04.12.08 · algebraic-geometry / tropical

Dual Intersection Complex; Tropical Manifold B

shipped3 tiersLean: partial

Anchor (Master): Gross-Siebert 2006 *Mirror symmetry via affine geometry, I*, J. Differential Geom. 72 (2), 169-338 — originator of the algebraic dual-intersection-complex framework; Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lectures 4-5 (canonical textbook); Kontsevich-Soibelman 2001 *Homological mirror symmetry and torus fibrations* (in *Symplectic Geometry and Mirror Symmetry*, World Scientific) and Kontsevich-Soibelman 2006 *Affine structures and non-archimedean analytic spaces* (in *The Unity of Mathematics*, Birkhäuser PM 244, 321-385) — non-archimedean SYZ; Strominger-Yau-Zaslow 1996 *Nuclear Phys. B* 479 — the SYZ-base motivation; Gross-Siebert 2011 *Annals of Math.* 174, 1301-1428 — reconstruction theorem consuming (B, P) as input

Intuition [Beginner]

A toric degeneration is a one-parameter family of complex varieties in which the smooth fibres are all isomorphic to a fixed target $X$ , but the central fibre at $t = 0$ breaks into a union of simpler pieces — finitely many toric strata glued along their common boundaries. The combinatorial pattern recording how those pieces fit together is a polyhedral cell complex, and that complex is the dual intersection complex $B$ of the degeneration.

The complex $B$ records one cell for each component, one shared edge for each pairwise intersection, one shared face for each triple intersection, and so on. The result is a finite combinatorial object you can draw on the page that captures the entire combinatorial skeleton of the central fibre.

A second piece of structure sits on top. Each cell of $B$ inherits an integer lattice from the toric stratum it came from, and you can patch these lattices together using transition matrices with integer entries. This is the integral affine structure on $B$ . Away from a small set of codimension-two cells where the transitions disagree, $B$ behaves like a flat manifold of the same dimension as $X$ , but with lengths and angles measured against integer rulers. The small set where the transitions disagree is the singular locus $Δ$ . Loops around the singular locus pick up an integer matrix called the monodromy of the affine structure.

The reason this combinatorial-plus-integer-affine structure matters is that it carries the entire mirror-symmetry data of the original family. When the family degenerates a Calabi-Yau variety, the monodromy is constrained to lie in a special subgroup of integer matrices — the unit-determinant subgroup — and the resulting object $(B, P)$ with the polyhedral decomposition $P$ is exactly the algebraic realisation of the base of the conjectural dual special-Lagrangian fibration in the Strominger-Yau-Zaslow picture. Gross and Siebert proved that you can read the entire mirror Calabi-Yau off $(B, P)$ by an order-by-order reconstruction procedure.

This unit defines $(B, P)$ from a toric degeneration carefully, describes its integral affine structure and singular locus, and sets up the Calabi-Yau monodromy condition that distinguishes the cases where mirror symmetry applies.

Visual [Beginner]

A three-panel schematic. Left: a one-parameter family of complex surfaces tilting as $t \to 0$ , breaking into three triangular toric strata glued along three edges meeting at a single triple-intersection vertex. Middle: the dual cell complex $B$ — three two-dimensional cells (one per component) glued along three edges (one per double intersection) meeting at one vertex (the triple intersection), assembled into a flat planar diagram. Right: the same $B$ with a dotted codimension-two locus drawn as a single interior point, around which the integer affine transitions wind by a non-identity matrix recording the monodromy of the structure.

The middle panel shows the combinatorial skeleton; the right panel adds the integer affine information that makes $B$ a tropical manifold rather than just a topological cell complex.

Worked example [Beginner]

Take the simplest possible example: a toric degeneration of the projective line $P^{1}$ to the union of two affine lines glued at a single point. The smooth fibre is a copy of $P^{1}$ for every $t \neq = 0$ . The central fibre at $t = 0$ is two copies of $P^{1}$ glued at a single point. The dual intersection complex records two zero-dimensional cells (one per component) joined by a single one-dimensional cell (the shared gluing point).

Step 1. Identify the toric strata. The two components of the central fibre are each $P^{1} = X_{[0, 1/2]}$ and $P^{1} = X_{[1/2, 1]}$ — the two halves of the polytope subdivision of the unit interval $[0, 1]$ at its midpoint.

Step 2. Assemble the dual complex. Each interval cell of the subdivision produces a zero-dimensional dual cell — one of the two vertices of $B$ . The shared interior vertex of the two intervals produces a one-dimensional dual cell — the single edge of $B$ joining the two vertices.

Step 3. Read off the integral affine structure. The lattice on each vertex is the zero integer lattice $Z^{0} = 0$ — there is nothing to patch in dimension zero. The edge inherits the integer structure from the interval: a single coordinate $u$ ranging from $0$ at one endpoint to $1$ at the other, lattice $Z$ . There is no codimension-two singular locus because the dimension of $B$ is $1$ .

What this tells us: $B$ is the closed interval $[0, 1]$ with the standard integer affine structure and the polyhedral decomposition into two vertices and one edge. The structure is degenerate in this baby example because the dimension is too low to carry interesting monodromy — but the construction template is the one that generalises in higher dimension to the rich integral-affine-with-singularities manifolds of Gross-Siebert.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The dual intersection complex $B$ of a toric degeneration records, for each component of the central fibre, one:

A. Vertex of $B$ . B. Top-dimensional cell of $B$ . C. Edge of $B$ . D. Codimension-two singular cell of $B$ .

Hint

The word "dual" is doing work. The combinatorial pattern reverses the dimensions: top-dimensional components produce top-dimensional cells of $B$ .

Answer

B. Each top-dimensional irreducible component of the central fibre $X_{0}$ — itself an $n$ -dimensional toric stratum — produces one top-dimensional cell of the dual intersection complex $B$ . Codimension-1 intersections of pairs of components produce $(n - 1)$ -dimensional cells of $B$ ; codimension- $k$ intersections produce $(n - k)$ -dimensional cells; and so on, with dimensions reversing under the "dual" terminology. Option A reverses the dual correspondence; options C and D point to lower-dimensional cells.

Exercise (easy, multiple choice).

The integral affine structure on $B$ is well-defined on the complement of a codimension-2 closed subset $Δ$ . The set $Δ$ is:

A. The boundary of $B$ as a topological manifold with corners. B. The codimension-1 skeleton of the polyhedral decomposition $P$ . C. The locus along which the integer affine transitions between adjacent charts fail to glue into a single global affine atlas — the monodromy locus. D. The fixed-point set of the torus action on the central fibre.

Hint

$Δ$ records the failure of the affine structure to extend globally. It is not a boundary.

Answer

C. The codimension-2 singular locus $Δ \subset B$ is precisely the locus around which the local integral affine charts fail to patch into a global affine atlas — a small loop encircling a codim-2 cell of $Δ$ picks up a non-identity matrix in $GL_{n} (Z)$ , the local monodromy. The complement $B ∖ Δ$ is an integral affine manifold in the standard sense; the full $B$ is only an integral affine manifold with singularities. Option A confuses singular locus with boundary; option B is too large (codim 1 versus codim 2); option D mixes up the algebraic and combinatorial sides.

Exercise (easy, true-false).

True or false: when the smooth generic fibre of the toric degeneration is a Calabi-Yau variety, the local monodromies of the integral affine structure on $B ∖ Δ$ lie in the unit-determinant integer subgroup $SL_{n} (Z)$ .

Hint

The Calabi-Yau condition is the existence of a nowhere-vanishing holomorphic volume form, equivalently the vanishing of the canonical bundle as an isomorphism class. Volume-preservation translates on $B$ to a determinant-one constraint on the monodromy.

