log structure of the singular locus. The log smooth condition is strictly more permissive than ordinary smoothness; this is exactly what allows log GW theory to handle degenerate fibres.
Contact orders are an essential part of the data. The moduli stack $\overline{M}_{g, m}^{l o g} (X / S, β)$ decomposes as a disjoint union over choices of contact data $c$ at each marked point. Two stable log maps to the same target with different contact orders at a marked point are different points in different components of the moduli stack; collapsing the contact data loses essential information.
The Abramovich-Chen and Gross-Siebert constructions are equivalent — but they were built in parallel. For several years (2010-2017) it was unclear whether they produced the same invariants. Abramovich-Chen-Marcus-Wise 2017 ^{[AbramovichChenMarcusWise2017]} proved equivalence. The two constructions emphasise different aspects (target log structure type, moduli compactness arguments) but agree on the resulting invariants.

Key theorem with proof [Intermediate+]

Theorem (degeneration formula for log GW; Gross-Siebert 2013). Let $π : X \to A^{1}$ be a flat family of fs log schemes with smooth generic fibre $X_{t}$ (for $t \neq = 0$ ) and log smooth central fibre $X_{0}$ . Let $β \in H_{2} (X, Z)$ be a curve class deforming compatibly across the family. Then for every $g, m \geq 0$ : $$ N_{g, m, \beta}(\mathcal{X}t) ;=; \sum{\tau} N^{\log, \tau}_{g, m, \beta}(\mathcal{X}_0), $$ where the right-hand sum is over combinatorial decoration types $τ$ — dual graphs of the central-fibre curve with vertex labels recording which irreducible component of $X_{0}$ each piece maps to and edge labels recording contact orders at the internal divisors — and $N^{l o g, τ}$ denotes the log GW invariant of $X_{0}$ of type $τ$ .

Proof (sketch at intermediate tier). The proof factors into four moves.

Step 1 (specialisation to the central fibre). The deformation invariance of GW invariants on a smooth one-parameter family says that $N_{g, m, β} (X_{t})$ is independent of $t \neq = 0$ . The specialisation map of the moduli stack $$ \overline{\mathcal{M}}_{g, m}(\mathcal{X}t, \beta) ;\rightsquigarrow; \overline{\mathcal{M}}{g, m}(\mathcal{X}_0, \beta) $$ sends a curve on the generic fibre to its limit on the central fibre, which (after stable reduction) is a log smooth curve on $X_{0}$ .

Step 2 (the moduli stack of stable log maps to $X_{0}$ ). The Abramovich-Chen and Gross-Siebert constructions provide the moduli stack $\overline{M}_{g, m}^{l o g} (X_{0}, β)$ as a proper Deligne-Mumford stack with perfect obstruction theory. The stratification by decoration type $τ$ — recording the dual graph and contact orders — gives a stratification $$ \overline{\mathcal{M}}{g, m}^{\log}(\mathcal{X}0, \beta) ;=; \bigsqcup{\tau} \overline{\mathcal{M}}{g, m}^{\log, \tau}(\mathcal{X}0, \beta) $$ with each stratum $\overline{\mathcal{M}}{g, m}^{\log, \tau}$ a closed substack with its own virtual fundamental class.

Step 3 (matching virtual classes). The specialisation of the virtual class on the generic fibre equals the sum of virtual classes on the strata: $$ \mathrm{sp}* [\overline{\mathcal{M}}{g, m}(\mathcal{X}t, \beta)]^{\mathrm{vir}} ;=; \sum\tau [\overline{\mathcal{M}}_{g, m}^{\log, \tau}(\mathcal{X}_0, \beta)]^{\mathrm{vir}}. $$ This step uses the Behrend-Fantechi machinery of perfect obstruction theories on the total family and a careful analysis of the boundary contributions at the stratum interfaces.

Step 4 (integration on each stratum). Integrating the constraint class against each stratum's virtual class gives the log GW invariant $N_{g, m, β}^{l o g, τ} (X_{0})$ . Summing over $τ$ recovers the generic-fibre GW invariant: $$ N_{g, m, \beta}(\mathcal{X}t) ;=; \int{[\overline{\mathcal{M}}_{g, m}(\mathcal{X}t, \beta)]^{\mathrm{vir}}} \prod \mathrm{ev}i^*(\gamma_i) ;=; \sum\tau N^{\log, \tau}{g, m, \beta}(\mathcal{X}_0). $$

Combining the four steps gives the degeneration formula. $□$

Bridge. The degeneration formula builds toward 04.12.06 Nishinou-Siebert correspondence, where the right-hand-side log GW sum on the central fibre of a toric degeneration is identified with a combinatorial tropical-curve count on the dual intersection complex, and appears again in 04.12.09 Gross-Siebert reconstruction theorem, where the log GW counts decorate the slab functions of a consistent scattering structure. The foundational reason is that log GW theory turns the singular central fibre — which ordinary GW theory cannot handle — into a moduli problem with a well-defined virtual fundamental class, and this is exactly the bridge from smooth enumerative geometry to combinatorial tropical enumeration.

Putting these together identifies the log GW count on the central fibre with the tropical curve count on the dual intersection complex, and the same pattern generalises to non-toric log Calabi-Yau pairs through the Gross-Hacking-Keel programme and the Mandel-Ruddat canonical scattering diagrams ^{[MandelRuddat2020]}.

Exercises [Intermediate+]

Exercise 2 (easy, multiple choice).

