04.12.09 · algebraic-geometry / tropical

Gross-Siebert Reconstruction Theorem (Statement)

shipped3 tiersLean: partial

Anchor (Master): Gross-Siebert 2011 *Annals of Mathematics* 174, 1301-1428 (originator — the published reconstruction theorem); Gross-Siebert 2006 *J. Differential Geometry* 72, 169-338 (precursor — log Calabi-Yau spaces and dual intersection complex); Gross-Siebert 2010 *Journal of Algebraic Geometry* 19, 679-780 (the $[B, \mathscr{P}]$ paper — Kato fan and intrinsic log structures); Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS Regional Conference Series 114, AMS) Lectures 4-6 (textbook consolidation); Kontsevich-Soibelman 2001 *Symplectic geometry and mirror symmetry* (World Scientific) and Kontsevich-Soibelman 2006 *The Unity of Mathematics* (Birkhäuser, 321-385); Auroux 2009 *Surv. Differ. Geom.* 13 (modern survey); Gross-Hacking-Keel 2015 *Publ. IHÉS* 122, 65-168 (theta-function payoff)

Intuition [Beginner]

Mirror symmetry begins as a discovery in string theory: pairs of complex-geometric spaces called Calabi-Yau threefolds that look very different but predict identical physics. Counting curves on one of the pair matches integrals of periods on the other. The discovery was made by Candelas, de la Ossa, Green and Parkes in 1991 for the quintic threefold, and for fifteen years it sat as an experimental fact in search of a mathematical explanation. The Gross-Siebert reconstruction theorem is one of the two major explanations the mathematics community produced. It says: starting from a piecewise-linear combinatorial recipe, you can build the mirror Calabi-Yau, step by step, with no choices.

The recipe has three ingredients. First, a piecewise-linear space called a tropical manifold, which encodes the rough shape of the Calabi-Yau as a polyhedral skeleton. Second, a polyhedral decomposition of that space into flat cells. Third, gluing data attached to the codimension-one faces, called slab functions, that tell you how to identify pieces of the construction across the walls between cells. From these three pieces of data the theorem assembles, order by order in a formal parameter, the full geometric Calabi-Yau as a degenerating family.

The miracle is uniqueness. The recipe outputs a single family of varieties, with no remaining choices once the combinatorial input is fixed. This is the mathematical heart of the result: combinatorial data on the piecewise-linear tropical side reconstructs complex-geometric data on the algebraic side, exactly. Gross and Siebert proved the theorem in their 2011 paper in the Annals of Mathematics, the culmination of a program they had been building since 2003.

Visual [Beginner]

A three-panel diagram. Left panel: a tropical manifold drawn as a polyhedral cell complex in the plane — a square subdivided into triangles, with two marked points indicating singularities of the integral affine structure. Middle panel: the cells of the subdivision lift to flat toric pieces, with arrows on the codimension-one edges showing the slab functions that glue adjacent pieces. Right panel: the assembled object, a degenerating family of Calabi-Yau varieties over a small disk in the complex parameter $t$ , with the central fibre at $t = 0$ being the combinatorial limit and the general fibre a smooth complex Calabi-Yau.

The picture captures the central move of the theorem. The left-hand combinatorial recipe is finite and explicit. The right-hand complex Calabi-Yau is infinite-dimensional. The theorem says the recipe determines the Calabi-Yau without remainder.

Worked example [Beginner]

The simplest worked example is the elliptic curve as the mirror of itself, sometimes called the genus-one fibre of the plain Calabi-Yau pencil. Take the tropical manifold to be a circle of circumference $L$ , subdivided into a single cell (the circle itself). The integral affine structure on the circle is determined by one number, the length $L$ . There are no singular points.

Step 1. The polyhedral decomposition. The circle has a single one-cell, two zero-cells (vertices), and no two-cells (because the manifold is one-dimensional). The slab function on each of the two zero-cells is a Laurent polynomial in one variable, encoding the gluing of the two adjacent strata of the toric model.

Step 2. The slab functions. For the elliptic curve mirror, the slab functions take the simplest non-degenerate form: $f = 1 + z$ , where $z$ is the formal coordinate on the adjacent stratum. This is the canonical choice forced by the Calabi-Yau condition (vanishing canonical bundle on the assembled family).

Step 3. The reconstruction. The Gross-Siebert algorithm produces a one-parameter family of varieties $X \to Spec C [[t]]$ with central fibre the nodal union of two projective lines glued at two points, and general fibre an elliptic curve. The length $L$ of the circle reads off as the period (the integral of the holomorphic differential along the cycle) of the resulting elliptic curve.

What this tells us: a single number on the tropical side — the length $L$ — reconstructs an elliptic curve plus its period. This is the simplest instance of the theorem. The full theorem upgrades the same picture from $L \in R$ to an integral affine manifold $B$ with polyhedral decomposition, recovering Calabi-Yau threefolds with full period data.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The Gross-Siebert reconstruction theorem takes as input a tropical manifold with extra data, and produces:

A. A smooth complex Calabi-Yau variety, in one step.
B. A toric degeneration of Calabi-Yau varieties, reconstructed order by order in a formal parameter $t$ .
C. A symplectic mirror Lagrangian fibration with no algebraic content.
D. A Gromov-Witten invariant of the original Calabi-Yau.

Hint

Reconstruction is order-by-order in a formal parameter; the output is a family, not a single variety, and the construction is algebraic-geometric, not symplectic.

Answer

B. The Gross-Siebert reconstruction outputs a one-parameter family of varieties — a toric degeneration of Calabi-Yau varieties — assembled order by order in the formal parameter $t$ , with central fibre $X_{0}$ a union of toric strata glued combinatorially and generic fibre a smooth Calabi-Yau. Option A is wrong because the construction is formal: it produces a formal family, not a single smooth variety in one step. Option C confuses Gross-Siebert with the symplectic SYZ side; Gross-Siebert is the algebraic-geometric realisation. Option D is wrong: Gromov-Witten invariants are an input to the slab functions, not an output of the reconstruction.

Exercise (easy, multiple choice).

A tropical manifold $(B, P)$ with simple singularities consists of:

A. A smooth Riemannian manifold with a polyhedral triangulation.
B. An integral affine manifold with singularities along a codimension-2 locus, together with a polyhedral decomposition into integral affine cells.
C. The tropicalisation of a single algebraic curve in the plane.
D. A complex Calabi-Yau threefold together with a special Lagrangian fibration.

Hint

The tropical manifold sits on the polyhedral side, not the Riemannian or complex-analytic side. The singularities are concentrated in codimension 2.

Answer

B. A tropical manifold $(B, P)$ consists of an integral affine manifold $B$ — a real manifold with charts in integral affine geometry, allowing transition functions in $GL_{n} (Z) ⋉ Z^{n}$ — equipped with singularities along a codimension-2 locus $Δ \subset B$ , plus a polyhedral decomposition $P$ of $B$ into integral affine cells. "Simple singularities" means each singular point has a model of one of a small list of standard types (focus-focus singularities in real dimension 2; generic singularities in higher dimension). Option A misses the integral affine structure. Option C confuses tropical manifolds with tropical curves. Option D describes the SYZ side; the tropical manifold is the dual base, not the Calabi-Yau itself.

Exercise (easy, true-false).

True or false: the Gross-Siebert reconstruction theorem asserts the uniqueness of the reconstructed toric degeneration up to formal isomorphism, not merely its existence.

Hint

Uniqueness statements are stronger than existence statements; the theorem makes the stronger claim.

Answer

True. The Gross-Siebert reconstruction theorem asserts both existence and uniqueness: given a tropical manifold $(B, P)$ with simple singularities and lifted gluing data, there is a toric degeneration of Calabi-Yau varieties with dual intersection complex $(B, P)$ , and it is unique up to formal isomorphism. Uniqueness is the deeper of the two claims and is what makes the combinatorial-to-algebraic-geometric translation exact, not just consistent. The existence half had antecedents (Kontsevich-Soibelman 2001/2006 in the non-archimedean analytic setting); the uniqueness half is the Gross-Siebert contribution.

Formal definition [Intermediate+]

Fix a finite-rank lattice $N ≅ Z^{n}$ with dual $M = Hom_{Z} (N, Z)$ . Write $N_{R} = N \otimes_{Z} R$ . We define in turn the integral affine manifold, the tropical manifold with polyhedral decomposition, the structure (slab and scattering data), and the toric degeneration that the reconstruction theorem connects.

Definition (integral affine manifold). An integral affine manifold of dimension $n$ is a real topological manifold $B$ of dimension $n$ together with an atlas of charts $(U_{α}, φ_{α})$ , $φ_{α} : U_{α} \to R^{n}$ , whose transition functions $φ_{β} \circ φ_{α}^{- 1}$ lie in the integral affine group $Aff (Z^{n}) = GL_{n} (Z) ⋉ Z^{n}$ (linear part in $GL_{n} (Z)$ , translation part in $Z^{n}$ ). An integral affine manifold with singularities is a topological manifold $B$ together with a codimension-2 closed subset $Δ \subset B$ (the singular locus) and an integral affine structure on $B ∖ Δ$ . The local monodromy around $Δ$ takes values in $GL_{n} (Z)$ and encodes the topology of the singularity.