Answer

True. Gross-Siebert 2006 §1.5 establishes the equivalence: the toric degeneration's generic fibre is Calabi-Yau (canonical bundle isomorphic to the structure sheaf) if and only if the local monodromy representation $π_{1} (B ∖ Δ) \to GL_{n} (Z)$ factors through $SL_{n} (Z)$ . Geometrically, the integral affine structure on $B ∖ Δ$ admits a parallel volume form, which is exactly the affine-side image of the holomorphic volume form on the Calabi-Yau generic fibre. The SL-condition is the defining feature of the Calabi-Yau case and the prerequisite for the Gross-Siebert reconstruction theorem to apply.

Formal definition [Intermediate+]

Fix a lattice $N ≅ Z^{n}$ of rank $n$ with dual $M = Hom_{Z} (N, Z)$ , and a smooth projective toric variety $X_{Σ}$ over $C$ associated to a fan $Σ$ in $N_{R}$ in the sense of [04.11.04]. Let $π : X \to A_{C}^{1}$ be a toric degeneration of $X_{Σ}$ — a flat $T$ -equivariant family with smooth generic fibre $X_{t} ≅ X_{Σ}$ for $t \neq = 0$ and central fibre $X_{0} = ⋃_{Δ_{i} \in Ξ} X_{Δ_{i}}$ a reduced union of irreducible toric strata indexed by a polyhedral subdivision $Ξ$ of a lattice polytope $Δ \subseteq M_{R}$ (cf. [04.12.06]).

Definition (dual intersection complex of $π$ ). The dual intersection complex $B (Ξ)$ of $π$ is the polyhedral cell complex defined as follows. For each cell $Δ_{i} \in Ξ$ of dimension $k$ , let $σ_{Δ_{i}}^{\lor} \subseteq N_{R}$ denote the dual cone of $Δ_{i}$ — the cone of integer-affine functionals constant on $Δ_{i}$ in the sense $$ \sigma_{\Delta_i}^\vee ;:=; {u \in N_\mathbb{R} ;:; \langle u, m\rangle \geq \langle u, m'\rangle \text{ for all } m, m' \in \Delta_i}. $$ This is a strongly convex rational polyhedral cone of dimension $n - k$ in $N_{R}$ . The dual intersection complex is the quotient $$ B(\Xi) ;:=; \bigsqcup_{\Delta_i \in \Xi} \sigma_{\Delta_i}^\vee ;\Big/; \sim, $$ where $\sim$ glues $σ_{Δ_{i}}^{\lor}$ and $σ_{Δ_{j}}^{\lor}$ along the dual of the common face $Δ_{i} \cap Δ_{j}$ whenever $Δ_{i}$ and $Δ_{j}$ share a face. The resulting topological space is a polyhedral cell complex of dimension $n$ .

Definition (polyhedral decomposition $P$ ). The polyhedral decomposition $P$ of $B (Ξ)$ has one cell per cell of $Ξ$ : the cell of $P$ corresponding to $Δ_{i} \in Ξ$ is the image of $σ_{Δ_{i}}^{\lor}$ in $B (Ξ)$ . The face order on $P$ is reverse to the face order on $Ξ$ — a cell of $P$ is a face of another cell of $P$ iff the corresponding cells of $Ξ$ stand in the opposite face relation. The pair $(B, P) = (B (Ξ), P (Ξ))$ packages the dual intersection complex with its combinatorial cell structure.

Definition (integral affine structure). An integral affine structure on an open subset $U \subseteq B (Ξ)$ is an atlas of charts $φ_{α} : U_{α} \to N_{R}$ with $⋃_{α} U_{α} = U$ whose transition functions $φ_{β} \circ φ_{α}^{- 1}$ lie in the integral affine group $N ⋊ GL_{n} (Z) \subseteq Aff (N_{R})$ . The natural integral affine structure on the interior of each top-dimensional cell of $B (Ξ)$ is induced by the dual cone $σ_{Δ_{i}}^{\lor}$ 's lattice $N \cap σ_{Δ_{i}}^{\lor}$ .

Definition (singular locus $Δ_{B}$ ). The integral affine structure on the top-dimensional cells of $B (Ξ)$ extends across cells of codimension 1 by gluing along the common face's lattice. The extension across cells of codimension 2 may fail: the gluings around a small loop encircling a codim-2 cell may compose to a non-identity element of $GL_{n} (Z)$ . The singular locus $Δ_{B} \subset B (Ξ)$ is the union of cells of codimension 2 around which the integral affine structure has non-identity local monodromy: $$ \Delta_B ;=; \bigcup_{\substack{\tau \in P,; \mathrm{codim}(\tau) = 2 \ \mathrm{mon}(\tau) \neq \mathrm{id}}} \tau ;\subseteq; B(\Xi). $$ The complement $B (Ξ) ∖ Δ_{B}$ is an integral affine manifold; $B (Ξ)$ itself is an integral affine manifold with singularities in the Gross-Siebert sense.

Definition (tropical manifold). A tropical manifold is a triple $(B, P, s)$ where $B$ is a polyhedral cell complex of dimension $n$ , $P$ is a polyhedral decomposition of $B$ into closed cells, and $s$ is an integral affine structure on $B ∖ Δ_{B}$ for a codimension-2 closed subset $Δ_{B} \subseteq B$ called the singular locus. The construction $(B (Ξ), P (Ξ), s_{nat})$ from a polyhedral subdivision $Ξ$ as above produces a tropical manifold whose singular locus is determined by the codim-2 cells with non-identity monodromy.

Definition (Calabi-Yau monodromy condition). A tropical manifold $(B, P)$ satisfies the Calabi-Yau monodromy condition if the monodromy representation $$ \rho_B : \pi_1(B \setminus \Delta_B) ;\longrightarrow; \mathrm{GL}_n(\mathbb{Z}) $$ factors through $SL_{n} (Z)$ , equivalently if every local monodromy preserves a parallel volume form on $B ∖ Δ_{B}$ .

Counterexamples to common slips

The dual intersection complex is not the fan, and not the polytope. The fan $Σ$ of $X_{Σ}$ lives in $N_{R}$ but describes the affine charts of $X_{Σ}$ as a whole, not the polyhedral subdivision of any line bundle. The polytope $Δ$ associated to an ample line bundle lives in $M_{R}$ . The polyhedral subdivision $Ξ$ of $Δ$ is the toric-degeneration input. The dual intersection complex $B (Ξ)$ is built from $Ξ$ by dualising cells into cones in $N_{R}$ and gluing — it is again in $N_{R}$ but assembled with reversed face order.
The singular locus is genuinely codimension 2, not codimension 1. In dimension $n = 2$ , the singular locus consists of isolated points; in dimension $n = 3$ , a one-dimensional graph; and so on. The integral affine structure always extends uniquely across codimension-1 cells of $P$ because the gluing on the codim-1 face determines the transition, and there is no holonomy to be picked up.
Not every tropical manifold satisfies the Calabi-Yau condition. The Calabi-Yau condition is a constraint, picked out by Gross-Siebert as the case for which the reconstruction theorem applies. Tropical manifolds with $GL_{n} (Z)$ -valued monodromy generalising beyond $SL_{n} (Z)$ correspond to non-Calabi-Yau toric degenerations and are studied separately, but the Gross-Siebert programme is specifically about the SL-monodromy case.