Which of the following is the virtual dimension of $\overline{M}_{g, m}^{l o g} (X / S, β, c)$ when $X$ is a smooth projective log scheme of complex dimension $n$ with a normal-crossings boundary divisor?

A. $(1 - g) (n - 3) + ⟨ c_{1} (T X^{l o g}), β ⟩ + m$ , where $T X^{l o g}$ is the log tangent bundle. B. $(1 - g) (n + 3) + ⟨ c_{1} (T X), β ⟩$ . C. $n - 3 + m$ . D. $g + m$ .

Hint

The virtual dimension of the log GW moduli stack mirrors the ordinary GW formula but with the log tangent bundle in place of the ordinary tangent bundle.

Answer

A. The virtual dimension of the log GW moduli stack is $(1 - g) (n - 3) + ⟨ c_{1} (T X^{l o g}), β ⟩ + m$ . The log tangent bundle $T X^{l o g}$ is the tangent bundle adjusted by twisting by the boundary divisor: $T X^{l o g} = T X (- lo g D)$ on a smooth target with simple-normal-crossings boundary. Imposing contact-order constraints reduces the moduli dimension by the total contact order at each marking, in a manner that matches the ordinary GW dimension formula in the empty-boundary case.

Exercise 3 (medium, short-answer).

Explain why ordinary Gromov-Witten theory cannot make sense of curve counts on a reducible variety obtained as the central fibre of a toric degeneration of a smooth toric variety, and why log Gromov-Witten theory can.

Hint

Ordinary GW theory requires the target to be a smooth Deligne-Mumford stack. The central fibre is reducible and singular as a scheme. Log smoothness is weaker than scheme-theoretic smoothness.

Answer

Ordinary Gromov-Witten theory is defined via the virtual fundamental class on the moduli of stable maps $\overline{M}_{g, m} (X, β)$ , and the standard construction (Behrend-Fantechi 1997) requires the target $X$ to be a smooth proper Deligne-Mumford stack with a well-behaved tangent complex. A toric-degeneration central fibre $X_{0} = ⋃ X_{Δ_{i}}$ is reducible (multiple irreducible components meeting along toric divisors), and as a scheme it is singular along the intersection loci. The ordinary moduli stack of stable maps to $X_{0}$ does not carry the right virtual class structure; its dimension is not well-defined, and intersections of constraint classes need not produce integer counts.

Log Gromov-Witten theory of Abramovich-Chen 2014 and Gross-Siebert 2013 repairs this. The central fibre $X_{0}$ equipped with its log structure (induced from the inclusion in the total family $X$ , or equivalently from its divisorial structure) is log smooth in the sense of Kato 1989. The moduli stack $\overline{M}_{g, m}^{l o g} (X_{0} / S, β, c)$ of stable log maps to the log smooth target $X_{0}$ is then a proper Deligne-Mumford stack with a perfect obstruction theory and a virtual fundamental class of well-defined dimension. The contact data $c$ at the marked points records tangency orders at the toric divisors of each component, and the resulting log GW invariants are integers. The degeneration formula then says that the smooth-generic-fibre ordinary GW count equals a sum of log GW counts on $X_{0}$ .

Exercise 4 (medium, numeric).

Consider the log Calabi-Yau pair $(P^{2}, L)$ where $L$ is a fixed line. State the log Gromov-Witten invariant counting genus- $0$ degree- $1$ stable log maps with one marked point of contact order $1$ at $L$ , and zero contact at all other markings.

Hint

A degree- $1$ rational curve in $P^{2}$ is a line. The count of lines through a generic point with simple tangency to $L$ is a classical degree count.

Answer

$1$ . A line in $P^{2}$ tangent to $L$ at a single point and passing through a fixed generic point is unique: the line determined by the generic point and the tangency point on $L$ . The virtual count contributes $1$ . (Note: the choice of which exact marked points are imposed and how the contact data is encoded shifts the answer; this exercise specifies a single tangency marked point and the count is $1$ under the natural normalisation.)

Exercise 6 (hard, short-answer).

State the equivalence theorem of Abramovich-Chen-Marcus-Wise 2017 and explain its significance for the log GW programme.

Hint

Two parallel log GW theories existed (Abramovich-Chen 2014; Gross-Siebert 2013). The 2017 paper resolves their relationship.

Answer

Abramovich-Chen-Marcus-Wise 2017 (Compos. Math. 153) proves that the moduli stacks of stable log maps constructed by Abramovich-Chen 2014 and by Gross-Siebert 2013 agree as Deligne-Mumford stacks, and that their virtual fundamental classes agree. The two constructions emphasise different aspects: Abramovich-Chen targets Deligne-Faltings pairs (simple normal-crossings divisor on a smooth target) and uses a moduli problem parameterising the log structure on the source curve via "minimality" arguments; Gross-Siebert targets fully general log smooth schemes and uses an obstruction-theoretic approach via the basic monoid of the log structure.

The significance for the log GW programme is that the equivalence proves a single well-defined invariant: the log GW number $N_{g, β, c}^{l o g} (X)$ is independent of the construction used. This was a load-bearing piece of theoretical infrastructure; without it, results in the literature using one construction could not be directly combined with results using the other. After 2017, the field uses "the" log GW invariant without further qualification.

Exercise 7 (hard, symbolic).

Write the degeneration formula for the ordinary genus- $0$ degree- $d$ Gromov-Witten invariant of $P^{2}$ in terms of log Gromov-Witten invariants of the central fibre of a toric degeneration whose dual intersection complex is the standard simplex of dimension $2$ .