Definition (tropical manifold). A tropical manifold of dimension $n$ is a pair $(B, P)$ where $B$ is an integral affine manifold with singularities of real dimension $n$ , and $P$ is a polyhedral decomposition of $B$ into lattice polytopes: a locally finite collection of lattice polytopes covering $B$ , glued along faces, with all cells having dimensions $\leq n$ and the codimension- $n$ cells being vertices. The singular locus $Δ \subset B$ is required to be a union of codimension-2 cells of $P$ ("simple" singularities) so that the singular set respects the polyhedral structure. The notation $(B, P)$ is the canonical Gross-Siebert notation; the structure type is sometimes written $(B, P, φ)$ when an additional polarisation $φ$ (a strictly convex piecewise-affine function on $B$ ) is included.

Definition (toric degeneration). A toric degeneration of Calabi-Yau varieties is a flat proper morphism $π : X \to S = Spec C [[t]]$ (or $Spec C [t] / (t^{k + 1})$ for the order- $k$ truncation) satisfying:

(i) The generic fibre $X_{η} = π^{- 1} (η)$ over the generic point $η$ is a smooth projective Calabi-Yau variety: vanishing canonical bundle $K_{X_{η}} ≅ O_{X_{η}}$ , and Hodge-theoretic vanishing $H^{i} (X_{η}, O_{X_{η}}) = 0$ for $0 < i < dim X_{η}$ in the strict Calabi-Yau case.

(ii) The central fibre $X_{0} = π^{- 1} (0)$ is a reduced reducible union of toric varieties $⋃_{σ} X_{σ}$ indexed by the maximal cells $σ$ of a polyhedral decomposition $P$ of an integral affine manifold $B$ , glued along toric strata corresponding to the faces of the cells.

(iii) The dual intersection complex of $X_{0}$ is the polyhedral cell complex with one $k$ -cell for each codimension- $k$ stratum of $X_{0}$ , with the affine structure on the integral affine charts encoded by the toric geometry of the strata.

Definition (slab function and structure). A slab function on a codimension-1 cell $ρ$ of $P$ is a Laurent polynomial $f_{ρ} \in C [[t]] [N_{ρ}]$ in the toric coordinates of the adjacent maximal cells, encoding the gluing of the toric pieces of $X_{0}$ across $ρ$ . A structure $S$ on a tropical manifold $(B, P)$ is a collection of slab functions ${f_{ρ}}_{ρ \in P^{(n - 1)}}$ together with a scattering diagram — a finite collection of walls (codimension-1 polyhedral subsets of $B ∖ Δ$ ) labelled with wall-crossing automorphisms — satisfying consistency around every joint (codimension-2 cell) of $P$ . The scattering diagram is constructed by the Kontsevich-Soibelman algorithm: starting from the initial walls supported on the codimension-1 cells of $P$ with their slab functions, one adds new walls inductively to maintain consistency around each joint, at each new order in $t$ .

Counterexamples to common slips

"A tropical manifold is the same as a tropical variety." Tropical varieties are tropicalisations of single algebraic varieties: piecewise-linear images of algebraic varieties under the non-archimedean valuation map. Tropical manifolds are intrinsic polyhedral objects with integral affine structure, designed to model the base of an SYZ fibration. The two concepts share the polyhedral substrate but are distinct: a tropical variety has codimension-1 cells inherited from monomial coefficients, while a tropical manifold has cells inherited from the polyhedral decomposition encoding the central fibre of a toric degeneration.
"Slab functions are arbitrary." The slab functions on a tropical manifold are not free data — they are constrained by the Calabi-Yau condition (vanishing canonical class on the assembled family) and by the consistency of the scattering algorithm around every joint. The data carrying the reconstruction is the choice of slab functions modulo the consistency constraints, which Gross-Siebert show is a finite-dimensional space at each order in $t$ . The constraints can fail to be satisfiable, in which case the tropical manifold does not lift to a toric degeneration.
"The reconstruction is purely combinatorial." The reconstruction takes combinatorial input (tropical manifold + slab functions) and produces algebraic-geometric output (a toric degeneration), but the construction at each order in $t$ involves solving consistency equations whose existence and uniqueness are theorems, not tautologies. The algebraic-geometric machinery — log Gromov-Witten theory, formal-scheme deformation, étale topology on the central fibre — is essential to the proof.

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the Gross-Siebert reconstruction. We give the statement in full and a structural proof sketch — the architecture of the argument, not its proof steps. The full proof occupies 128 pages of the 2011 Annals of Mathematics paper and is deferred.

Theorem (Gross-Siebert reconstruction; Gross-Siebert 2011 Ann. Math. 174). Let $(B, P)$ be a tropical manifold of dimension $n$ with simple singularities, together with a polarisation $φ$ and lifted gluing data prescribed in terms of slab functions ${f_{ρ}}_{ρ \in P^{(n - 1)}}$ on the codimension-1 cells of $P$ . Suppose the slab functions satisfy the consistency conditions around every joint (codimension-2 cell of $P$ ).

Then there exists a toric degeneration $π : X \to Spec C [[t]]$ of Calabi-Yau varieties of relative dimension $n$ whose dual intersection complex is the given $(B, P)$ and whose slab gluing data are the given ${f_{ρ}}$ . The toric degeneration is unique up to formal isomorphism over $Spec C [[t]]$ .

The reconstruction is algorithmic: order by order in $t$ , the reconstruction at order $k$ is determined by the reconstruction at order $k - 1$ together with the data of the scattering diagram at order $k$ , which is built from the slab functions and the inductive consistency constraints by the Kontsevich-Soibelman scattering algorithm.

Structural proof outline. The full proof is 128 pages and is deferred to the originating Annals paper. We give the structural architecture of the argument — what each move is, not how to execute it — in four parts.

Move I (formal infinitesimal lift). The reconstruction is built inductively. At order $0$ , the construction reduces to the bare central fibre $X_{0}$ as the union of toric strata indexed by $P$ , glued via the prescribed combinatorial data. The order- $0$ result is straightforward: the toric pieces are toric varieties, the gluing is along toric strata, and the central fibre is well-defined as a reduced reducible union. The deformation-theoretic content begins at order $1$ : lift $X_{0}$ to a flat family over $Spec C [t] / (t^{2})$ realising the slab functions on the codimension-1 cells.

The order- $1$ lift exists because the slab functions $f_{ρ}$ encode first-order deformation data on each codimension-1 stratum, and the Calabi-Yau condition (vanishing relative canonical bundle) is satisfied by construction once the slab functions are taken to be of the prescribed form. The obstruction theory of Gross-Siebert (2010) shows that the order- $1$ lift is unobstructed when the slab functions are consistent at the joints.

Move II (the scattering algorithm). The inductive step from order $k - 1$ to order $k$ is the central technical content. At each order, the consistency of the formal lift around every joint (codimension-2 cell) of $P$ may fail unless additional gluing data — new walls in the scattering diagram — are added. The Kontsevich-Soibelman algorithm prescribes exactly which walls to add, in terms of a recursive procedure that scans the joints of $P$ at each order, identifies the deficits in consistency, and inserts new walls labelled by automorphisms compensating for the deficit. The algorithm terminates at each order (only finitely many walls are added at each order) and is uniquely determined by the input data.

The new walls produced at order $k$ may not lie on cells of $P$ — they can be arbitrary codimension-1 polyhedral subsets of $B ∖ Δ$ . The scattering diagram $S$ accumulates over all orders into an infinite collection of walls, but at each order only finitely many walls contribute.

Move III (consistency around joints). The Gross-Siebert theorem in (2010) and (2011) gives a precise consistency criterion at each joint: the composition of wall-crossing automorphisms around a joint of $P$ must equal the identity, computed modulo $t^{k + 1}$ at order $k$ . The scattering algorithm produces additional walls at each order so as to maintain this consistency. The proof of termination at each order, and of the uniqueness of the additional walls, is the heart of the Gross-Siebert 2011 argument; it relies on the Kato fan structure on the central fibre (introduced in Gross-Siebert 2010) and on the affine geometry of the singular locus $Δ$ .

Move IV (uniqueness up to formal isomorphism). The order- $k$ reconstruction is unique because any two formal lifts of $X_{k - 1}$ over $Spec C [t] / (t^{k + 1})$ differing only in scattering-diagram detail at order $k$ are isomorphic via a formal automorphism of $X_{k}$ supported in the scattering walls. The uniqueness at each order, combined with the inductive existence, yields uniqueness of the entire formal family $X \to Spec C [[t]]$ up to formal isomorphism. The construction of the isomorphism is a Čech-cohomology argument on a covering of $X_{k}$ by formal neighbourhoods of the joints, with the cocycle conditions following from the scattering consistency.

The full proof in Gross-Siebert 2011 is deferred here. $□$

Bridge. The Gross-Siebert reconstruction builds toward the theta-function basis of regular functions on the mirror Calabi-Yau (Gross-Hacking-Keel 2015) and appears again in the SYZ identification of the tropical manifold $B$ with the symplectic SYZ base. The foundational reason that the reconstruction is exact is the algebro-geometric Kato-fan structure on the central fibre, which records the combinatorial gluing data without analytic obstructions; this is exactly the bridge from the symplectic SYZ heuristic (where T-duality on torus fibres is an analytic operation) to the algebraic-geometric mirror construction (where T-duality becomes the slab-function gluing). The central insight is that the formal-scheme deformation around the central fibre is controlled by combinatorial data alone, and the scattering algorithm generalises the Kontsevich-Soibelman wall-crossing formula from a heuristic to a constructive procedure. Putting these together with the enumerative input from the Nishinou-Siebert correspondence 04.12.06 identifies the slab functions with tropical-disk count generating series, completing the bridge from the Mikhalkin correspondence 04.12.05 to the full mirror-symmetry programme.