Key theorem with proof [Intermediate+]

Theorem (Gross-Siebert 2006 §1: the dual intersection complex is a tropical manifold). Let $π : X \to A^{1}$ be a toric degeneration of a smooth projective toric variety $X_{Σ}$ of dimension $n$ , with central fibre $X_{0} = ⋃_{Δ_{i} \in Ξ} X_{Δ_{i}}$ indexed by a polyhedral subdivision $Ξ$ . Then the dual intersection complex $B (Ξ)$ with its induced polyhedral decomposition $P (Ξ)$ and the cell-induced integral affine structure $s_{nat}$ is a tropical manifold: $(B (Ξ), P (Ξ), s_{nat})$ is a polyhedral cell complex of dimension $n$ carrying an integral affine structure on the complement of a closed codimension-2 singular locus $Δ_{B}$ . Moreover, the generic fibre $X_{t}$ ( $t \neq = 0$ ) is Calabi-Yau if and only if the local monodromy around every codim-2 cell of $Δ_{B}$ lies in $SL_{n} (Z)$ .

Proof. The proof factors into four moves.

Step 1 (assembly as a topological cell complex). For each cell $Δ_{i} \in Ξ$ of dimension $k$ , the dual cone $σ_{Δ_{i}}^{\lor} \subseteq N_{R}$ is a strongly convex rational polyhedral cone of dimension $n - k$ . The face relation $Δ_{i} \subseteq Δ_{j}$ corresponds to the inclusion of dual cones $σ_{Δ_{j}}^{\lor} \subseteq σ_{Δ_{i}}^{\lor}$ (face-order reversal). Gluing the dual cones along the dual identifications produces a CW-complex of dimension $max_{Δ_{i} \in Ξ} (n - dim Δ_{i}) = n$ (achieved when $Δ_{i}$ is a vertex of $Ξ$ ). The polyhedral decomposition $P (Ξ)$ is the cell structure inherited from the dual cones.

Step 2 (integral affine charts on the smooth strata). The top-dimensional cells of $P (Ξ)$ are the dual cones $σ_{v}^{\lor} ≅ R^{n}$ associated to the vertices $v$ of $Ξ$ . Each carries the standard integral affine structure inherited from the lattice $N$ . The interior of each top-dimensional cell is therefore an integral affine manifold in the strict sense.

For a codimension-1 cell $τ$ of $P (Ξ)$ separating two top-dimensional cells $σ_{v}^{\lor}$ and $σ_{v^{'}}^{\lor}$ , the dual edge $\overline{v v^{'}}$ of $Ξ$ supplies the gluing: the integral affine chart on $σ_{v}^{\lor}$ extends across $τ$ to an integral affine chart on the union $σ_{v}^{\lor} \cup τ \cup σ_{v^{'}}^{\lor}$ . The extension is unique because the gluing along $τ$ is determined by the lattice $N \cap τ$ on the common face and there is no holonomy in codimension 1.

Step 3 (monodromy in codimension 2). For a codim-2 cell $ρ$ of $P (Ξ)$ , the link of $ρ$ in $B (Ξ)$ is a small circle in the transverse 2-plane. Choose a sequence of top-dimensional cells $σ_{v_{1}}^{\lor}, σ_{v_{2}}^{\lor}, \dots, σ_{v_{k}}^{\lor}$ encountered in order around the link. The composition of the codim-1 gluings $σ_{v_{1}}^{\lor} ⇝ σ_{v_{2}}^{\lor} ⇝ \dots ⇝ σ_{v_{k}}^{\lor} ⇝ σ_{v_{1}}^{\lor}$ is an element $mon (ρ) \in GL_{n} (Z)$ — the local monodromy around $ρ$ . The cell $ρ$ belongs to $Δ_{B}$ iff $mon (ρ) \neq = id$ .

The integral affine structure on $B (Ξ) ∖ Δ_{B}$ is the unique structure extending the chart on each top-dimensional cell across all codim-1 faces and all codim- $\geq 2$ cells with identity monodromy. This is well-defined because the codim- $\geq 3$ links are simply connected and any potential higher-codimension monodromy is determined by the codim-2 data.

Step 4 (Calabi-Yau condition). The toric degeneration's central fibre $X_{0} = ⋃ X_{Δ_{i}}$ is log smooth in the sense of Kato when equipped with its natural divisorial log structure. The relative canonical bundle $ω_{X / A^{1}}$ restricts to the central fibre as a line bundle determined combinatorially by the polyhedral data. Gross-Siebert 2006 §1.5 computes: $$ \omega_{\mathcal{X}/\mathbb{A}^1}\big|_{\mathcal{X}0} ;\cong; \mathcal{O}{\mathcal{X}0}\Big(\sum{i} a_i D_i\Big), \qquad a_i \in \mathbb{Z}, $$ where the sum runs over codim-1 toric strata and $a_{i}$ is determined by the local monodromy of the integral affine structure at the codim-2 cell dual to $D_{i}$ .

Vanishing of $ω_{X / A^{1}} ∣_{X_{0}}$ as a non-isomorphism-class — equivalently, the Calabi-Yau condition on the generic fibre $X_{t}$ — translates into $a_{i} = 0$ for every $i$ , which in turn translates into the local monodromy lying in $SL_{n} (Z)$ (since determinant detects the deviation from volume preservation). The converse is also true: if every monodromy lies in $SL_{n} (Z)$ , the integral affine structure on $B ∖ Δ_{B}$ admits a parallel volume form, which lifts to a holomorphic volume form on $X_{t}$ , witnessing the Calabi-Yau condition.

Combining the four steps establishes the theorem. $□$

Bridge. The dual intersection complex builds toward [04.12.09] Gross-Siebert reconstruction theorem, where $(B, P)$ becomes the input to an order-by-order smoothing producing a mirror Calabi-Yau, and appears again in [04.12.11] slab functions, where the codim-1 cells of $P$ carry the gluing data that records the deformation parameters. The foundational reason is that the integral affine structure on $B ∖ Δ_{B}$ encodes the entire local model of the toric degeneration, and this is exactly the bridge from the algebraic side — a one-parameter family of complex varieties — to the combinatorial side — a polyhedral manifold with integer-affine charts. Putting these together identifies the algebraic Calabi-Yau geometry with the combinatorial integral-affine geometry of $(B, P)$ , and this identification is the central insight of the Gross-Siebert programme.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Verify the face-order reversal: if $Δ_{i}, Δ_{j} \in Ξ$ with $Δ_{i} \subseteq Δ_{j}$ , show that the dual cones satisfy $σ_{Δ_{j}}^{\lor} \subseteq σ_{Δ_{i}}^{\lor}$ in $N_{R}$ .

Hint

Use the definition $σ_{Δ}^{\lor} = {u : ⟨ u, m ⟩ \geq ⟨ u, m^{'} ⟩ for all m, m^{'} \in Δ}$ . The constraint set grows as $Δ$ grows, so the dual cone shrinks.

Answer

By definition, $$ \sigma_{\Delta}^\vee ;=; {u \in N_\mathbb{R} ;:; \langle u, m\rangle \geq \langle u, m'\rangle \text{ for all } m, m' \in \Delta}. $$ The set of constraints ${m, m^{'} \in Δ : ⟨ u, m ⟩ \geq ⟨ u, m^{'} ⟩}$ is larger when $Δ$ is larger, because more pairs $(m, m^{'})$ are available. Therefore the cone of $u$ satisfying all constraints is smaller: if $Δ_{i} \subseteq Δ_{j}$ , every constraint on $σ_{Δ_{i}}^{\lor}$ is also a constraint on $σ_{Δ_{j}}^{\lor}$ , plus extra constraints from pairs not entirely in $Δ_{i}$ . So $σ_{Δ_{j}}^{\lor} \subseteq σ_{Δ_{i}}^{\lor}$ . This is the face-order reversal of the dual-cone construction, and is the combinatorial reason the polyhedral decomposition $P$ of $B (Ξ)$ has reversed face order relative to $Ξ$ .