Hint

The central fibre of the standard simplex degeneration of $P^{2}$ is the union of three coordinate planes glued along three coordinate lines.

Answer

The toric degeneration of $P^{2}$ with dual intersection complex the standard $2$ -simplex has central fibre $X_{0} = X_{1} \cup X_{2} \cup X_{3}$ — three copies of $P^{2}$ (the toric strata of three lattice triangles in the simplex) glued along three $P^{1}$ 's (the corresponding lattice edges) meeting at the central point. The degeneration formula reads $$ N_{0, d}(\mathbb{P}^2) ;=; \sum_{\tau} N^{\log, \tau}_{0, d}(\mathcal{X}0), $$ where $τ$ runs over combinatorial types: dual graphs of the central-fibre curve, recording which of $X_{1}, X_{2}, X_{3}$ each irreducible component of the source curve maps to, the contact orders at the internal $P^{1}$ gluing divisors, and the leg structure at the toric boundary. Each $N^{l o g, τ}$ is computable by the Nishinou-Siebert correspondence as a multiplicity-weighted count of parametrised tropical curves on the standard simplex. For $d = 3$ , the sum equals the classical Severi degree $N{0, 3}(\mathbb{P}^2) = 12 $— r eco v er e df r o m t h e t r o p i c a l co u n t o f t r i v a l e n t ba l an ce d g r a p h s t h r o ug h$ 8$ generic points in the simplex.

Exercise 8 (hard, short-answer).

Explain how log Gromov-Witten counts feed the Gross-Siebert reconstruction theorem of 04.12.09, with attention to the canonical scattering diagram construction.

Hint

The wall functions $f_{ρ}$ of the canonical scattering diagram are determined by log GW counts of curves passing through the singular locus of the tropical manifold $B$ .

Answer

The Gross-Siebert reconstruction theorem of 04.12.09 builds a toric degeneration of a Calabi-Yau variety from a polarised tropical manifold $(B, P)$ with simple singularities together with consistent slab gluing data ${f_{ρ}}_{ρ}$ on the codimension- $1$ cells of $P$ . The slab function $f_{ρ} \in C [[t]] [N_{ρ}]$ on a slab $ρ$ has constant term $1$ and higher-order corrections determined by log Gromov-Witten counts of curves on the central fibre $X_{0}$ in the dual intersection complex $B$ that touch the singular locus of $B$ at codimension- $2$ cells with prescribed contact orders.

Explicitly: the order- $k$ coefficient $α_{ρ}^{(k)}$ of $f_{ρ} = 1 + \sum_{k} α_{ρ}^{(k)} t^{k}$ is a sum over log GW invariants of $X_{0}$ of genus $0$ curves with $1$ marked point on the slab $ρ$ , contact order $k$ at the singular locus on the boundary of $ρ$ , and class data determined by the polyhedral structure of $ρ$ . The Mandel-Ruddat 2020 ^{[MandelRuddat2020]} canonical scattering diagram construction realises this dictionary: every wall function of the canonical scattering diagram is a generating series of log GW counts. The Gross-Hacking-Keel-Kontsevich-Soibelman ordered-product formulas for wall-crossing become statements about products of log GW generating functions, providing both the combinatorial structure of the reconstruction and substantive enumerative content for the smooth mirror Calabi-Yau.

Lean formalisation [Intermediate+]

The Lean module Codex.AlgGeom.Tropical.LogGromovWitten schematises the data and statements of log Gromov-Witten theory. Current Mathlib provides the scheme-theoretic infrastructure through Mathlib.AlgebraicGeometry.Scheme and the moduli-stack scaffolding through the categorical machinery, but the specific log GW package — fine saturated log structures, log smooth morphisms, the moduli stack of stable log maps with its perfect obstruction theory — is absent.

The module declares typed placeholders for the seven essential structures: a log scheme LogScheme recording a scheme together with an étale sheaf of monoids and a structure morphism; a log smooth morphism LogSmoothMap capturing the étale-local chart structure; a log smooth curve LogSmoothCurve over a base, with marked points and nodal log structure; a stable log map StableLogMap from a log smooth curve to a log smooth target with prescribed genus, marked points, class, and contact data; the moduli stack ModuliStableLogMaps of stable log maps; the virtual fundamental class virtualFundamentalClass on the moduli stack; and the log GW invariant logGWInvariant as the integral of constraint classes against the virtual class.

The module records three named theorems with sorry-stubbed proof bodies: logGW_well_defined recording that the log GW invariant is a well-defined integer independent of the choice of (Abramovich-Chen vs Gross-Siebert) construction; degeneration_formula recording the equality of generic-fibre ordinary GW with the sum of central-fibre log GW invariants over decoration types; and nishinou_siebert_application recording the specialisation of the degeneration formula to the tropical-correspondence setting of 04.12.06. The Mathlib gap enumerated in the frontmatter lean_mathlib_gap field — log structures, log smooth schemes and morphisms, moduli of stable log maps, perfect obstruction theory in the logarithmic category, virtual fundamental class — is the upstream-contribution roadmap for porting log GW theory to Mathlib.

Foundational log-scheme setup (Kato 1989) [Master]

The technical foundation of log Gromov-Witten theory is the framework of logarithmic geometry introduced by Fontaine-Illusie and developed in detail by Kato in 1989 ^{[Kato 1989]}. We summarise the essentials needed to state log GW theory in arbitrary dimension.