Exercises [Intermediate+]

Exercise 1 (easy, numeric).

A tropical manifold $(B, P)$ has integral affine structure with monodromy around its singular locus $Δ$ . For a simple focus-focus singularity in dimension $n = 2$ , the local monodromy is the integer matrix $(1011)$ . What is the determinant of this monodromy matrix?

Hint

Compute the determinant of the matrix directly.

Answer

1. The determinant of $(1011)$ is $1 \cdot 1 - 1 \cdot 0 = 1$ . Monodromy matrices on integral affine manifolds with singularities take values in $SL_{n} (Z)$ (volume-preserving): the determinant is always $\pm 1$ , and for orientation-preserving monodromy on Calabi-Yau bases the determinant is $+ 1$ . The focus-focus matrix is the simplest non-identity unipotent element of $SL_{2} (Z)$ and is the canonical local model for codimension-2 singularities of integral affine structures arising in dimension 2.

Exercise 3 (medium, symbolic).

A slab function on a codimension-1 cell $ρ$ of a tropical manifold takes the form $f_{ρ} = 1 + c_{1} z^{m_{1}} + c_{2} z^{m_{2}} + \dots + c_{r} z^{m_{r}}$ , where $z^{m_{i}}$ are monomials in the toric coordinates of the adjacent strata with $m_{i} \in N$ , and $c_{i} \in C [[t]]$ are formal-series coefficients. The constant term $1$ is forced by the Calabi-Yau condition (vanishing relative canonical bundle). State why the constant term must equal $1$ and write the simplest non-degenerate slab function with one extra monomial.

Hint

The slab function's constant term records the leading-order behaviour of the toric coordinate change across the codimension-1 stratum; setting it to 1 normalises the change to be the identity at $t = 0$ .

Answer

Why the constant term is 1. The slab function $f_{ρ}$ records the gluing automorphism of the toric coordinates on the two adjacent strata of $X_{0}$ across $ρ$ . At order $t^{0}$ , the gluing must reduce to the identity — the two strata meet along the codimension-1 toric stratum $X_{ρ} \subset X_{0}$ , and across this stratum the coordinates extend without ambiguity. Setting $f_{ρ} ∣_{t = 0} = 1$ encodes this normalisation. Higher-order corrections to the constant term would change the central fibre, contradicting the prescribed shape of $X_{0}$ .

Simplest non-degenerate slab function. Take $r = 1$ , $m_{1} = e_{1} \in N$ a primitive lattice vector, and $c_{1} = t$ : $$ f_\rho = 1 + t \cdot z^{e_1}. $$ This is the slab function for the simplest non-degenerate gluing, encoding a first-order deformation across $ρ$ in the direction of the primitive lattice vector $e_{1}$ . It is the slab function for the elliptic curve mirror discussed in the Beginner Worked Example (with $z = z^{e_{1}}$ the single coordinate).

Exercise 4 (medium, numeric).

The Gross-Siebert reconstruction operates on tropical manifolds with simple singularities. In real dimension $n = 2$ , the simple singularities are exactly the focus-focus singularities. How many real dimensions does the singular locus $Δ$ of an integral affine manifold of dimension $n = 2$ have?

Hint

Singular loci of integral affine structures live in codimension 2.

Answer

0. The singular locus $Δ$ of an integral affine manifold with singularities is of codimension 2 in $B$ . For $n = 2$ , codimension 2 is $2 - 2 = 0$ — the singular locus is a discrete set of points. Focus-focus singularities are isolated points in a real surface; the monodromy around a focus-focus point is the unipotent $SL_{2} (Z)$ matrix $(1011)$ from Exercise 1. For $n = 3$ (the case of Calabi-Yau threefolds), the singular locus has real dimension $3 - 2 = 1$ — a one-dimensional graph of singular curves in the threefold base. The graph structure of $Δ$ in dimension 3 is the substantive combinatorial input of the Gross-Siebert reconstruction for Calabi-Yau threefolds.

Exercise 5 (medium, symbolic).

State the consistency condition that slab functions ${f_{ρ}}$ on a tropical manifold $(B, P)$ must satisfy in order for the reconstruction theorem to apply. Explain in words why the consistency condition takes the form of a product of wall-crossing automorphisms around a joint equalling the identity.

Hint

A joint is a codimension-2 cell; the slab functions on the codimension-1 cells incident to a joint each give a wall-crossing automorphism. Taking the product of these automorphisms in cyclic order around the joint should give the identity, otherwise the reconstruction at the joint fails to lift past first order.

Answer

Consistency condition. Let $τ$ be a joint (codimension-2 cell) of $P$ , and let $ρ_{1}, ρ_{2}, \dots, ρ_{k}$ be the codimension-1 cells of $P$ incident to $τ$ , listed in cyclic order around $τ$ . Each $f_{ρ_{i}}$ defines a wall-crossing automorphism $θ_{ρ_{i}}$ — a formal automorphism of the local ring of $X_{0}$ near $τ$ , given by an exponential of an integer combination of the monomials in $f_{ρ_{i}}$ . The consistency condition is $$ \theta_{\rho_1} \circ \theta_{\rho_2} \circ \cdots \circ \theta_{\rho_k} = \text{id} $$ as formal automorphisms of $C [[t]] [N_{τ}^{\lor}]$ , the local ring near the joint $τ$ .

Why this form. The product of wall-crossing automorphisms around a joint records the holonomy of the formal-scheme structure around $τ$ . If the product is not the identity, the formal-scheme structure on the central fibre $X_{0}$ is not consistent at $τ$ — the local ring of $X_{0}$ near $τ$ has a non-identity monodromy that obstructs the existence of the order- $1$ lift. The identity condition is the cocycle condition for a formal-scheme gluing; it generalises the standard cocycle condition for sheaves on a polyhedral covering. When the slab functions do not satisfy the identity condition, the Kontsevich-Soibelman scattering algorithm adds new walls at each successive order in $t$ to compensate, until the consistency condition is restored at every order modulo $t^{k + 1}$ .

Exercise 6 (medium, multiple choice).

The Gross-Siebert reconstruction theorem reconstructs a toric degeneration of Calabi-Yau varieties. Why is the family a toric degeneration specifically — what restricts the central fibre to be a union of toric pieces?

A. Because the input $(B, P)$ is a polyhedral object, and polyhedra naturally index toric varieties.
B. Because the Calabi-Yau condition forces the central fibre to be toric.
C. Because the reconstruction theorem is only stated for toric ambient varieties.
D. Because the inverse-limit completion at $t = 0$ collapses any non-toric structure.

Hint

The toric structure on the central fibre comes from the polyhedral structure of $P$ , not from a Calabi-Yau-specific constraint.

Answer

A. The polyhedral decomposition $P$ of the tropical manifold $B$ has cells which are lattice polytopes, and each lattice polytope $σ \in P$ indexes a toric variety $X_{σ}$ via the standard polytope-to-toric-variety construction (the toric variety with Newton polytope $σ$ ). The central fibre $X_{0}$ is built as the union of these toric pieces glued along their toric strata, and is therefore a reduced union of toric varieties by construction. Option B is wrong: the Calabi-Yau condition restricts the generic fibre and the slab functions, not the central-fibre shape. Option C confuses Gross-Siebert (which reconstructs the central fibre, with toric pieces) with statements about ambient toric varieties (a separate concept). Option D misidentifies the source of the toric structure — the limit $t \to 0$ is a flat degeneration that preserves the polyhedral indexing, not a process that produces toric structure where none existed.

Exercise 7 (hard, symbolic).

The Gross-Siebert reconstruction uses the Kontsevich-Soibelman scattering algorithm to add walls to the scattering diagram $S$ at each order in $t$ . State the structural form of the scattering algorithm: given the scattering diagram $S_{k - 1}$ at order $k - 1$ satisfying consistency modulo $t^{k}$ , describe what walls are added to produce $S_{k}$ satisfying consistency modulo $t^{k + 1}$ .

Hint

The algorithm scans every joint, computes the deficit at order $t^{k}$ (the failure of the product of wall-crossings to be the identity), and adds a single new wall per joint to compensate the deficit — emanating from the joint and supported in a specific direction determined by the deficit's monomial content.

Answer

Statement of the scattering algorithm (Kontsevich-Soibelman 2006; Gross-Siebert 2011 §3). Suppose $S_{k - 1}$ is a scattering diagram on $(B, P)$ satisfying consistency modulo $t^{k}$ at every joint $τ \in P^{(n - 2)}$ . To produce $S_{k}$ :

For each joint $τ$ , compute the product of wall-crossing automorphisms around $τ$ modulo $t^{k + 1}$ . By the order- $(k - 1)$ consistency, this product is congruent to the identity modulo $t^{k}$ , but may differ from the identity at order $t^{k}$ by an automorphism $$ \theta_\tau \equiv \exp\left(\sum_j a_{\tau, j} t^k z^{m_{\tau, j}} \partial_{z^{m_{\tau, j}}}\right) \pmod{t^{k+1}} $$ for some integer combination of monomials $z^{m_{τ, j}}$ with primitive lattice directions $m_{τ, j} \in N_{τ}^{*}$ .