Exercise 4 (medium, short-answer).

Describe the codimension-2 cells of $P (Ξ)$ for the polyhedral subdivision $Ξ$ of the unit cube $[0, 1]^{3} \subseteq R^{3}$ into six tetrahedra (the standard subdivision via the long diagonal). How many codim-2 cells are there, and which integer-affine monodromies do they carry?

Hint

Codim-2 cells of $P (Ξ)$ correspond to 1-dimensional cells (edges) of $Ξ$ that are interior to the cube. Count the interior edges and inspect their stars.

Answer

The standard subdivision of the unit cube into 6 tetrahedra (each sharing the long diagonal from $(0, 0, 0)$ to $(1, 1, 1)$ ) has $19$ edges total: $12$ on the boundary of the cube + $1$ interior long diagonal + $6$ interior face-diagonals. Codim-2 cells of $P (Ξ)$ correspond to edges $e$ of $Ξ$ via the dual-cell construction: $σ_{e}^{\lor}$ is a 1-dimensional cell in $N_{R}$ . The interior edges (the long diagonal and the six face-diagonals) give codim-2 cells of $P (Ξ)$ that may carry non-identity monodromy; the boundary edges give cells on the boundary of $B (Ξ)$ where the integral affine structure does not need to glue around them.

The local monodromy around the codim-2 cell dual to the long diagonal is a non-identity element of $GL_{3} (Z)$ — concretely, the composition of the six face-pairings around the diagonal, which can be computed as a $3 \times 3$ integer matrix with determinant $\pm 1$ . For a Calabi-Yau choice of subdivision, the monodromy lies in $SL_{3} (Z)$ ; for generic subdivisions it lies in $GL_{3} (Z) ∖ SL_{3} (Z)$ .

Exercise 5 (medium, short-answer).

Why is the singular locus $Δ_{B}$ codimension exactly 2, not codimension 1?

Hint

Around a codim-1 cell, the gluing between two charts is determined by the lattice on the common face, leaving no freedom for non-identity monodromy. Monodromy first appears when you have a loop around a cell, which requires codim 2.

Answer

In codimension 1, the link of a cell $τ \in P$ in the polyhedral complex is a pair of points (one in each of the two top-dimensional cells adjacent to $τ$ ). There is no loop in the link, hence no possibility of monodromy: the integral affine charts on the two adjacent top-cells extend uniquely across $τ$ by the gluing determined by the lattice $N \cap τ$ on the common face.

In codimension 2, the link of a cell $ρ \in P$ is a circle — the small transverse 2-plane circle going around $ρ$ . The composition of integer affine gluings around this circle defines the local monodromy $mon (ρ) \in GL_{n} (Z)$ . The codim-2 cells of $P$ with non-identity monodromy form the singular locus $Δ_{B}$ .

In codimension $\geq 3$ , the link is simply connected (a sphere $S^{k - 1}$ for $k \geq 3$ ), so any loop around a codim- $\geq 3$ cell is contractible in the link and the monodromy is determined entirely by the codim-2 monodromies of cells in the closure. This is why $Δ_{B}$ is supported on codim-2 cells specifically.

Exercise 6 (hard, short-answer).

State the discrete Legendre duality of Gross-Siebert 2006 §1.4 and explain its role in mirror symmetry on $(B, P)$ .

Hint

Discrete Legendre duality interchanges the integral affine structure on $B$ with a dual structure $\overset{ˇ}{B}$ supported on the same underlying space; cell-wise, top-dimensional cells of $P$ swap with vertices of $\overset{ˇ}{P}$ via the dual-cone construction.

Answer

Discrete Legendre duality (Gross-Siebert 2006 §1.4): given a tropical manifold $(B, P, s)$ , the discrete Legendre dual is the tropical manifold $(\overset{ˇ}{B}, \overset{ˇ}{P}, \overset{s}{ˇ})$ on the same underlying space $B$ , with $\overset{ˇ}{P}$ obtained by cell-wise dualisation:

$k$ -dimensional cells of $P$ correspond bijectively to $(n - k)$ -dimensional cells of $\overset{ˇ}{P}$ ;
the integral affine structure $\overset{s}{ˇ}$ on $\overset{ˇ}{B} ∖ Δ_{B}$ is the dual of $s$ under the perfect pairing $N \times M \to Z$ , locally interchanging $N_{R}$ -charts with $M_{R}$ -charts;
the singular locus $Δ_{B}$ is the same in both structures.

The construction is involutive: $(\overset{ˇ}{\overset{ˇ}{B}}, \overset{ˇ}{\overset{ˇ}{P}}, \overset{ˇ}{\overset{s}{ˇ}}) ≅ (B, P, s)$ canonically.

The role in mirror symmetry is structural. The Strominger-Yau-Zaslow heuristic says mirror Calabi-Yau pairs $(X, X^{\lor})$ arise as dual special-Lagrangian torus fibrations over a common base $B$ . The discrete Legendre duality is the algebraic-side analogue of this fibrewise T-duality: the Gross-Siebert reconstruction theorem applied to $(B, P, s)$ produces the Calabi-Yau $X$ , while reconstruction applied to $(\overset{ˇ}{B}, \overset{ˇ}{P}, \overset{s}{ˇ})$ produces the mirror $X^{\lor}$ . The triplet $(B, P)$ thus carries both halves of the mirror pair, with the Legendre dual swapping them.

Exercise 7 (hard, symbolic).

Compute a local model of the codim-2 monodromy in dimension $n = 2$ : for two adjacent top-dimensional cells $σ_{1}, σ_{2}$ of $P$ meeting along an edge, suppose the integral affine chart on $σ_{1}$ uses coordinates $(x_{1}, y_{1}) \in R^{2}$ and the chart on $σ_{2}$ uses coordinates $(x_{2}, y_{2}) = A (x_{1}, y_{1}) + b$ for $A \in GL_{2} (Z)$ and $b \in Z^{2}$ . If $σ_{1}, σ_{2}, σ_{3}, σ_{4}$ are four top-cells meeting at a codim-2 point with transition matrices $A_{12}, A_{23}, A_{34}, A_{41}$ , write the local monodromy at the codim-2 point.

Hint

The monodromy at the codim-2 point is the product $A_{41} A_{34} A_{23} A_{12}$ of the four transitions traversed around the point. Compatibility of the integral affine structure on the union $σ_{1} \cup \dots \cup σ_{4} ∖ {point}$ requires the product to be $id$ ; in the singular case the product is a non-identity element of $GL_{2} (Z)$ .

Answer

The local monodromy is $$ \mathrm{mon}(\rho) ;=; A_{41} \cdot A_{34} \cdot A_{23} \cdot A_{12} ;\in; \mathrm{GL}2(\mathbb{Z}). $$ For the structure to extend across the codim-2 point $ρ$ as an integral affine manifold, the product must equal $id$ . When the product is non-identity, $ρ$ lies in $Δ_{B}$ . The Calabi-Yau condition demands $det mon (ρ) = 1$ for every codim-2 cell, i.e., the product of the determinants $\det A{12} \cdot \det A_{23} \cdot \det A_{34} \cdot \det A_{41} = 1 $; t hi sco n s t r ain s t h ec h o i ceo f p o l y h e d r a l s u b d i v i s i o n$ \Xi$ used to build the toric degeneration.

A concrete non-identity Calabi-Yau example is the local model around a focus-focus singularity in dimension 2: $$ \mathrm{mon} ;=; \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}), $$ which is conjugate to the unique non-identity unipotent in $SL_{2} (Z)$ up to inversion. This monodromy appears as the local model at every node of the discriminant locus of a K3 surface viewed through its SYZ fibration.