Fine saturated log structures. A prelog structure on a scheme $X$ is an étale sheaf of monoids $P$ together with a morphism $α : P \to (O_{X}, \cdot)$ . The associated log structure $P^{a}$ is the pushout $P^{a} := P \oplus_{α^{- 1} O_{X}^{\times}} O_{X}^{\times}$ — the smallest log structure containing $P$ . A log structure $M_{X}$ is fine if it is étale-locally generated by a finitely generated prelog structure, and saturated if its quotient $\overline{M}_{X} := M_{X} / O_{X}^{\times}$ is a sheaf of saturated monoids ( $x \in M^{gp}, n x \in M \Rightarrow x \in M$ for all $n \geq 1$ ). Fine saturated (fs) log structures are the standard setting; we work in this category throughout.

Charts and log smooth morphisms. A chart for an fs log scheme $(X, M_{X})$ is an étale-local presentation by a single fs monoid $P$ — that is, an étale map $U \to X$ with $M_{X} ∣_{U}$ induced from a monoid morphism $P \to Γ (U, O_{X})$ . A morphism $f : (X, M_{X}) \to (Y, M_{Y})$ of fs log schemes is log smooth if it is étale-locally on the source representable by a chart morphism $Spec Z [P] \to Spec Z [Q]$ induced by a monoid injection $Q ↪ P$ whose cokernel $P^{gp} / Q^{gp}$ is a finitely generated free abelian group ^{[Kato 1989]}. The cokernel records the "log tangent space" along the chart, generalising the usual relative dimension.

Log differentials and the log cotangent complex. Kato defines the sheaf of log differentials $Ω_{X / Y}^{1} (lo g)$ as the universal sheaf of additive maps $δ : O_{X} \to Ω_{X / Y}^{1} (lo g)$ together with a multiplicative map $δ_{l o g} : M_{X} \to Ω_{X / Y}^{1} (lo g)$ satisfying $δ_{l o g} (α) = d α / α$ when $α \in O_{X}^{\times}$ . The log cotangent complex $L_{X / Y}^{l o g}$ extends $Ω_{X / Y}^{1} (lo g)$ to the derived category and provides the obstruction theory for log GW.

Log smooth curves and the Deligne-Mumford boundary. A log smooth curve $π : (C, M_{C}) \to (S, M_{S})$ is a log smooth proper flat morphism of relative dimension $1$ whose geometric fibres are nodal reduced curves of arithmetic genus $g$ . The log structure $M_{C}$ records both the marked points (as smooth boundary points, contributing $N$ -charts) and the nodes (as singular points, contributing $N^{2}$ -charts of the form $x y = t$ for $t \in M_{S}$ ). This is precisely the log enhancement of the moduli of pointed nodal curves $\overline{M}_{g, m}$ , and the resulting log moduli stack maps to $\overline{M}_{g, m}$ forgetfully.

The basic monoid and minimality. A subtle technical point in log GW theory is the minimality condition for stable log maps. Given a candidate stable log map $f : (C, M_{C}) \to (X, M_{X})$ , the source log structure $M_{C}$ is determined only up to enlargement; Gross-Siebert 2013 ^{[Gross-Siebert 2013]} resolve this ambiguity by imposing the condition that $M_{C}$ is minimal among those making $f$ log compatible. The minimality condition equips the moduli stack with a well-defined log structure of its own and is essential for properness of the moduli problem.

Two parallel constructions: Abramovich-Chen vs Gross-Siebert [Master]

Log Gromov-Witten theory was developed essentially in parallel by two groups during 2010-2014. We describe both constructions, their differences in emphasis, and the equivalence theorem.

Abramovich-Chen 2014 ^{[AbramovichChen2014]}. The Abramovich-Chen-Chen approach targets Deligne-Faltings pairs $(X, D)$ — a smooth projective scheme $X$ with a simple normal-crossings divisor $D = \sum D_{i}$ and the associated divisorial log structure. The moduli problem parameterises stable log maps from a log smooth curve $(C, M_{C})$ to $(X, D)$ with prescribed genus, marked points, class, and contact orders at the marked points along the components $D_{i}$ . The construction's key technical innovation is the moduli of minimal log structures: the source log structure $M_{C}$ is minimal in the sense that it is uniquely determined by the underlying map and the contact data, removing the ambiguity that would otherwise plague the moduli problem.

The Abramovich-Chen moduli stack $\overline{M}_{g, m}^{l o g, AC} ((X, D), β, c)$ is constructed as a proper Deligne-Mumford stack with a perfect obstruction theory, hence carries a virtual fundamental class. The companion paper Chen 2014 ^[Chen2014] (Annals 180) develops the foundational moduli theory and the deformation theory of stable log maps in the Deligne-Faltings setting.

Gross-Siebert 2013 ^{[Gross-Siebert 2013]}. The Gross-Siebert approach targets arbitrary log smooth schemes $X$ , including singular fs log schemes such as normal-crossings degenerations and toric varieties with their natural divisorial log structures. The moduli problem parameterises stable log maps from log smooth curves to $X$ in the same way, with the same contact-order data. The key technical innovation is the basic monoid construction: for every candidate stable log map, Gross-Siebert construct a minimal monoid $Q$ called the basic monoid that captures the combinatorial type of the curve, the dual graph, and the contact-order distribution; the moduli of stable log maps decomposes as a disjoint union over isomorphism classes of basic monoids.

The Gross-Siebert moduli stack $\overline{M}_{g, m}^{l o g, GS} (X, β, c)$ is again a proper Deligne-Mumford stack with a perfect obstruction theory and a virtual fundamental class. The construction handles a broader class of targets than Abramovich-Chen — the central fibre of an arbitrary log smooth degeneration, for instance, may have several irreducible components meeting along arbitrary log smooth strata, not only smooth normal-crossings divisors.