For each non-zero summand $a_{τ, j} t^{k} z^{m_{τ, j}}$ in the deficit, add a new wall $d_{τ, j}$ to $S$ : a codimension-1 polyhedral subset of $B ∖ Δ$ emanating from $τ$ in the direction $m_{τ, j}^{⊥}$ , labelled with the wall-crossing automorphism $θ_{τ, j}^{- 1}$ that compensates the deficit $a_{τ, j} t^{k} z^{m_{τ, j}}$ .

The new scattering diagram $S_{k} := S_{k - 1} \cup {d_{τ, j}}_{τ, j}$ then satisfies consistency modulo $t^{k + 1}$ at every joint. The walls added at order $k$ are finite in number (the deficit at each joint has finitely many monomial summands), and the new walls themselves can be joints at higher orders, triggering further wall additions at subsequent orders. Termination at any fixed order is guaranteed; the infinite scattering diagram is the inverse limit.

The algorithm is uniquely determined by the input: at each order, the deficits are uniquely computed, and the compensating walls are uniquely prescribed by the requirement that they emanate from the joint in the direction of the deficit's monomial. This uniqueness is what makes the reconstruction theorem produce a unique formal family.

Exercise 8 (hard, symbolic).

State the enumerative content of the Gross-Siebert reconstruction: the slab functions $f_{ρ}$ on the codimension-1 cells of $(B, P)$ can be expressed as generating series of tropical-disk counts on $B$ . Describe this enumerative interpretation and explain its origin via the Nishinou-Siebert correspondence 04.12.06.

Hint

The slab function $f_{ρ}$ is a formal-series Laurent polynomial whose coefficients enumerate tropical Maslov-index-2 disks ending on $ρ$ , by the Gross-Siebert ansatz. The Nishinou-Siebert correspondence identifies these tropical-disk counts with log Gromov-Witten invariants on the central fibre, giving an algebraic-geometric interpretation of the formal-series coefficients.

Answer

Enumerative content. The slab function $f_{ρ}$ on a codimension-1 cell $ρ$ of $(B, P)$ takes the form $$ f_\rho = 1 + \sum_{\beta \in H_2(\mathfrak{X}\rho, \mathbb{Z})} N\beta^{\text{trop}}(\rho) \cdot t^{\omega(\beta)} \cdot z^{\partial \beta}, $$ where $β$ ranges over relative-homology classes of tropical Maslov-index-2 disks ending on the codimension-1 stratum $X_{ρ} \subset X_{0}$ , $N_{β}^{trop} (ρ)$ is the count of tropical disks in $B$ with combinatorial type $β$ and ending on $ρ$ , $ω (β)$ is the symplectic area of the disk (a positive integer encoding the order in $t$ ), and $z^{\partial β} \in C [N_{ρ}]$ is the boundary monomial of the disk in the toric coordinates of $X_{ρ}$ .

Origin via Nishinou-Siebert. The tropical-disk counts $N_{β}^{trop} (ρ)$ are computed combinatorially on $(B, P)$ : enumerate balanced metric trees in $B$ with one unbounded leg ending on $ρ$ and prescribed combinatorial type $β$ , weighted by the Mikhalkin multiplicity generalised to higher dimension. The Nishinou-Siebert correspondence theorem 04.12.06 identifies these tropical counts with log Gromov-Witten invariants on the central fibre $X_{0}$ : the count of log-smooth holomorphic disks in $X_{ρ}$ with boundary on the codimension-2 strata of $X_{ρ}$ , ending on $ρ$ with the prescribed combinatorial profile. This identification gives the slab functions a precise enumerative meaning: the coefficients of $f_{ρ}$ count log Gromov-Witten holomorphic disks ending on $ρ$ , computed combinatorially by tropical enumeration.

The Calabi-Yau condition (vanishing relative canonical bundle) is the geometric reason that Maslov-index-2 disks are the relevant invariants: in a Calabi-Yau manifold, holomorphic disks with Maslov index 2 are the dimension-zero moduli, contributing point-counts to the slab functions. Disks with higher Maslov index would contribute higher-dimensional moduli and would not enter the slab function in the simple monomial form above.

The Gross-Hacking-Keel 2015 paper makes this enumerative interpretation completely explicit in the surface case $n = 2$ , identifying the slab functions with "broken-line counts" on the integral affine surface $B$ .

Lean formalization [Intermediate+]

The companion file lean/Codex/AlgGeom/Tropical/GrossSiebertReconstruction.lean records the statement of the reconstruction theorem and its load-bearing structural definitions as sorry-stubbed declarations, with proof bodies pending the full Mathlib infrastructure on formal-scheme deformation theory, log Gromov-Witten invariants, and integral affine manifolds with singularities.

First, a TropicalManifold structure: a pair $(B, P)$ consisting of an integral affine manifold $B$ with simple singularities along a codimension-2 locus, together with a polyhedral decomposition $P$ of $B$ into lattice polytopes. The structure records the integral affine charts, the monodromy data around the singular locus, and the polyhedral incidence relations. In the formalisation, the integral affine structure is encoded as a sheaf of $Aff (Z^{n})$ -valued transition functions on $B ∖ Δ$ .

Second, a Structure (in the Gross-Siebert sense) on a TropicalManifold: a collection of slab functions ${f_{ρ}}_{ρ \in P^{(n - 1)}}$ together with a scattering diagram of walls labelled by wall-crossing automorphisms, satisfying consistency around every joint. The consistency condition is recorded as a Prop-valued predicate on the collection.

Third, the statement of the reconstruction theorem as a sorry-stubbed declaration: given a TropicalManifold with consistent Structure, there exists a formal toric degeneration ToricDegenerationOfCY whose dual intersection complex is the given tropical manifold, and the formal toric degeneration is unique up to formal isomorphism. The full proof requires Kato fan theory, formal-scheme deformation around log-smooth central fibres, and the scattering algorithm — none of which are in Mathlib at present.

Fourth, the reduction to Mikhalkin-Nishinou-Siebert enumerative input: a sorry-stubbed equation expressing the slab functions as generating series of tropical-disk counts, identified via the Nishinou-Siebert correspondence with log Gromov-Witten invariants.

The combinatorial skeleton — tropical manifold, polyhedral decomposition, slab function as a formal Laurent series with constant term 1 — is formalisable in Lean today. The full theorem requires the Mathlib gap described in the frontmatter's lean_mathlib_gap.

Advanced results [Master]

Historical context — Gross-Siebert 2011 and its antecedents

In Gross-Siebert 2011 (Annals of Mathematics 174, pp. 1301-1428), the authors establish: a tropical manifold $(B, P)$ with simple singularities and consistent lifted gluing data lifts uniquely to a formal toric degeneration of Calabi-Yau varieties realising $(B, P)$ as its dual intersection complex. The paper carries the title "From real affine geometry to complex geometry" and runs to 128 pages; it is the published consolidation of a research programme Gross and Siebert had been developing since their 2003 joint announcement and pursued through three major papers — the 2006 Journal of Differential Geometry paper (log Calabi-Yau spaces and dual intersection complexes), the 2010 Journal of Algebraic Geometry paper (the $[B, P]$ paper on intrinsic log structures and Kato fans), and the 2011 Annals paper (the reconstruction theorem in final form).

The 2011 Annals paper has the explicit statement, restated here in its period-correct form: "Given a polarised tropical manifold $(B, P, φ)$ with simple singularities, equipped with lifted gluing data ${f_{ρ}}$ satisfying the joint-consistency conditions, there exists a unique-up-to-formal-isomorphism formal toric degeneration $X \to Spec C [[t]]$ whose dual intersection complex is $(B, P)$ and whose slab gluing recovers ${f_{ρ}}$ . The construction is by induction on the order in $t$ : the order- $k$ reconstruction $X_{k} \to Spec C [t] / (t^{k + 1})$ is built from $X_{k - 1}$ by the Kontsevich-Soibelman scattering algorithm, which adds at most finitely many new walls at order $k$ to maintain consistency at every joint of $P$ ." This formulation is the canonical Gross-Siebert statement; the modern reformulation in terms of log Gromov-Witten invariants and intrinsic log structures came later, after Abramovich-Chen 2014 and Chen 2014 established the log Gromov-Witten foundations.

The scope of the 2011 paper is conceptually narrower than the full Gross-Siebert programme: it proves the reconstruction theorem under simple-singularities hypotheses and for tropical manifolds with polarisations, leaving certain natural extensions (non-simple singularities, the unpolarised case, infinite-volume bases) to subsequent work. The narrow scope was necessary to make the consistency arguments at joints tractable; the broader extensions have followed in later work by Gross-Hacking-Keel (the surface case, 2015), Gross-Hacking-Keel-Siebert (the canonical-coordinates extension), and Mandel-Ruddat (specific examples).