Exercise 8 (hard, short-answer).

Explain why the dimension of $B (Ξ)$ equals $dim X_{Σ}$ , not $dim Ξ$ .

Hint

Look at the dual-cone construction. A vertex of $Ξ$ (dimension 0) produces a top-dimensional cone in $N_{R}$ of dimension $n$ — and this top-dimensional cone is a top-dimensional cell of $B (Ξ)$ .

Answer

The polyhedral subdivision $Ξ$ lives in $M_{R} ≅ R^{n}$ and has dimension equal to $dim Δ = n$ (since $Δ$ is the polytope of an ample line bundle on the $n$ -dimensional toric variety $X_{Σ}$ ). The cells of $Ξ$ have dimensions ranging from $0$ (vertices) to $n$ (top cells).

The dual-cone construction sends a cell $Δ_{i}$ of dimension $k$ to a cone $σ_{Δ_{i}}^{\lor}$ in $N_{R}$ of dimension $n - k$ . The maximum is achieved at $k = 0$ , i.e., vertices of $Ξ$ : $σ_{v}^{\lor}$ has dimension $n$ . The dual intersection complex $B (Ξ)$ is therefore $n$ -dimensional, matching $dim X_{Σ} = dim N_{R}$ , not $dim Ξ$ .

The face-order reversal here is the combinatorial fingerprint of duality: the cell structure of $B (Ξ)$ inverts the cell dimensions of $Ξ$ . Geometrically, top-dimensional components of the central fibre $X_{0}$ correspond to top-dimensional cells of $B (Ξ)$ ; codimension- $k$ intersections of central-fibre components correspond to codim- $k$ cells of $B (Ξ)$ . This is what "dual" means in "dual intersection complex".

Lean formalisation [Intermediate+]

The Lean module Codex.AlgGeom.Tropical.DualIntersectionComplex schematises the integral-affine manifold $(B, P)$ and the Calabi-Yau monodromy condition. Mathlib currently supplies the basic free-ℤ-module API through Mathlib.LinearAlgebra.FreeModule.Basic, the tropical semiring through Mathlib.Algebra.Tropical.Basic, and matrix infrastructure through Mathlib.Data.Matrix.Basic; what is absent is the polyhedral-complex API, integral-affine charts, monodromy representations of fundamental groups of cell complexes, and the SL-subgroup constraint distinguishing the Calabi-Yau case.

The module declares typed placeholders for seven essential structures: a finite-rank lattice Lattice, integral-affine charts IntegralAffineChart, polyhedral subdivisions PolyhedralSubdivision (re-declared so downstream files need not import NishinouSiebertCorrespondence), the dual intersection complex DualIntersectionComplex Ξ, the polyhedral decomposition PolyhedralDecomposition Ξ, the codimension-2 singular locus singularLocus Ξ, and the tropical manifold TropicalManifold N packaging the data.

It then names the construction fromSubdivision : PolyhedralSubdivision M → TropicalManifold N, the discrete Legendre dual legendreDual, the local monodromy localMonodromy, and the Calabi-Yau predicate isCalabiYauMonodromy. Five theorems are recorded with sorry-equivalent (rfl-style) proof bodies: fromSubdivision_isTropicalManifold (the construction yields a tropical manifold in the Gross-Siebert sense), monodromy_calabi_yau (the SL-condition characterises the Calabi-Yau case), legendreDual_involutive (Legendre duality squares to the identity), and syz_base_identification (the constructed $(B, P)$ is the SYZ base of the corresponding toric degeneration). The Mathlib gap is the polyhedral-complex + integral-affine-chart + monodromy-representation package — none of which is upstream — together with the comparison theorem relating the integral-affine volume form to the relative canonical bundle of the toric degeneration.

Construction from a toric degeneration [Master]

The technical assembly of $(B, P)$ from a toric degeneration $π : X \to A^{1}$ proceeds in four explicit steps, each tied to a concrete piece of toric data. We make the construction layer-by-layer to expose the role of every choice.

Layer 1 — the polytope $Δ$ and its polyhedral subdivision $Ξ$ . A smooth projective toric variety $X_{Σ}$ together with a $T$ -equivariant ample line bundle $L$ corresponds (per [04.11.10]) to a lattice polytope $Δ \subseteq M_{R}$ — the Newton polytope of the global sections $H^{0} (X_{Σ}, L)$ . The dimension of $Δ$ equals $dim X_{Σ} = n$ , and the integer points $Δ \cap M$ are the $T$ -weights of the section module. A polyhedral subdivision $Ξ$ of $Δ$ is a finite collection of lattice subpolytopes $Δ_{i} \subseteq Δ$ meeting along common faces, with vertices in $M$ .

The combinatorial data of $Ξ$ that matters for the dual intersection complex is the face poset $(Ξ, \subseteq)$ — the lattice of all cells with their face-containment order. Two different subdivisions of the same polytope $Δ$ can produce different toric degenerations and different dual intersection complexes; the choice of $Ξ$ is part of the input data.

Layer 2 — the toric degeneration $π : X \to A^{1}$ . Following Mumford 1972 ^{[Mumford 1972]} and Kempf-Knudsen-Mumford-Saint-Donat 1973 ^{[KKMS 1973]}, a polyhedral subdivision $Ξ$ of $Δ$ together with a concave piecewise-linear function $φ : Δ \to R$ — concave on $Δ$ , linear on each $Δ_{i} \in Ξ$ , integer-valued at lattice points — produces a toric degeneration $π : X \to A^{1}$ via the Mumford-KKMS construction. The total space $X$ sits in $P_{A^{1}}^{∣Δ \cap M ∣ - 1}$ as the zero locus of binomials lifted from the binomials of $X_{Σ}$ via $t$ -weights $φ (m)$ .

The generic fibre $X_{t}$ for $t \neq = 0$ is isomorphic to $X_{Σ}$ — setting $t = 1$ recovers the original binomials. The central fibre $X_{0}$ is the reduced union $⋃_{Δ_{i} \in Ξ} X_{Δ_{i}}$ of irreducible toric strata, with each $X_{Δ_{i}}$ the projective toric variety with Newton polytope $Δ_{i}$ . Two strata $X_{Δ_{i}}, X_{Δ_{j}}$ intersect along $X_{Δ_{i} \cap Δ_{j}}$ — the toric stratum corresponding to the polyhedral face $Δ_{i} \cap Δ_{j}$ .

Layer 3 — the dual intersection complex $B (Ξ)$ . For each cell $Δ_{i}$ of dimension $k$ , the dual cone $σ_{Δ_{i}}^{\lor} \subseteq N_{R}$ has dimension $n - k$ . Gluing the dual cones along their dual-of-face identifications produces $B (Ξ)$ as a polyhedral cell complex of dimension $n$ : $$ B(\Xi) ;=; \bigsqcup_{\Delta_i \in \Xi} \sigma_{\Delta_i}^\vee ;\Big/; \sim, $$ with $\sim$ identifying $σ_{Δ_{i}}^{\lor} \supseteq σ_{Δ_{j}}^{\lor}$ when $Δ_{i} \subseteq Δ_{j}$ . The polyhedral decomposition $P (Ξ)$ has one cell per cell of $Ξ$ , with reversed face order.

Top-dimensional cells of $P (Ξ)$ are the dual cones $σ_{v}^{\lor} ≅ R^{n}$ associated to vertices $v$ of $Ξ$ ; they form the smooth strata of $B (Ξ)$ . Codim-1 cells of $P (Ξ)$ are associated to edges of $Ξ$ . Codim-2 cells are associated to 2-cells of $Ξ$ — these are the candidate sites of the singular locus.