The Abramovich-Chen-Marcus-Wise comparison theorem (2017) ^{[AbramovichChenMarcusWise2017]}. The two constructions a priori produce two different moduli stacks. Their equivalence is the subject of Abramovich-Chen-Marcus-Wise 2017 (Compos. Math. 153):

Theorem (Abramovich-Chen-Marcus-Wise 2017). For a smooth projective Deligne-Faltings pair $(X, D)$ , there is a canonical isomorphism of moduli stacks $$ \overline{\mathcal{M}}{g, m}^{\log, \mathrm{AC}}((X, D), \beta, \mathbf{c}) ;\cong; \overline{\mathcal{M}}{g, m}^{\log, \mathrm{GS}}((X, D), \beta, \mathbf{c}), $$ and the perfect obstruction theories agree under this isomorphism. In particular, the resulting log Gromov-Witten invariants agree.

The proof goes through a careful comparison of the basic monoid construction with the minimal-log-structure construction, plus a delicate verification that the two perfect obstruction theories match. After 2017, the field uses "the log Gromov-Witten invariant" without specifying construction, and the choice of which formalism to apply to a given problem becomes a matter of computational convenience.

Where they differ in practice. The Abramovich-Chen formalism is cleaner for problems where the target is fixed and smooth, with a single normal-crossings divisor — most enumerative log Calabi-Yau calculations follow this pattern. The Gross-Siebert formalism is essential for degeneration problems where the target is the singular central fibre of a flat family, with irreducible components meeting along arbitrary log smooth strata — the Nishinou-Siebert and Gross-Siebert reconstruction settings depend on it. The two formalisms together cover all log GW computations of interest.

The Nishinou-Siebert correspondence as a tropical computation [Master]

The most striking computational consequence of log Gromov-Witten theory is the Nishinou-Siebert correspondence theorem of 04.12.06, which expresses log GW counts on the central fibre of a toric degeneration as combinatorial tropical-curve counts on the dual intersection complex.

The setup. Let $X_{Σ}$ be a smooth projective toric variety of dimension $n$ , and let $π : X \to A^{1}$ be a toric degeneration with central fibre $X_{0} = ⋃_{i} X_{Δ_{i}}$ indexed by a polyhedral subdivision $Ξ$ of an associated lattice polytope. The central fibre carries its natural log structure inherited from the inclusion in $X$ , making $X_{0}$ log smooth. The dual intersection complex $B (Ξ)$ is the cell complex obtained by gluing the dual cones over the cells of $Ξ$ ; it carries an integral affine structure with monodromy along its codimension- $2$ singular locus.

The tropicalisation map. Given a stable log map $f : (C, M_{C}) \to (X_{0}, M_{X_{0}})$ on the central fibre, the tropicalisation $(Γ_{f}, h_{f})$ is a parametrised tropical curve in $B (Ξ)$ ^{[NishinouSiebert2006]}:

Vertices of $Γ_{f}$ correspond to irreducible components of $C$ , located at the dual-cone point of the toric stratum each component maps to.
Edges correspond to nodes of $C$ , with direction vectors recording the contact orders at the nodes.
Legs correspond to marked points of $C$ , with direction vectors recording contact orders at the toric boundary.
Balancing at each internal vertex holds as a consequence of the toric residue theorem applied to the corresponding component.

The tropicalisation map is functorial on the moduli stack: families of stable log maps tropicalise to families of parametrised tropical curves, and the moduli of stable log maps of decoration type $τ$ tropicalises onto the moduli of parametrised tropical curves of the dual combinatorial type.

The correspondence theorem. The Nishinou-Siebert theorem states that the log GW count of stable log maps of decoration type $τ$ on $X_{0}$ equals the multiplicity of the corresponding parametrised tropical curve $(Γ_{τ}, h_{τ})$ in $B (Ξ)$ . Summing over decoration types and applying the degeneration formula: $$ N_{g, \beta}(X_\Sigma) ;=; \sum_\tau N^{\log, \tau}{g, \beta}(\mathcal{X}0) ;=; \sum{(\Gamma, h)} m(\Gamma, h) ;=; N^{\mathrm{trop}}{g, \beta}(\Xi), $$ where the last equality identifies the multiplicity-weighted tropical count with the integer count $N_{g, β}^{trop} (Ξ)$ defined directly on $B (Ξ)$ .

Witnesses. For $X_{Σ} = P^{2}$ and degree- $d$ rational curves, the count is the Severi degree $N_{0, d} (P^{2})$ : $N_{0, 1} = 1, N_{0, 2} = 1, N_{0, 3} = 12, N_{0, 4} = 620, N_{0, 5} = 87, 304$ . Each of these is recovered tropically as a Mikhalkin-multiplicity-weighted count of balanced trivalent graphs in the standard simplex of dimension $2$ — and the log GW degeneration formula is what relates the Severi count on $P^{2}$ to the tropical count on the simplex.

Beyond toric varieties. The Nishinou-Siebert correspondence specialises log GW theory to the toric-degeneration setting. The full strength of log GW theory extends to non-toric log Calabi-Yau pairs: the Gross-Hacking-Keel programme for log Calabi-Yau surfaces uses log GW counts to construct theta-function bases on the mirror Calabi-Yau, and the Mandel-Ruddat 2020 ^{[MandelRuddat2020]} canonical scattering diagram construction identifies the wall functions of the Gross-Hacking-Keel-Kontsevich-Soibelman scattering diagrams with log GW generating series. The picture is that log GW theory provides the universal enumerative input to mirror symmetry, with the tropical correspondence as the most explicit computational specialisation.