The antecedent stack is well-defined. Mikhalkin 2005 (Journal of the American Mathematical Society 18) proved the tropical-classical correspondence for toric surfaces — the foundational identification of classical enumerative geometry with tropical-combinatorial counts. Nishinou-Siebert 2006 (Duke Mathematical Journal 135) extended Mikhalkin's correspondence to toric varieties of arbitrary dimension via toric degenerations, providing the enumerative input the Gross-Siebert reconstruction uses. Strominger-Yau-Zaslow 1996 (Nuclear Physics B 479) proposed the conjectural physical-geometric origin of mirror symmetry as T-duality on special-Lagrangian torus fibrations, which Gross-Siebert algebraised. Kontsevich-Soibelman 2001 (in Symplectic geometry and mirror symmetry) and Kontsevich-Soibelman 2006 (in The Unity of Mathematics) developed the non-archimedean SYZ proposal and the scattering-diagram formalism, complementary to the algebraic-geometric realisation Gross-Siebert pursued. By the time of Gross-Siebert 2011, the conjectural status of the SYZ programme had stabilised into a programme with multiple parallel realisations (non-archimedean analytic via Kontsevich-Soibelman, real-analytic via Gross-Wilson, algebro-geometric via Gross-Siebert); the reconstruction theorem closed the algebro-geometric branch with a constructive theorem rather than a conjecture.

The statement of the theorem — period-correct form

The Gross-Siebert reconstruction theorem in its 2011 Annals form requires a careful setup. We give the statement in three layers: the data, the hypotheses, and the conclusion.

Data. A polarised tropical manifold $(B, P, φ)$ where $B$ is an integral affine manifold of real dimension $n$ with singular locus $Δ \subset B$ of real dimension $n - 2$ , $P$ is a polyhedral decomposition of $B$ into lattice polytopes with $Δ \subset ⋃ P^{(n - 2)}$ , and $φ : B \to R$ is a strictly convex piecewise-affine function (the polarisation) on $B$ realising the polyhedral decomposition as the maximal-cell decomposition. The lifted gluing data is a collection ${f_{ρ}}$ of slab functions on the codimension-1 cells $ρ \in P^{(n - 1)}$ , each a formal Laurent polynomial of the form $f_{ρ} = 1 + \sum_{i} c_{ρ, i} z^{m_{ρ, i}}$ in the local toric coordinates with $c_{ρ, i} \in C [[t]]$ .

Hypotheses. (i) The singularities of $B$ at $Δ$ are simple: each component of $Δ$ has a standard local model of one of a small list of types (focus-focus in dimension 2; generic Type- $I$ and Type- $II$ singularities in dimension 3, as classified in Gross-Siebert 2006). (ii) The slab functions satisfy the joint-consistency conditions: for every joint $τ$ (codimension-2 cell of $P$ ), the product of wall-crossing automorphisms around $τ$ , computed from the slab functions on the codimension-1 cells incident to $τ$ , is the identity modulo $t$ . (iii) The polarisation $φ$ is non-degenerate: the cone over $B$ from the apex of $φ$ is a strongly convex polyhedral cone.

Conclusion. There exists a formal toric degeneration $π : X \to Spec C [[t]]$ of Calabi-Yau varieties of relative dimension $n$ such that:

(a) The central fibre $X_{0}$ has dual intersection complex $(B, P)$ , with each maximal cell $σ \in P^{(n)}$ corresponding to a toric component $X_{σ}$ of $X_{0}$ and each codimension- $k$ face of $P$ corresponding to a codimension- $k$ toric stratum of $X_{0}$ .

(b) The slab gluing data of $X_{0}$ — the local automorphism of the toric coordinates of the adjacent components across each codimension-1 stratum — recovers the prescribed slab functions ${f_{ρ}}$ .

(c) The formal toric degeneration is unique up to formal isomorphism over $Spec C [[t]]$ : any two such degenerations are isomorphic via a formal automorphism extending the identity on the central fibre.

(d) The construction is algorithmic: the order- $k$ reconstruction $X_{k}$ is uniquely determined by the order- $(k - 1)$ reconstruction and the scattering diagram at order $k$ , with the scattering diagram constructed by the Kontsevich-Soibelman algorithm.

This is the canonical period-correct statement. The modern reformulation, after Abramovich-Chen 2014 and Gross-Siebert 2013, recasts the slab functions as log Gromov-Witten generating series and the scattering diagram as a wall-crossing structure on the moduli of log-smooth degenerations, but the underlying theorem is unchanged.

Structural proof sketch — the architecture only

The full proof in Gross-Siebert 2011 is 128 pages; the proof is deferred. We give the architecture of the argument, identifying the four conceptual moves and their interdependencies. Detailed proof steps are not provided.

Move A (formal-infinitesimal lift; order 0 to order 1). The order- $0$ central fibre is built directly from the combinatorial data $(B, P)$ : each maximal cell $σ$ contributes a toric component $X_{σ}$ , and the gluing along codimension-1 strata is determined by the polyhedral incidences. The order- $1$ lift to $Spec C [t] / (t^{2})$ is built by applying the slab functions ${f_{ρ}}$ as first-order deformations of the codimension-1 gluings. The Calabi-Yau condition (vanishing relative canonical bundle) is automatic at this order if the slab functions have constant term 1. The existence of the order-1 lift uses Kato's logarithmic deformation theory of toric varieties and was proved in Gross-Siebert 2006.

Move B (the scattering algorithm; order $k - 1$ to order $k$ ). This is the technical heart. Suppose the order- $(k - 1)$ reconstruction $X_{k - 1}$ exists, satisfying consistency at every joint modulo $t^{k}$ . To produce $X_{k}$ satisfying consistency modulo $t^{k + 1}$ , the scattering algorithm scans every joint $τ$ , computes the deficit (the difference from the identity of the product of wall-crossings around $τ$ , modulo $t^{k + 1}$ ), and adds new walls to $S$ emanating from $τ$ to compensate the deficit. The new walls carry wall-crossing automorphisms of the form $θ = exp (a t^{k} z^{m} \partial_{z^{m}})$ where $a$ is the deficit coefficient and $m$ is the monomial direction.

The termination of the algorithm at each fixed order $k$ requires controlling the number of walls produced: the deficit at each joint has finitely many monomial components, and each component produces one new wall. Hence at each order finitely many walls are added; over all orders the scattering diagram is countably infinite. The construction is uniquely determined at each order, which is the key step in the uniqueness half of the theorem.

Move C (consistency around joints). The order- $k$ lift's consistency at joints relies on a delicate compatibility check: the wall-crossing automorphisms added at order $k$ at one joint must be compatible with the wall-crossing automorphisms added at neighbouring joints. The Gross-Siebert 2011 paper proves a "global consistency" theorem: the local additions at each joint, computed independently, are compatible globally, so that the scattering diagram $S_{k}$ extends consistently across all joints. The proof uses the Kato fan structure on the central fibre and a finite-dimensional cohomology vanishing computation at each order.

Move D (uniqueness via formal automorphism). The uniqueness half follows from a separate Čech-cohomology argument. Any two order- $k$ lifts of $X_{k - 1}$ that satisfy the prescribed slab gluing are isomorphic via a formal automorphism supported in the scattering walls: the difference of the two lifts is recorded by a 1-cocycle on the covering of $X_{k}$ by formal neighbourhoods of the joints, and the cocycle is shown to be a coboundary by the joint-consistency condition. The argument is purely formal-scheme-theoretic and uses no analytic input.

The interdependencies are: Move A is the base case, Moves B and C run inductively in tandem at each order, and Move D establishes the uniqueness once each order's reconstruction is in place. The full proof in Gross-Siebert 2011 occupies §3-§8 and is deferred here.

Reconstruction via scattering — outline

The scattering diagram $S$ of the reconstruction is itself an object of interest. Its structure encodes the entire formal family $X$ via the slab functions on its walls. The scattering construction has three layers.

Layer 1 (initial walls). The initial scattering diagram $S_{0}$ consists of the codimension-1 cells $ρ \in P^{(n - 1)}$ , each carrying the prescribed slab function $f_{ρ}$ as its wall-crossing automorphism. The initial diagram corresponds to the order- $0$ scattering data: the gluing of the toric pieces of $X_{0}$ .

Layer 2 (scattered walls at each order). At order $k$ , the Kontsevich-Soibelman algorithm adds new walls emanating from the joints of $P$ — but the new walls may not lie on cells of $P$ . They are arbitrary codimension-1 polyhedral subsets of $B ∖ Δ$ , characterised by their initial points (the joints they emanate from) and their primitive directions (determined by the deficits at the joints). The cumulative scattering diagram $S = ⋃_{k} S_{k}$ is countably infinite.

Layer 3 (consistency around joints). The defining property of the scattering diagram is the joint-consistency: at every joint $τ$ , the product of wall-crossing automorphisms around $τ$ — taken over all walls (initial or scattered) incident to $τ$ — is the identity. This consistency is automatic at every order by construction (the new walls at each order are added precisely to cancel the deficit), and the limit consistency is a tautological consequence.

The scattering diagram is a Berkovich-non-archimedean-analytic object in the Kontsevich-Soibelman formulation, and an algebraic-geometric Kato-fan structure in the Gross-Siebert formulation. The two viewpoints are equivalent for the reconstruction problem; the Gross-Siebert 2011 paper uses the algebro-geometric viewpoint to prove the reconstruction, while subsequent work (Mandel-Ruddat, Gross-Hacking-Keel-Siebert) has translated the construction back into the non-archimedean analytic setting for specific worked examples.

Connections to mirror symmetry — the wider programme

The reconstruction theorem is the central technical move of the Gross-Siebert programme for mirror symmetry. The wider programme assembles three additional ingredients on top of the reconstruction.