Layer 4 — the integral affine structure $s_{nat}$ and singular locus $Δ_{B}$ . Each top-dimensional cell $σ_{v}^{\lor}$ inherits the standard integral affine structure from the lattice $N \cap σ_{v}^{\lor} ≅ Z^{n}$ . The gluing across a codim-1 cell $τ \subseteq σ_{v}^{\lor} \cap σ_{v^{'}}^{\lor}$ is the unique integer affine map identifying the two charts on $τ$ , determined by the polyhedral data. There is no freedom in this gluing because $τ$ is a single $(n - 1)$ -cell and the lattice on $τ$ pins down the chart compatibility.

The composition of codim-1 gluings around a codim-2 cell $ρ$ is the local monodromy $mon (ρ) \in GL_{n} (Z)$ . The singular locus is $$ \Delta_B ;=; \bigcup_{\substack{\rho \in P,; \mathrm{codim}(\rho) = 2 \ \mathrm{mon}(\rho) \neq \mathrm{id}}} \rho. $$ On $B (Ξ) ∖ Δ_{B}$ , the integral affine structure extends uniquely to all codimensions; on $B (Ξ)$ itself, the structure has codim-2 singularities indexed by $Δ_{B}$ .

Monodromy, simple singularities, and the Calabi-Yau condition [Master]

The local monodromy $mon (ρ) \in GL_{n} (Z)$ around a codim-2 cell encodes the topological obstruction to extending the integral affine structure of $(B, P)$ across $ρ$ . We make the structure of the monodromy explicit and identify the Calabi-Yau case.

Simple singularities (Gross-Siebert 2006 §1.6). A codim-2 cell $ρ \in Δ_{B}$ is a simple singularity if a local model around $ρ$ exists in which the monodromy takes the focus-focus form: $$ \mathrm{mon}(\rho) ;=; \begin{pmatrix} I_{n - 2} & 0 \ 0 & \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} \end{pmatrix} ;\in; \mathrm{SL}_n(\mathbb{Z}), $$ in suitable integer affine coordinates transverse to $ρ$ . The focus-focus block is the unique non-identity unipotent in $SL_{2} (Z)$ up to inversion. A tropical manifold $(B, P)$ has only simple singularities if every codim-2 cell of $Δ_{B}$ is a simple singularity. Gross-Siebert 2006 proves the reconstruction theorem under this hypothesis.

The Calabi-Yau monodromy condition. The monodromy representation $$ \rho_B : \pi_1(B \setminus \Delta_B) ;\longrightarrow; \mathrm{GL}_n(\mathbb{Z}) $$ is the obstruction to a global integral affine atlas on $B$ . The image is contained in $SL_{n} (Z)$ iff the integral affine structure preserves a parallel volume form. By Gross-Siebert 2006 §1.5, the toric degeneration $π : X \to A^{1}$ has Calabi-Yau generic fibre $X_{t}$ (i.e., $ω_{X_{t}} ≅ O_{X_{t}}$ ) iff the monodromy $ρ_{B}$ factors through $SL_{n} (Z)$ . The proof passes through the computation $$ \omega_{\mathcal{X}/\mathbb{A}^1}\big|_{\mathcal{X}0} ;\cong; \mathcal{O}{\mathcal{X}_0}\Big(\sum_i a_i D_i\Big), \qquad a_i = a_i(\mathrm{mon}), $$ with the integers $a_{i}$ extracted from the determinant deviations of the codim-2 monodromies; vanishing of all $a_{i}$ is equivalent to the SL-condition.

Maximally unipotent degenerations. A Calabi-Yau toric degeneration is maximally unipotent at $t = 0$ if every local monodromy is unipotent in $SL_{n} (Z)$ (i.e., a simple singularity in the focus-focus form). The Gross-Siebert reconstruction theorem [04.12.09] applies to maximally unipotent Calabi-Yau degenerations; the corresponding tropical manifolds $(B, P)$ have only simple singularities and SL-monodromy.

Discrete Legendre duality. The cell-wise dualisation interchanges $Ξ$ -cells with $Ξ^{\lor}$ -cells via the perfect pairing $N \times M \to Z$ , producing a dual tropical manifold $(\overset{ˇ}{B}, \overset{ˇ}{P}, \overset{s}{ˇ})$ on the same underlying space. Gross-Siebert 2006 §1.4 establishes that discrete Legendre duality is an involution and that it intertwines the SL-monodromy condition with itself. The two sides $(B, P)$ and $(\overset{ˇ}{B}, \overset{ˇ}{P})$ of a Legendre-dual pair input separately into the Gross-Siebert reconstruction to produce the mirror Calabi-Yau pair $(X, X^{\lor})$ .

Relation to Strominger-Yau-Zaslow and the SYZ base [Master]

The dual intersection complex $(B, P)$ is the algebraic realisation of the Strominger-Yau-Zaslow base. We trace the identification and its consequences.

The Strominger-Yau-Zaslow heuristic. Strominger-Yau-Zaslow 1996 ^{[Strominger-Yau-Zaslow 1996]} proposed that mirror Calabi-Yau pairs $(X, X^{\lor})$ admit dual special-Lagrangian torus fibrations $$ X \xrightarrow{f} B \qquad \text{and} \qquad X^\vee \xrightarrow{\check{f}} B $$ over the same real base $B$ , with the fibres of $f$ being special-Lagrangian tori in $X$ and the fibres of $\overset{ˇ}{f}$ being their dual tori in $X^{\lor}$ . The base $B$ inherits an integral affine structure from the Arnold-Liouville theorem applied fibrewise — on the smooth locus of $f$ , the base has natural integer affine coordinates from the action coordinates of the Lagrangian fibration. The discriminant locus of $f$ , where fibres degenerate, contributes the singular locus $Δ_{B} \subseteq B$ of the integral affine structure.

Gross-Siebert's algebraic realisation. Gross-Siebert 2006 ^{[Gross-Siebert 2006]} provides the algebraic-geometric construction of the SYZ base. Starting from a toric degeneration $π : X \to A^{1}$ of a Calabi-Yau variety $X_{t}$ , the dual intersection complex $B (Ξ)$ is exactly the algebraic-side image of the SYZ base $B$ : the polyhedral decomposition $P$ corresponds to the discriminant stratification of $f$ , the integral affine structure corresponds to the action-coordinate structure on the smooth locus, and the singular locus $Δ_{B}$ corresponds to the SYZ discriminant.

The identification is made precise by Castaño-Bernard-Matessi 2009 ^{[Castano-Bernard-Matessi 2009]} in dimension 3, where they construct symplectic Lagrangian torus fibrations on Calabi-Yau threefolds whose integral affine bases match the algebraic $(B, P)$ from Gross-Siebert. The dimension-3 case is the smallest in which the singular locus $Δ_{B}$ is one-dimensional (a graph), exposing the rich combinatorial structure that distinguishes Calabi-Yau threefolds.

Mirror symmetry via Legendre duality. Once $(B, P, s)$ is identified with the SYZ base of $X = X_{t}$ , the discrete Legendre dual $(\overset{ˇ}{B}, \overset{ˇ}{P}, \overset{s}{ˇ})$ supplies the SYZ base of the mirror $X^{\lor}$ . The Gross-Siebert reconstruction theorem [04.12.09] produces $X$ from $(B, P, s)$ and $X^{\lor}$ from $(\overset{ˇ}{B}, \overset{ˇ}{P}, \overset{s}{ˇ})$ via an order-by-order smoothing procedure. The mirror map between $X$ and $X^{\lor}$ — relating Kähler moduli of one to complex moduli of the other — is the algebraic incarnation of the SYZ fibrewise T-duality at the level of integral affine geometry on $B$ .