Connections [Master]

Nishinou-Siebert correspondence theorem 04.12.06. The log Gromov-Witten degeneration formula on the central fibre of a toric degeneration is identified by the Nishinou-Siebert correspondence with a multiplicity-weighted tropical-curve count on the dual intersection complex. The present pointer unit records the foundational log GW package; 04.12.06 consumes it to produce the explicit enumerative correspondence in arbitrary dimension. The Abramovich-Chen and Gross-Siebert log GW constructions are exactly the technical input that makes the Nishinou-Siebert proof go through; without them the degeneration argument on the singular central fibre would have no foothold.
Toric degeneration of a Calabi-Yau variety 04.12.07. Toric degenerations of Calabi-Yau varieties produce log smooth central fibres on which log Gromov-Witten counts are well-defined. The unit 04.12.07 supplies the algebraic-geometric setup — the flat family $π : X \to A^{1}$ with smooth generic fibre and degenerate Calabi-Yau central fibre — while the present unit records the log GW machinery applied to those central fibres. The two units pair: 04.12.07 builds the geometric stage, and the present pointer records the enumerative dynamics.
Gross-Siebert reconstruction theorem 04.12.09. The Gross-Siebert reconstruction theorem of 04.12.09 consumes log Gromov-Witten counts as the enumerative input to its scattering-diagram construction. The slab functions $f_{ρ}$ on codimension- $1$ cells and the wall functions on the scattering diagram are determined by log GW counts of curves on the central fibre $X_{0}$ passing through the singular locus of the dual intersection complex with prescribed contact orders. The present unit records the log GW infrastructure; 04.12.09 applies it to build the mirror Calabi-Yau via order-by-order smoothing.
Strominger-Yau-Zaslow conjecture 04.12.10. The SYZ heuristic of dual special-Lagrangian torus fibrations is the physical motivation behind the entire log GW + tropical-correspondence + Gross-Siebert reconstruction edifice. The log GW counts encoded in the present unit are the algebraic-enumerative shadow of the symplectic disk-counts of the SYZ programme, and the Gross-Siebert / Gross-Hacking-Keel / Mandel-Ruddat constructions are the algebraic realisations of the SYZ heuristic. The present pointer records the algebraic infrastructure that the SYZ programme requires for a precise statement.
Period integral on a Calabi-Yau (pointer) 04.12.13. The period-integral computations on a smooth Calabi-Yau mirror $X_{t}^{\lor}$ — central to mirror symmetry in its earliest formulations through Candelas-de la Ossa-Green-Parkes and beyond — are enriched in the Gross-Siebert programme by log GW generating series: theta functions arising from log GW counts contribute to the mirror's period integrals via the wall-crossing structure of the scattering diagram. The present pointer records the log GW machinery that, combined with 04.12.13's period-integral framework, produces the analytic content of the mirror Calabi-Yau.
Forward to Gromov-Witten foundations (future unit). A foundational unit on ordinary Gromov-Witten invariants on a smooth projective target — defining the moduli stack of stable maps, the Behrend-Fantechi virtual fundamental class, the Givental-Lian-Liu-Yau formulation, the WDVV equations — is the natural prerequisite for the present pointer. As of the current Codex state, no such unit exists; the present unit therefore stands somewhat above its natural foundations. When a Gromov-Witten foundations unit is produced (provisionally to live in a future symplectic or algebraic-geometry chapter), the present pointer will be retrofitted with a forward prereq link, and the Behrend-Fantechi virtual-fundamental-class machinery currently sketched in the Master tier will be cross-linked rather than restated.

Historical & philosophical context [Master]

Logarithmic geometry was introduced by Fontaine and Illusie in unpublished notes circulated in the late 1980s and was developed systematically by Kato 1989 ^[Kato1989] in his paper Logarithmic structures of Fontaine-Illusie. Kato's paper supplied the foundational definitions of fine saturated log structures, log smooth morphisms, log differentials, and the étale-local chart theory; subsequent work by Fontaine-Illusie, Illusie, Nakayama, Olsson, and others extended the theory to the derived category, to stacks, and to higher Hodge structures.

Gromov-Witten theory itself dates to the early 1990s with Gromov's symplectic-side work ^[Gromov1985] (Inventiones 82) and the algebraic-side development by Kontsevich, Manin, Behrend, Fantechi, and others through the mid-1990s. The Behrend-Fantechi 1997 paper ^{[BehrendFantechi1997]} (Invent. Math. 128) constructing the intrinsic normal cone is the technical heart of all modern algebraic GW theory and is essential prerequisite infrastructure for the log GW package.

Relative Gromov-Witten invariants — the predecessor of log GW theory, handling tangency conditions to a smooth divisor — were developed by Li-Ruan 2001 and by J. Li 2001-02 in the symplectic and algebraic settings respectively. These constructions handled the case of a smooth boundary divisor only; they did not extend cleanly to normal-crossings boundaries or to singular log smooth targets. The push to log GW theory in the late 2000s was motivated by the need for an enumerative theory adapted to degenerating families with non-smooth central fibres, particularly in the context of mirror symmetry.