Theta functions (Gross-Hacking-Keel 2015 Publ. IHÉS 122). The slab functions and scattering diagram of the reconstruction encode not only the toric degeneration of the Calabi-Yau but also a canonical $Z$ -basis ${ϑ_{m}}_{m \in B (Z)}$ of the algebra of regular functions on the generic fibre $X_{η}$ , indexed by integral points of the base $B$ . These are the theta functions of the polarised tropical manifold; they generalise the lattice-point basis $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} χ^{m}$ of the smooth-toric case to the degenerate Calabi-Yau case. The theta functions are constructed by "broken-line counts" on $B$ : a broken line is a piecewise-linear path in $B$ from a fixed initial point to $m$ , with breaks at scattering walls and lifts of the broken line to multiplicities encoded by the wall-crossing automorphisms.

Mirror map and periods (Gross-Siebert programme, surveyed in Auroux 2009). The reconstructed family $X \to Spec C [[t]]$ carries a relative holomorphic volume form $Ω_{t}$ , and the periods $\int_{γ_{i}} Ω_{t}$ along a basis of cycles $γ_{i} \in H_{n} (X_{η}, Z)$ satisfy a Picard-Fuchs equation whose leading-order behaviour is computed by tropical-disk counts on the A-side mirror. The mirror map is the change-of-coordinates relating the formal parameter $t$ on the B-side (algebraic-geometric reconstruction) to the symplectic parameter (Kähler class) on the A-side. The mirror map is read off from the leading-order period via a standard calculation in Hodge theory.

Homological mirror conjecture connection (Kontsevich 1994; Auroux 2009 survey). The reconstructed family $X$ has a derived category $D^{b} (X)$ that, conjecturally, is equivalent to the Fukaya category of the A-side mirror. Gross-Siebert do not prove this conjecture, but their construction is consistent with it: the theta-function basis on $X$ corresponds to specific Lagrangian objects in the Fukaya category, and the wall-crossing automorphisms in $S$ correspond to symplectic isotopies of the Lagrangians.

Synthesis. The Gross-Siebert reconstruction is the foundational reason that mirror symmetry has an algebro-geometric constructive proof, not just a string-theoretic prediction. The central insight is that the tropical manifold $(B, P)$ with its slab data is a complete combinatorial encoding of a Calabi-Yau degeneration, and the scattering algorithm assembles this combinatorial data into a formal family with no remaining choices. This is exactly the bridge from the symplectic SYZ heuristic (where T-duality on torus fibres is an analytic operation on a real manifold) to the algebraic-geometric mirror construction (where T-duality becomes the slab-function gluing on a formal scheme). Putting these together with the Mikhalkin and Nishinou-Siebert correspondences identifies the slab functions with tropical-disk count generating series and identifies the theta functions with broken-line counts on $B$ , completing the bridge from the Mikhalkin enumerative result to the full mirror-symmetry programme. The reconstruction generalises Batyrev's mirror duality (Batyrev 1994) — which works for reflexive polytopes and the toric case — to the full Calabi-Yau case with substantive singular locus, and the pattern recurs in the Gross-Hacking-Keel theta-function story for surfaces and in the Gross-Hacking-Keel-Siebert canonical-coordinates extension.

Full proof set [Master]

The Gross-Siebert reconstruction proper is deferred. We collect three load-bearing structural propositions whose proofs lie within scope and feed the architecture of the main theorem.

Proposition 1 (existence of the order-0 central fibre). Let $(B, P)$ be a tropical manifold of dimension $n$ with simple singularities, and let ${f_{ρ}}_{ρ \in P^{(n - 1)}}$ be a collection of slab functions with constant term 1. Then the order- $0$ central fibre $X_{0}$ , defined as the reduced reducible union $⋃_{σ \in P^{(n)}} X_{σ}$ glued along the codimension- $k$ toric strata indexed by codimension- $k$ cells of $P$ , is a well-defined reduced scheme.

Proof. For each maximal cell $σ \in P^{(n)}$ , the toric variety $X_{σ}$ is the toric variety with Newton polytope $σ$ (equivalently, the toric variety whose fan is the normal fan of $σ$ ). The gluing of $X_{σ}$ and $X_{σ^{'}}$ along their common codimension-1 stratum corresponds to the codimension-1 cell $ρ = σ \cap σ^{'} \in P^{(n - 1)}$ . The slab function $f_{ρ}$ has constant term 1, so at order $0$ the gluing along $ρ$ reduces to the identity on the toric stratum $X_{ρ}$ . Hence the codimension-1 gluings are well-defined and the central fibre is the union of the $X_{σ}$ along the prescribed toric strata.

Reducedness: each $X_{σ}$ is a normal toric variety, hence reduced; the gluings along reduced strata preserve reducedness; the union is therefore a reduced scheme. The Calabi-Yau condition at order $0$ (vanishing canonical class on $X_{0}$ ) is automatic for a union of toric varieties glued along the codimension-1 strata, by a standard Cartier divisor calculation: the relative dualising sheaf of $X_{0}$ restricted to each $X_{σ}$ is identity-class (since each $X_{σ}$ is Gorenstein toric with its canonical divisor a sum of toric boundary divisors, and the gluing accounts for the boundary contributions). $□$

Proposition 2 (joint-consistency at order 1 is necessary for the order-1 lift). Suppose ${f_{ρ}}$ admits an order- $1$ lift $X_{1} \to Spec C [t] / (t^{2})$ recovering the prescribed slab data. Then for every joint $τ \in P^{(n - 2)}$ , the product of wall-crossing automorphisms around $τ$ computed from ${f_{ρ}}_{ρ \supset τ}$ is the identity modulo $t$ .

Proof. Suppose $X_{1}$ exists. The local structure of $X_{1}$ near the toric stratum $X_{τ}$ corresponding to a joint $τ$ is encoded by a formal-scheme gluing of the formal neighbourhoods of $X_{τ}$ in the components ${X_{σ}}_{σ \supset τ}$ via the slab functions on the codimension-1 cells ${ρ \supset τ}$ .

The cocycle condition for this gluing is: the composition of the formal-coordinate-change automorphisms along a closed loop around $τ$ — taking the loop $σ_{1} \to ρ_{1} \to σ_{2} \to ρ_{2} \to \dots \to σ_{k} \to ρ_{k} \to σ_{1}$ where the $σ_{i}$ are the maximal cells incident to $τ$ and the $ρ_{i}$ are the codimension-1 cells separating them — is the identity. Each transition $σ_{i} \to σ_{i + 1}$ across $ρ_{i}$ contributes the wall-crossing automorphism $θ_{ρ_{i}}$ derived from $f_{ρ_{i}}$ . Hence the cocycle condition reads $$ \theta_{\rho_1} \circ \theta_{\rho_2} \circ \cdots \circ \theta_{\rho_k} = \text{id} \pmod{t}. $$ If the cocycle condition fails modulo $t$ , the formal-scheme gluing is inconsistent and $X_{1}$ does not exist. $□$

Proposition 3 (the scattering algorithm produces unique additional walls). Let $S_{k - 1}$ be a scattering diagram on $(B, P)$ satisfying joint consistency modulo $t^{k}$ . The Kontsevich-Soibelman scattering algorithm produces walls ${d_{τ, j}}$ to add to $S_{k - 1}$ such that the augmented diagram $S_{k}$ satisfies joint consistency modulo $t^{k + 1}$ , and the additional walls are uniquely determined by the deficits at each joint modulo $t^{k + 1}$ .

Proof. By the order- $(k - 1)$ consistency, the product of wall-crossing automorphisms around each joint $τ$ is the identity modulo $t^{k}$ . The deviation from the identity at order $t^{k}$ is the deficit at $τ$ , a wall-crossing-like automorphism of the form $θ_{τ} \equiv exp (\sum_{j} a_{τ, j} t^{k} z^{m_{τ, j}} \partial_{z^{m_{τ, j}}}) (mod t^{k + 1})$ for finitely many integer coefficients $a_{τ, j}$ and primitive lattice directions $m_{τ, j} \in N_{τ}^{*}$ .

For each non-zero summand of the deficit, the algorithm prescribes a single wall $d_{τ, j}$ emanating from $τ$ in the direction $m_{τ, j}^{⊥}$ and carrying the inverse wall-crossing automorphism $θ_{τ, j}^{- 1}$ that exactly cancels the deficit summand. Adding all such walls to $S_{k - 1}$ produces $S_{k}$ with the deficit at every joint equal to the identity modulo $t^{k + 1}$ , hence consistency at the new order.

Uniqueness: any two collections of walls ${d_{τ, j}}$ and ${d_{τ, j}^{'}}$ that restore consistency modulo $t^{k + 1}$ differ by walls whose composition is the identity at order $t^{k}$ . By the wall-crossing automorphism convention (each wall carries a specific direction-and-coefficient pair determined by the deficit), the difference walls must themselves be vanishing (zero-coefficient), so the two collections agree. The additional walls are uniquely prescribed by the deficits. $□$

These three propositions are the architectural framework for the four moves of the structural proof sketch. The remaining content of the Gross-Siebert 2011 proof — global consistency across all joints simultaneously at each order, the Čech-cohomology argument for uniqueness up to formal isomorphism, the inverse-limit completion to the full formal family — is deferred to the originating paper.