Kontsevich-Soibelman non-archimedean SYZ. Kontsevich-Soibelman 2001 ^{[Kontsevich-Soibelman 2001]} and 2006 ^{[Kontsevich-Soibelman 2006]} develop a parallel programme in which $(B, P)$ is identified with the Berkovich skeleton of a maximally degenerate Calabi-Yau over a non-archimedean field $K = C ((t))$ . The Berkovich analytic space $X_{K}^{an}$ admits a deformation retract onto $B$ , and the retraction equips $B$ with the integral affine structure of the dual intersection complex. The non-archimedean and algebraic descriptions of $(B, P)$ agree, providing two routes to the same tropical manifold and reinforcing the identification with the SYZ base.

Synthesis. The dual intersection complex $(B, P)$ is the foundational reason that mirror symmetry has a combinatorial-geometric description, and the central insight is that the entire smoothing of a degenerate Calabi-Yau — including its mirror partner — is determined by the polyhedral and integral-affine data on $B$ . Putting these together identifies the algebraic toric-degeneration framework of Gross-Siebert with the symplectic SYZ framework of Strominger-Yau-Zaslow and the non-archimedean framework of Kontsevich-Soibelman; the bridge is the integral affine structure with simple singularities, and this same structure generalises in [04.12.09] Gross-Siebert reconstruction theorem, in [04.12.11] slab functions and tropical-manifold structure, in [04.12.12] theta functions on polarised tropical manifolds, and in the broader landscape of cluster algebras and homological mirror symmetry. The pattern recurs: the SL-monodromy integral affine geometry on $(B, P)$ is exactly the combinatorial mirror-symmetry data.

Connections [Master]

Toric degeneration of a Calabi-Yau variety 04.12.07. The dual intersection complex is constructed directly from a toric degeneration, which is itself defined in [04.12.07]. The pair $(B, P)$ is the combinatorial output of the degeneration; the degeneration itself is the algebraic input. Reversing this construction is the content of [04.12.09] Gross-Siebert reconstruction theorem.
Gross-Siebert reconstruction theorem 04.12.09. The reconstruction theorem takes $(B, P)$ with simple singularities and lifted gluing data and produces a toric degeneration of a Calabi-Yau variety whose dual intersection complex is $(B, P)$ . The two units pair: [04.12.08] defines the combinatorial object, [04.12.09] recovers the algebraic geometry from it.
Nishinou-Siebert correspondence 04.12.06. The dual intersection complex first appears in [04.12.06] as the polyhedral home for tropical curves enumerated by the Nishinou-Siebert correspondence. In the present unit, $(B, P)$ is upgraded from "combinatorial home for tropical curves" to "tropical manifold with integral affine structure", and the Calabi-Yau monodromy condition added. The Nishinou-Siebert curve counts on $(B, P)$ supply the scattering-diagram input for the Gross-Siebert reconstruction.
Strominger-Yau-Zaslow conjecture 04.12.10. The SYZ heuristic of dual special-Lagrangian torus fibrations is the symplectic-side picture behind $(B, P)$ . The dual intersection complex is the algebraic realisation of the SYZ base; the discrete Legendre dual is the algebraic side of fibrewise T-duality. A lateral pointer also reaches 05-symplectic/lagrangian/ for the special-Lagrangian fibration construction in the symplectic category.
Polytope-fan dictionary 04.11.10. The construction of $(B, P)$ from a toric degeneration starts with a lattice polytope $Δ \subseteq M_{R}$ associated to an ample line bundle on $X_{Σ}$ — exactly the polytope-fan dictionary of [04.11.10]. The polyhedral subdivision $Ξ$ refines $Δ$ , and the dual intersection complex is the cone-complex dual of $Ξ$ . The toric-geometry language of [04.11.04] and [04.11.10] is the combinatorial scaffold on which the tropical manifold is built.
Slab function and structure of a tropical manifold 04.12.11. The codim-1 cells of $P$ carry slab functions — Laurent polynomials in the toric coordinates of the adjacent chambers — that record the deformation gluing data used in the Gross-Siebert reconstruction. The unit [04.12.11] is the downstream development of the slab structure on the present $(B, P)$ .
Fan and the toric variety 04.11.04. The dual cones $σ_{Δ_{i}}^{\lor}$ that assemble $B (Ξ)$ are exactly the cones whose union is the (non-fan) cone complex dual to $Ξ$ . The toric language of fans from [04.11.04] supplies the combinatorial primitive — strongly convex rational polyhedral cones with face relations — used to build $B$ . The dual intersection complex is not itself a fan (it has more structure: a polyhedral decomposition and integral affine charts with singularities), but its building blocks are the fan-theoretic cones of [04.11.04].

Historical & philosophical context [Master]

The dual intersection complex sits at the convergence of three lineages. The toric-degeneration tradition originates with Mumford 1972 ^{[Mumford 1972]} Compositio Math. 24 (explicit construction of degenerating abelian varieties to a periodic tiling of $R^{n}$ , with the dual graph carrying an integral affine structure — the first appearance of the integral-affine-base idea in algebraic geometry), and is formalised in the toroidal-embedding language of Kempf-Knudsen-Mumford-Saint-Donat 1973 ^{[KKMS 1973]} Toroidal Embeddings I. The connection to enumerative geometry was developed in the 1990s and 2000s through Sturmfels, Gathmann, and others; the link to mirror symmetry crystallised in the 2000s through the Gross-Siebert programme.

The integral-affine-base lineage traces to Strominger-Yau-Zaslow 1996 ^{[Strominger-Yau-Zaslow 1996]} Nucl. Phys. B 479, who proposed the SYZ heuristic for mirror Calabi-Yau pairs as dual special-Lagrangian torus fibrations. The base of an SYZ fibration carries an integral affine structure from Arnold-Liouville action coordinates; identifying this with the dual intersection complex of a Calabi-Yau toric degeneration was the foundational insight of Gross-Siebert 2006 ^{[Gross-Siebert 2006]} J. Diff. Geom. 72 Mirror symmetry via affine geometry, I. Parallel development by Kontsevich-Soibelman 2001 ^{[Kontsevich-Soibelman 2001]} and 2006 ^{[Kontsevich-Soibelman 2006]} established the non-archimedean realisation of the same $(B, P)$ as the Berkovich skeleton of a maximally degenerate Calabi-Yau.

The Gross-Siebert programme proper unfolded in a sequence of papers between 2006 and the present. The foundational Gross-Siebert 2006 paper defined the dual intersection complex, the integral affine structure with simple singularities, and the Calabi-Yau monodromy condition; the Annals 2011 paper ^{[Gross-Siebert 2011]} proved the reconstruction theorem assembling a smoothing of a degenerate Calabi-Yau from $(B, P)$ and lifted gluing data; subsequent work by Gross, Hacking, Keel, Kontsevich, and collaborators extended the picture to log Calabi-Yau pairs, cluster algebras, and theta functions. The Gross 2011 CBMS lecture notes ^{[Gross 2011 CBMS]} provide the textbook synthesis. The symplectic side was filled in by Castaño-Bernard-Matessi 2009 ^{[Castano-Bernard-Matessi 2009]} J. Diff. Geom. 81, who constructed Lagrangian torus fibrations with discriminant matching the algebraic singular locus in dimension 3.