The two parallel log GW constructions appeared between 2010 and 2014: Abramovich-Chen 2014 ^{[AbramovichChen2014]} (Asian J. Math. 18) and the companion Chen 2014 ^[Chen2014] (Annals 180) on the Deligne-Faltings side; Gross-Siebert 2013 ^{[Gross-Siebert 2013]} (J. AMS 26) on the general log smooth side. Both constructions emerged from sustained programmes — Abramovich-Chen growing out of relative-GW theory and the Abramovich-Vistoli theory of stacks; Gross-Siebert growing out of the toric-degeneration / mirror-symmetry programme initiated by Gross-Siebert 2003 (Math. Ann.) and continued through the Annals 2011 reconstruction theorem ^{[GrossSiebert2011]}.

The Abramovich-Chen-Marcus-Wise 2017 ^{[AbramovichChenMarcusWise2017]} equivalence theorem is one of the most consequential technical results in modern algebraic geometry: a confirmation that two parallel constructions targeting overlapping but not identical settings produce the same invariants. Subsequent work by Battistella-Carocci-Manolache, Battistella-Nabijou, and others has continued to refine the log GW machinery; the Mandel-Ruddat 2020 ^{[MandelRuddat2020]} canonical scattering construction connects log GW directly to the wall-crossing structures of mirror symmetry. The Gross-Pandharipande-Siebert 2010 ^{[GrossPandharipandeSiebert2010]} (Publ. IHÉS 122) paper The tropical vertex extracts BPS-type integer invariants from log GW counts and identifies them with the structure constants of the tropical-vertex algebra — the modern algebraic substrate of the entire programme.

Bibliography [Master]

@incollection{Kato1989,
  author = {Kato, Kazuya},
  title = {Logarithmic structures of {Fontaine-Illusie}},
  booktitle = {Algebraic Analysis, Geometry, and Number Theory},
  publisher = {Johns Hopkins University Press},
  year = {1989},
  pages = {191--224},
}

@article{AbramovichChen2014,
  author = {Abramovich, Dan and Chen, Qile},
  title = {Stable logarithmic maps to {Deligne-Faltings} pairs {II}},
  journal = {Asian Journal of Mathematics},
  volume = {18},
  number = {3},
  year = {2014},
  pages = {465--488},
}

@article{Chen2014,
  author = {Chen, Qile},
  title = {Stable logarithmic maps to {Deligne-Faltings} pairs {I}},
  journal = {Annals of Mathematics},
  volume = {180},
  number = {2},
  year = {2014},
  pages = {455--521},
}

@article{GrossSiebert2013,
  author = {Gross, Mark and Siebert, Bernd},
  title = {Logarithmic {Gromov-Witten} invariants},
  journal = {Journal of the American Mathematical Society},
  volume = {26},
  number = {2},
  year = {2013},
  pages = {451--510},
}

@article{AbramovichChenMarcusWise2017,
  author = {Abramovich, Dan and Chen, Qile and Marcus, Steffen and Wise, Jonathan},
  title = {Boundedness of the space of stable logarithmic maps},
  journal = {Compositio Mathematica},
  volume = {153},
  number = {8},
  year = {2017},
  pages = {1633--1666},
}

@article{NishinouSiebert2006,
  author = {Nishinou, Takeo and Siebert, Bernd},
  title = {Toric degenerations of toric varieties and tropical curves},
  journal = {Duke Mathematical Journal},
  volume = {135},
  number = {1},
  year = {2006},
  pages = {1--51},
}

@article{GrossPandharipandeSiebert2010,
  author = {Gross, Mark and Pandharipande, Rahul and Siebert, Bernd},
  title = {The tropical vertex},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume = {122},
  year = {2010},
  pages = {65--168},
}

@article{MandelRuddat2020,
  author = {Mandel, Travis and Ruddat, Helge},
  title = {Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves},
  journal = {Algebraic Geometry},
  volume = {7},
  number = {5},
  year = {2020},
  pages = {591--624},
}

@article{Mikhalkin2005,
  author = {Mikhalkin, Grigory},
  title = {Enumerative tropical algebraic geometry in $\mathbb{R}^2$},
  journal = {Journal of the American Mathematical Society},
  volume = {18},
  number = {2},
  year = {2005},
  pages = {313--377},
}

@article{BehrendFantechi1997,
  author = {Behrend, Kai and Fantechi, Barbara},
  title = {The intrinsic normal cone},
  journal = {Inventiones Mathematicae},
  volume = {128},
  number = {1},
  year = {1997},
  pages = {45--88},
}

@article{Gromov1985,
  author = {Gromov, Mikhail},
  title = {Pseudoholomorphic curves in symplectic manifolds},
  journal = {Inventiones Mathematicae},
  volume = {82},
  number = {2},
  year = {1985},
  pages = {307--347},
}

@article{GrossSiebert2011,
  author = {Gross, Mark and Siebert, Bernd},
  title = {From real affine geometry to complex geometry},
  journal = {Annals of Mathematics},
  volume = {174},
  number = {3},
  year = {2011},
  pages = {1301--1428},
}

@book{Gross2011CBMS,
  author = {Gross, Mark},
  title = {Tropical Geometry and Mirror Symmetry},
  series = {CBMS Regional Conference Series in Mathematics},
  volume = {114},
  publisher = {American Mathematical Society},
  year = {2011},
}

@article{Abramovich2014BAMS,
  author = {Abramovich, Dan},
  title = {Logarithmic geometry and moduli},
  journal = {Bulletin of the AMS},
  volume = {51},
  number = {3},
  year = {2014},
  pages = {393--410},
}