Connections [Master]

Toric degeneration of a Calabi-Yau variety 04.12.07. The reconstruction theorem reconstructs a toric degeneration of Calabi-Yau varieties. The toric-degeneration unit supplies the precise definition of the output: a flat family $π : X \to Spec C [[t]]$ with smooth Calabi-Yau generic fibre, central fibre a union of toric varieties indexed by a polyhedral decomposition, and maximal unipotent monodromy at the central fibre. The reconstruction theorem inverts the dual-intersection-complex construction (which goes from a toric degeneration to a tropical manifold) by going from a tropical manifold back to a toric degeneration. The two units are inverse operations in the dictionary algebraic-geometric ↔ tropical.
Dual intersection complex; tropical manifold $B$ 04.12.08. The dual-intersection-complex unit defines the tropical manifold $(B, P)$ as a polyhedral combinatorial object built from a toric degeneration. The reconstruction theorem closes the loop: given a tropical manifold with the right combinatorial data, there is a unique toric degeneration realising it. The reciprocal cross-link is structural — the dual-intersection-complex construction is the input of the reconstruction theorem, and the reconstruction is the inverse of the dual-intersection-complex construction.
Slab function and structure of a tropical manifold 04.12.11. The slab functions on a tropical manifold are the lifted gluing data the reconstruction theorem uses. The slab-function unit supplies the precise definition of $f_{ρ}$ , the consistency conditions, and the wall-crossing automorphism each slab function defines. The reconstruction theorem reads off these slab functions as the order- $0$ content of the full scattering diagram, with higher-order content added by the Kontsevich-Soibelman algorithm. The slab-function unit is the prerequisite vocabulary for the reconstruction; the reconstruction is the existence-and-uniqueness statement that gives the slab-function data its constructive meaning.
Nishinou-Siebert correspondence 04.12.06. The Nishinou-Siebert correspondence supplies the enumerative content of the slab functions: their coefficients are tropical-disk counts on $(B, P)$ , identified with log Gromov-Witten invariants on the central fibre via the Nishinou-Siebert theorem. The reconstruction theorem assembles these enumerative coefficients into the formal Calabi-Yau family, so the Nishinou-Siebert correspondence is the foundational link from tropical-classical enumeration to the slab-function data the reconstruction needs.
Mikhalkin's correspondence theorem 04.12.05. The Mikhalkin correspondence is the dimension-2 prototype of the tropical-classical correspondence. The reconstruction theorem in dimension 2 (for tropical surfaces with focus-focus singularities) reconstructs K3 surfaces and Enriques surfaces, with the slab functions matching the Mikhalkin tropical-curve counts via the Nishinou-Siebert identification. Mikhalkin's correspondence is the foundational enumerative result that motivates the entire Gross-Siebert programme.
Fan and the toric variety $X_{Σ}$ 04.11.04. The polyhedral cells $σ \in P^{(n)}$ of the tropical manifold index toric varieties $X_{σ}$ via the standard fan-and-toric-variety construction. The reconstruction's central fibre $X_{0} = ⋃_{σ} X_{σ}$ is a union of these toric varieties glued along their toric strata. The fan-and-toric-variety unit supplies the construction of each $X_{σ}$ from the cell $σ$ ; the reconstruction theorem builds on this to produce the full formal family. The dictionary is: cells of $P \leftrightarrow$ toric strata of $X_{0}$ , with the dimension reversed (codimension- $k$ cell of $P$ ↔ codimension- $k$ stratum of $X_{0}$ ).
Period integral and the mirror map (pointer) 04.12.13. The Picard-Fuchs operator of the reconstructed Calabi-Yau family inherits the slab-function coefficients of the present unit's structure $S$ , and the higher-order mirror-map coefficients $r_{k}$ are read off as tropical-disk counts on $(B, P)$ . The pointer unit [04.12.13] records the period-integral / mirror-map apparatus on the smoothing produced by the present reconstruction theorem; the reconstruction is the constructive content of the Candelas-de la Ossa-Green-Parkes 1991 prediction in the Gross-Siebert setting.
Log Gromov-Witten invariants (pointer) 04.12.15. The reconstruction theorem of the present unit consumes log Gromov-Witten counts as the enumerative input to its scattering-diagram construction: the slab functions $f_{ρ}$ and the wall functions are determined by log GW counts of curves on the central fibre $X_{0}$ with prescribed contact orders along the singular locus of the dual intersection complex. The pointer unit [04.12.15] records the log GW infrastructure; the present unit applies it to build the mirror Calabi-Yau via order-by-order smoothing.

Historical & philosophical context [Master]

Gross-Siebert 2011 Annals of Mathematics 174, 1301-1428 ^{[GrossSiebert2011]} established the reconstruction theorem as the central technical result of the Gross-Siebert programme for mirror symmetry, a programme the authors had been developing since their 2003 joint announcement. The 2011 paper carries the title "From real affine geometry to complex geometry" and consolidates the constructive content of two earlier major papers: Gross-Siebert 2006 Journal of Differential Geometry 72, 169-338 ^{[GrossSiebert2006]} introducing log Calabi-Yau spaces and the dual intersection complex, and Gross-Siebert 2010 Journal of Algebraic Geometry 19, 679-780 ^{[GrossSiebert2010]} introducing intrinsic log structures on the central fibre via Kato fans (the so-called $[B, P]$ paper).

The conceptual lineage runs from Strominger-Yau-Zaslow 1996 Nuclear Physics B 479, 243-259 ^[SYZ1996], whose conjecture that mirror Calabi-Yau pairs admit dual special-Lagrangian torus fibrations over a common real base furnished the geometric intuition. Kontsevich-Soibelman 2001 Symplectic geometry and mirror symmetry 203-263 ^{[KontsevichSoibelman2001]} and Kontsevich-Soibelman 2006 The Unity of Mathematics 321-385 ^{[KontsevichSoibelman2006]} proposed the non-archimedean SYZ realisation, introducing scattering diagrams and wall-crossing as the algorithmic content of the construction. Gross-Wilson 2000 J. Differential Geom. 55 supplied the differential-geometric SYZ realisation in the K3 case. By 2006-2011 the Gross-Siebert programme had emerged as the algebraic-geometric branch of this triad, providing a constructive proof of mirror symmetry via the reconstruction theorem rather than a heuristic identification via T-duality or non-archimedean analytic skeleta.

The technical antecedents on the tropical side run from Mikhalkin 2005 J. Amer. Math. Soc. 18, 313-377 ^{[Mikhalkin2005]} (the tropical-classical correspondence for toric surfaces) through Nishinou-Siebert 2006 Duke Mathematical Journal 135, 1-51 ^{[NishinouSiebert2006]} (the higher-dimensional toric generalisation) and culminate in the enumerative interpretation of the slab functions as tropical-disk count generating series. The log Gromov-Witten foundations were completed by Abramovich-Chen 2014 Asian J. Math. 18 ^{[AbramovichChen2014]}, Chen 2014 Annals of Math. 180 ^[Chen2014], and Gross-Siebert 2013 J. Amer. Math. Soc. 26 ^{[GrossSiebert2013Log]}, giving the modern algebro-geometric framework in which the slab functions live.

Gross 2011 Tropical Geometry and Mirror Symmetry (CBMS Regional Conference Series 114) ^{[Gross2011CBMS]} is the textbook consolidation of the reconstruction theorem and its surrounding programme; Lecture 5 of the CBMS lectures is the canonical exposition. The Gross-Hacking-Keel 2015 Publ. IHÉS 122 paper ^{[GrossHackingKeel2015]} worked out the surface case in complete detail and identified the theta functions as the canonical $Z$ -basis on the reconstructed mirror, completing the enumerative interpretation in dimension 2. The Auroux 2009 Surveys in Differential Geometry 13 ^[Auroux2009] survey gives the canonical modern overview, situating the reconstruction inside the wider symplectic and Floer-theoretic mirror-symmetry landscape.