Bibliography [Master]

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

@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{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{StromingerYauZaslow1996,
  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},
}

@incollection{KontsevichSoibelman2001,
  author = {Kontsevich, Maxim and Soibelman, Yan},
  title = {Homological mirror symmetry and torus fibrations},
  booktitle = {Symplectic Geometry and Mirror Symmetry},
  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\"auser},
  year = {2006},
  pages = {321--385},
}

@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{Mumford1972,
  author = {Mumford, David},
  title = {An analytic construction of degenerating abelian varieties over complete rings},
  journal = {Compositio Mathematica},
  volume = {24},
  number = {3},
  year = {1972},
  pages = {239--272},
}

@book{KKMS1973,
  author = {Kempf, George and Knudsen, Finn and Mumford, David and Saint-Donat, Bernard},
  title = {Toroidal Embeddings {I}},
  series = {Lecture Notes in Mathematics},
  volume = {339},
  publisher = {Springer},
  year = {1973},
}

@article{CastanoBernardMatessi2009,
  author = {Casta\~no-Bernard, Ricardo and Matessi, Diego},
  title = {Lagrangian 3-torus fibrations},
  journal = {Journal of Differential Geometry},
  volume = {81},
  number = {3},
  year = {2009},
  pages = {483--573},
}

@article{Gross2005,
  author = {Gross, Mark},
  title = {Toric degenerations and {Batyrev-Borisov} duality},
  journal = {Mathematische Annalen},
  volume = {333},
  number = {3},
  year = {2005},
  pages = {645--688},
}

@article{TianZhu2002,
  author = {Tian, Gang and Zhu, Shengmao},
  title = {A note on the affine structure of toric varieties},
  journal = {Mathematical Research Letters},
  volume = {9},
  number = {5-6},
  year = {2002},
  pages = {689--700},
}

Prerequisites

04.11.04
04.12.06

Tier anchors

beginner: Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 4 informal opening; Gross-Siebert 2011 *Annals of Math.* 174 §1 introductory paragraphs
intermediate: Gross 2011 CBMS 114 Lecture 4 §4.1-§4.3; Gross-Siebert 2006 *J. Diff. Geom.* 72, §1.1-§1.2 (dual intersection complex, integral affine structure, simple singularities)
master: Gross-Siebert 2006 *Mirror symmetry via affine geometry, I*, J. Differential Geom. 72 (2), 169-338 — originator of the algebraic dual-intersection-complex framework; Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lectures 4-5 (canonical textbook); Kontsevich-Soibelman 2001 *Homological mirror symmetry and torus fibrations* (in *Symplectic Geometry and Mirror Symmetry*, World Scientific) and Kontsevich-Soibelman 2006 *Affine structures and non-archimedean analytic spaces* (in *The Unity of Mathematics*, Birkhäuser PM 244, 321-385) — non-archimedean SYZ; Strominger-Yau-Zaslow 1996 *Nuclear Phys. B* 479 — the SYZ-base motivation; Gross-Siebert 2011 *Annals of Math.* 174, 1301-1428 — reconstruction theorem consuming (B, P) as input

References

TODO_REF
Gross, M. & Siebert, B. — Mirror symmetry via affine geometry, I · *Journal of Differential Geometry* 72 (2), 169-338 (2006). §1 (dual intersection complex of a toric degeneration; integral affine manifold with simple singularities; polyhedral decomposition P); §1.4 (discrete Legendre duality between (B, P) and its dual (B̌, P̌)); §1.5 (the Calabi-Yau monodromy condition: local SL_n(ℤ) holonomy around codim-2 singular cells).
TODO_REF
Gross, M. — Tropical Geometry and Mirror Symmetry · CBMS Regional Conference Series in Mathematics 114, American Mathematical Society (2011). Lecture 4 (the dual intersection complex of a toric degeneration; cell complex assembly from the polyhedral subdivision Ξ); Lecture 5 (the polyhedral decomposition P; integral affine structure; singular locus Δ; the SYZ-base identification).
TODO_REF
Gross, M. & Siebert, B. — From real affine geometry to complex geometry · *Annals of Mathematics* 174 (3), 1301-1428 (2011). §1 (the reconstruction theorem: starting from a tropical manifold (B, P) with simple singularities and lifted gluing data, an order-by-order smoothing produces a toric degeneration of a Calabi-Yau variety with dual intersection complex (B, P)); §2-§4 (scattering diagrams and slab functions on (B, P)).
TODO_REF
Kontsevich, M. & Soibelman, Y. — Affine structures and non-archimedean analytic spaces · in *The Unity of Mathematics*, Progress in Mathematics 244, Birkhäuser, 321-385 (2006). The non-archimedean SYZ programme complementary to Gross-Siebert; supplies the analytic interpretation of (B, P) as the Berkovich skeleton of a maximally degenerate Calabi-Yau over a non-archimedean field, with the integral affine structure recovered from the Berkovich retraction.
TODO_REF
Kontsevich, M. & Soibelman, Y. — Homological mirror symmetry and torus fibrations · in *Symplectic Geometry and Mirror Symmetry*, World Scientific, 203-263 (2001). The original Kontsevich-Soibelman proposal: the SYZ base carries an integral affine structure with codim-2 singularities, and mirror symmetry is the discrete Legendre transform on (B, P).
TODO_REF
Strominger, A., Yau, S.-T. & Zaslow, E. — Mirror symmetry is T-duality · *Nuclear Physics B* 479 (1-2), 243-259 (1996). The SYZ heuristic: mirror Calabi-Yau pairs arise as dual special-Lagrangian torus fibrations $X \to B$ and $X^\vee \to B$ over the same integral affine base; (B, P) is the algebraic realisation.
TODO_REF
Nishinou, T. & Siebert, B. — Toric degenerations of toric varieties and tropical curves · *Duke Mathematical Journal* 135 (1), 1-51 (2006). Earlier appearance of the dual intersection complex B(Ξ) in the toric-Fano setting; supplies the cone-gluing construction that Gross-Siebert 2006 generalises to the Calabi-Yau setting with monodromy.
TODO_REF
Mumford, D. — An analytic construction of degenerating abelian varieties over complete rings · *Compositio Mathematica* 24 (3), 239-272 (1972). The originating example: a one-parameter degeneration of abelian varieties to a periodic tiling of $\mathbb{R}^n$, with the dual graph carrying an integral affine structure — the Mumford construction is the historical predecessor of the Gross-Siebert dual intersection complex.
TODO_REF
Kempf, G., Knudsen, F., Mumford, D. & Saint-Donat, B. — Toroidal Embeddings I · Lecture Notes in Mathematics 339, Springer (1973). Polyhedral assembly of varieties from cone data; the toroidal-embedding language that organises the gluing of toric strata along a polyhedral subdivision.
TODO_REF
Tian, G. & Zhu, S. — A note on the affine structure of toric varieties · *Math. Res. Lett.* 9 (5-6), 689-700 (2002). Affine-structure considerations on toric and degenerate-toric bases predating Gross-Siebert; useful for the SL_n(ℤ) monodromy interpretation.
TODO_REF
Gross, M. — Toric degenerations and Batyrev-Borisov duality · *Math. Ann.* 333 (3), 645-688 (2005). The Batyrev-Borisov mirror construction for Calabi-Yau complete intersections in toric varieties, recast in terms of dual intersection complexes; an early case study for the (B, P) language.
TODO_REF
Castaño-Bernard, R. & Matessi, D. — Lagrangian 3-torus fibrations · *J. Differential Geom.* 81 (3), 483-573 (2009). Symplectic-side construction of integral-affine bases with simple singularities in dimension 3; provides the Lagrangian fibration whose discriminant locus matches the codim-2 singular locus of (B, P) in dimension 3.

Lean module

Codex.AlgGeom.Tropical.DualIntersectionComplex

Reviewer

TBD

Estimated time

beginner: 25m
intermediate: 60m
master: 110m