Prerequisites

04.12.06
04.12.07
04.12.09

Tier anchors

beginner: Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 6 informal opening; Abramovich 2014 *Bull. AMS* 51 *Logarithmic geometry and moduli* informal §1
intermediate: Abramovich-Chen 2014 *Asian J. Math.* 18 §1-§3; Gross-Siebert 2013 *J. AMS* 26 §0-§4; Kato 1989 *Logarithmic structures of Fontaine-Illusie* §1-§2; Behrend-Fantechi 1997 *Invent. Math.* 128 (virtual fundamental class)
master: Gross-Siebert 2013 *J. AMS* 26 *Logarithmic Gromov-Witten invariants*; Abramovich-Chen 2014 *Asian J. Math.* 18 *Stable logarithmic maps to Deligne-Faltings pairs II*; Chen 2014 *Annals of Math.* 180 *Stable logarithmic maps to Deligne-Faltings pairs I*; Abramovich-Chen-Marcus-Wise 2017 *Compos. Math.* 153 *Boundedness of the space of stable logarithmic maps* (equivalence of the two constructions); Kato 1989 *Logarithmic structures of Fontaine-Illusie* (foundations); Gross-Pandharipande-Siebert 2010 *Publ. IHÉS* 122 *The tropical vertex* (BPS invariants); Mandel-Ruddat 2020 *Algebr. Geom.* 7 (canonical scattering diagrams from log GW); Nishinou-Siebert 2006 *Duke* 135 (tropical-correspondence application); Behrend-Fantechi 1997 *Invent. Math.* 128 (virtual fundamental class); Mikhalkin 2005 *J. AMS* 18 (the $n = 2$ predecessor of the tropical correspondence)

References

TODO_REF
Kato, K. — Logarithmic structures of Fontaine-Illusie · in *Algebraic Analysis, Geometry, and Number Theory* (Johns Hopkins, 1989), pp. 191-224. The foundational paper of log geometry: fine saturated log structures on schemes, log smooth morphisms, the étale-local chart $\mathrm{Spec}\,\mathbb{Z}[P]$ for $P$ a fine saturated monoid, and log differentials.
TODO_REF
Abramovich, D. & Chen, Q. — Stable logarithmic maps to Deligne-Faltings pairs II · *Asian Journal of Mathematics* 18 (3), 465-488 (2014). One of the two parallel constructions of log Gromov-Witten theory: stable log maps to a Deligne-Faltings (= simple normal-crossings divisor) pair, moduli stack with virtual fundamental class.
TODO_REF
Chen, Q. — Stable logarithmic maps to Deligne-Faltings pairs I · *Annals of Mathematics* 180 (2), 455-521 (2014). Companion paper to Abramovich-Chen II; the moduli-of-log-curves setup, log GW invariants for log smooth schemes.
TODO_REF
Gross, M. & Siebert, B. — Logarithmic Gromov-Witten invariants · *Journal of the AMS* 26 (2), 451-510 (2013). The Gross-Siebert construction: stable log maps to a log smooth target, the moduli stack with its perfect obstruction theory, the degeneration formula.
TODO_REF
Abramovich, D., Chen, Q., Marcus, S. & Wise, J. — Boundedness of the space of stable logarithmic maps · *Compositio Mathematica* 153 (8), 1633-1666 (2017). Proves that the Abramovich-Chen and Gross-Siebert moduli stacks coincide, settling the equivalence of the two constructions.
TODO_REF
Nishinou, T. & Siebert, B. — Toric degenerations of toric varieties and tropical curves · *Duke Math. J.* 135 (1), 1-51 (2006). The application of log GW theory to the tropical correspondence; the degeneration-formula argument runs through the log GW invariant of the central fibre.
TODO_REF
Gross, M., Pandharipande, R. & Siebert, B. — The tropical vertex · *Publ. Math. IHÉS* 122, 65-168 (2010). BPS-type integer invariants extracted from log GW counts; tropical-vertex algebra; ordered-product factorisation of wall-crossing automorphisms in scattering diagrams.
TODO_REF
Mandel, T. & Ruddat, H. — Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves · *Algebraic Geometry* 7 (5), 591-624 (2020). Canonical scattering diagrams from log GW invariants; identifies the wall functions of Gross-Hacking-Keel-Kontsevich with tropical-curve generating series.
TODO_REF
Mikhalkin, G. — Enumerative tropical algebraic geometry in $\mathbb{R}^2$ · *Journal of the AMS* 18 (2), 313-377 (2005). The $n = 2$ correspondence-theorem predecessor; recovered as the surface case of the Nishinou-Siebert / log GW degeneration formula.
TODO_REF
Behrend, K. & Fantechi, B. — The intrinsic normal cone · *Invent. Math.* 128 (1), 45-88 (1997). The virtual fundamental class on moduli stacks with perfect obstruction theory; the technical heart of all modern Gromov-Witten counts, including log GW.
TODO_REF
Gross, M. — Tropical Geometry and Mirror Symmetry · CBMS Regional Conference Series in Mathematics 114, AMS (2011). Lecture 6 — log Gromov-Witten invariants and their use in the Gross-Siebert mirror programme.
TODO_REF
Abramovich, D. — Logarithmic geometry and moduli · *Bull. AMS* 51 (3), 393-410 (2014). Survey article on log geometry; reaches log GW theory at §4.

Lean module

Codex.AlgGeom.Tropical.LogGromovWitten

Reviewer

TBD

Estimated time

beginner: 12m
intermediate: 30m
master: 60m