Bibliography [Master]

@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}
}

@article{GrossSiebert2006,
  author  = {Gross, Mark and Siebert, Bernd},
  title   = {Mirror symmetry via logarithmic degeneration data {I}},
  journal = {Journal of Differential Geometry},
  volume  = {72},
  number  = {2},
  year    = {2006},
  pages   = {169--338}
}

@article{GrossSiebert2010,
  author  = {Gross, Mark and Siebert, Bernd},
  title   = {Mirror symmetry via logarithmic degeneration data {II}},
  journal = {Journal of Algebraic Geometry},
  volume  = {19},
  number  = {4},
  year    = {2010},
  pages   = {679--780}
}

@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}
}

@incollection{KontsevichSoibelman2001,
  author    = {Kontsevich, Maxim and Soibelman, Yan},
  title     = {Homological mirror symmetry and torus fibrations},
  booktitle = {Symplectic geometry and mirror symmetry (Seoul 2000)},
  publisher = {World Scientific},
  year      = {2001},
  pages     = {203--263}
}

@incollection{KontsevichSoibelman2006,
  author    = {Kontsevich, Maxim and Soibelman, Yan},
  title     = {Affine structures and non-{A}rchimedean analytic spaces},
  booktitle = {The Unity of Mathematics},
  series    = {Progress in Mathematics},
  volume    = {244},
  publisher = {Birkh{\"a}user},
  year      = {2006},
  pages     = {321--385}
}

@article{SYZ1996,
  author  = {Strominger, Andrew and Yau, Shing-Tung and Zaslow, Eric},
  title   = {Mirror symmetry is {T}-duality},
  journal = {Nuclear Physics B},
  volume  = {479},
  number  = {1-2},
  year    = {1996},
  pages   = {243--259}
}

@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{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{Auroux2009,
  author  = {Auroux, Denis},
  title   = {Special {L}agrangian fibrations, wall-crossing, and mirror symmetry},
  journal = {Surveys in Differential Geometry},
  volume  = {13},
  year    = {2009},
  pages   = {1--47}
}

@article{GrossHackingKeel2015,
  author  = {Gross, Mark and Hacking, Paul and Keel, Sean},
  title   = {Mirror symmetry for log {C}alabi-{Y}au surfaces {I}},
  journal = {Publications math{\'e}matiques de l'IH{\'E}S},
  volume  = {122},
  number  = {1},
  year    = {2015},
  pages   = {65--168}
}

@article{AbramovichChen2014,
  author  = {Abramovich, Dan and Chen, Qile},
  title   = {Stable logarithmic maps to {D}eligne-{F}altings 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 {D}eligne-{F}altings pairs {I}},
  journal = {Annals of Mathematics},
  volume  = {180},
  number  = {2},
  year    = {2014},
  pages   = {455--521}
}

@article{GrossSiebert2013Log,
  author  = {Gross, Mark and Siebert, Bernd},
  title   = {Logarithmic {G}romov-{W}itten invariants},
  journal = {Journal of the American Mathematical Society},
  volume  = {26},
  number  = {2},
  year    = {2013},
  pages   = {451--510}
}

@article{Batyrev1994,
  author  = {Batyrev, Victor},
  title   = {Dual polyhedra and mirror symmetry for {C}alabi-{Y}au hypersurfaces in toric varieties},
  journal = {Journal of Algebraic Geometry},
  volume  = {3},
  year    = {1994},
  pages   = {493--535}
}

@article{Kato1989,
  author    = {Kato, Kazuya},
  title     = {Logarithmic structures of {F}ontaine-{I}llusie},
  booktitle = {Algebraic Analysis, Geometry, and Number Theory},
  publisher = {Johns Hopkins University Press},
  year      = {1989},
  pages     = {191--224}
}

@article{Candelas1991,
  author  = {Candelas, Philip and de la Ossa, Xenia and Green, Paul S. and Parkes, Linda},
  title   = {A pair of {C}alabi-{Y}au manifolds as an exactly soluble superconformal theory},
  journal = {Nuclear Physics B},
  volume  = {359},
  number  = {1},
  year    = {1991},
  pages   = {21--74}
}

Prerequisites

04.11.04
04.12.06

Tier anchors

beginner: Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 5 informal opening — reconstruct a Calabi-Yau from a combinatorial recipe
intermediate: Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 5; Gross-Siebert 2011 *Annals of Mathematics* 174, 1301-1428 §1-§4 (setup of $(B, \mathscr{P}, \mathscr{S})$ and statement)
master: Gross-Siebert 2011 *Annals of Mathematics* 174, 1301-1428 (originator — the published reconstruction theorem); Gross-Siebert 2006 *J. Differential Geometry* 72, 169-338 (precursor — log Calabi-Yau spaces and dual intersection complex); Gross-Siebert 2010 *Journal of Algebraic Geometry* 19, 679-780 (the $[B, \mathscr{P}]$ paper — Kato fan and intrinsic log structures); Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS Regional Conference Series 114, AMS) Lectures 4-6 (textbook consolidation); Kontsevich-Soibelman 2001 *Symplectic geometry and mirror symmetry* (World Scientific) and Kontsevich-Soibelman 2006 *The Unity of Mathematics* (Birkhäuser, 321-385); Auroux 2009 *Surv. Differ. Geom.* 13 (modern survey); Gross-Hacking-Keel 2015 *Publ. IHÉS* 122, 65-168 (theta-function payoff)

References

TODO_REF
Gross, M. & Siebert, B. — From real affine geometry to complex geometry · *Annals of Mathematics* (2) 174 (3), 1301-1428 (2011). The published reconstruction theorem: given a tropical manifold $(B, \mathscr{P})$ with simple singularities and lifted gluing data, there is a unique (up to formal isomorphism) toric degeneration of Calabi-Yau varieties realising $(B, \mathscr{P})$ as its dual intersection complex. §1-§2 (setup), §3-§4 (scattering algorithm and consistency), §5 (slab functions), §6-§7 (consistency around joints), §8 (uniqueness), §9 (the statement).
TODO_REF
Gross, M. & Siebert, B. — Mirror symmetry via logarithmic degeneration data I · *Journal of Differential Geometry* 72 (2), 169-338 (2006). The precursor: log Calabi-Yau spaces, the dual intersection complex of a toric degeneration, the integral affine structure with singularities, the polyhedral decomposition $\mathscr{P}$.
TODO_REF
Gross, M. & Siebert, B. — Mirror symmetry via logarithmic degeneration data II · *Journal of Algebraic Geometry* 19 (4), 679-780 (2010). The $[B, \mathscr{P}]$ paper: intrinsic log structures on the central fibre, Kato fan, the unobstructed-smoothing criterion.
TODO_REF
Gross, M. — Tropical Geometry and Mirror Symmetry · CBMS Regional Conference Series in Mathematics 114, American Mathematical Society (2011). Lecture 5 — the reconstruction theorem; Lecture 4 — toric degenerations; Lecture 6 — theta functions. The CBMS lectures consolidating the 2006/2010/2011 papers.
TODO_REF
Kontsevich, M. & Soibelman, Y. — Homological mirror symmetry and torus fibrations · In *Symplectic geometry and mirror symmetry* (Seoul 2000), World Scientific (2001), 203-263. The non-archimedean SYZ proposal that motivated the algebraic reconstruction picture; introduces the scattering / wall-crossing formalism in the mirror-symmetry setting.
TODO_REF
Kontsevich, M. & Soibelman, Y. — Affine structures and non-Archimedean analytic spaces · In *The Unity of Mathematics*, Progress in Mathematics 244, Birkhäuser (2006), 321-385. The Berkovich-analytic interpretation of the SYZ base and the scattering-diagram formalism; complementary to Gross-Siebert's algebraic-geometric realisation.
TODO_REF
Strominger, A., Yau, S.-T. & Zaslow, E. — Mirror symmetry is T-duality · *Nuclear Physics B* 479 (1-2), 243-259 (1996). The original SYZ conjecture: mirror Calabi-Yau pairs admit dual special-Lagrangian $T^n$-fibrations over a common integral-affine base. The conjectural physical input that Gross-Siebert algebraise.
TODO_REF
Nishinou, T. & Siebert, B. — Toric degenerations of toric varieties and tropical curves · *Duke Mathematical Journal* 135 (1), 1-51 (2006). The tropical-classical correspondence in arbitrary toric dimension, generalising Mikhalkin 2005. The enumerative input the Gross-Siebert reconstruction uses to read off slab functions from tropical-disk counts.
TODO_REF
Mikhalkin, G. — Enumerative tropical algebraic geometry in $\mathbb{R}^2$ · *Journal of the American Mathematical Society* 18 (2), 313-377 (2005). The originating dim-2 correspondence theorem; supplies the tropical multiplicity formula generalised in Nishinou-Siebert.
TODO_REF
Auroux, D. — Special Lagrangian fibrations, wall-crossing, and mirror symmetry · *Surveys in Differential Geometry* 13, International Press (2009), 1-47. The canonical modern survey of the SYZ-and-Gross-Siebert programme; situates the reconstruction theorem inside symplectic and Floer-theoretic mirror symmetry.
TODO_REF
Gross, M., Hacking, P. & Keel, S. — Mirror symmetry for log Calabi-Yau surfaces I · *Publications mathématiques de l'IHÉS* 122 (1), 65-168 (2015). Theta functions as the canonical $\mathbb{Z}$-basis of regular functions on the Gross-Siebert mirror; the surface case worked out in complete detail.
TODO_REF
Abramovich, D. & Chen, Q. — Stable logarithmic maps to Deligne-Faltings pairs II · *Asian Journal of Mathematics* 18 (3), 465-488 (2014). Log Gromov-Witten foundations on the central fibre of a toric degeneration; one of two parallel constructions (with Chen 2014 and Gross-Siebert 2013) underpinning the enumerative input of the reconstruction.
TODO_REF
Chen, Q. — Stable logarithmic maps to Deligne-Faltings pairs I · *Annals of Mathematics* (2) 180 (2), 455-521 (2014). Companion paper to Abramovich-Chen 2014; jointly with Gross-Siebert 2013 supplies the log Gromov-Witten theory.
TODO_REF
Gross, M. & Siebert, B. — Logarithmic Gromov-Witten invariants · *Journal of the American Mathematical Society* 26 (2), 451-510 (2013). The alternative log Gromov-Witten foundation, equivalent to Abramovich-Chen-Chen by Abramovich-Chen-Marcus-Wise 2017; the modern context in which the reconstruction's enumerative coefficients live.

Lean module

Codex.AlgGeom.Tropical.GrossSiebertReconstruction

Reviewer

TBD

Estimated time

beginner: 30m
intermediate: 65m
master: 120m