04.11.10 · algebraic-geometry / toric

Polytope-fan dictionary; the line bundle $L_{P}$

shipped3 tiersLean: partial

Anchor (Master): Fulton §1.5, §3.4; Cox-Little-Schenck §6.1, §6.2, §6.3; Oda *Convex Bodies and Algebraic Geometry* Ch. 1 §1.5, Ch. 2 §2.3; Demazure 1970 *Sous-groupes algébriques de rang maximum*; Kempf-Knudsen-Mumford-Saint-Donat *Toroidal Embeddings I* Ch. I §3; Mumford-Suominen 1972 *Introduction to the theory of moduli*; Brion 1996 *Piecewise polynomial functions, convex polytopes and enumerative geometry*; Batyrev 1994 *Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties*

Intuition [Beginner]

A lattice polytope $P$ is a convex shape in $R^{n}$ whose corner points lie on the integer grid $Z^{n}$ . The polytope-fan dictionary says that every such polytope encodes a projective algebraic variety together with an ample line bundle, in a way that is fully reversible: any projective toric variety with a chosen ample line bundle corresponds to a unique lattice polytope. The polytope is the visible combinatorial fingerprint of the algebraic geometry hiding inside.

The recipe in one sentence is this: rotate the polytope so that each vertex points outward, and read off the inward-normal cones at the vertices. The collection of these inward-normal cones, together with all their faces, is a fan in the dual lattice. The toric variety of that fan is the projective variety associated to the polytope, and the line bundle is determined by recording how the polytope sits inside the dual space. The global sections of the line bundle are counted by the lattice points of the polytope itself — a striking identity that turns sheaf cohomology into integer-grid counting.

A concrete example makes this vivid. The unit simplex with vertices at the origin and the standard basis vectors of $R^{n}$ produces $P^{n}$ together with the line bundle $O (1)$ . The $n + 1$ lattice points of the simplex are exactly the coordinate basis of $H^{0} (P^{n}, O (1))$ .

The unit square produces $P^{1} \times P^{1}$ with the diagonal line bundle $O (1, 1)$ , and its four corner lattice points are the four bilinear basis elements of that line bundle's sections. Batyrev's 1994 application of the dictionary to reflexive polytopes — those whose polar dual is also a lattice polytope — opens the door to mirror symmetry, with pairs $(P, P^{\circ})$ of reflexive polytopes corresponding to mirror pairs of Calabi-Yau hypersurfaces.

Visual [Beginner]

A two-panel diagram of the polytope-fan dictionary. Left panel: a lattice polytope $P \subseteq M_{R} = R^{2}$ drawn as a unit square with vertices at $(0, 0), (1, 0), (1, 1), (0, 1)$ . The four lattice points of $P$ are marked, and the four inward-pointing normal arrows at each vertex are drawn. Middle: the normal fan $Σ_{P}$ in $N_{R} = R^{2}$ , with four maximal cones (the four quadrants) and four rays in directions $\pm e_{1}, \pm e_{2}$ . Right panel: the projective toric variety $X_{P} = P^{1} \times P^{1}$ with its four toric divisors $D_{1}, D_{2}, D_{3}, D_{4}$ labelled, and the four lattice points of $P$ identified with the four global sections of the line bundle $L_{P} = O (1, 1)$ .

$A three-panel schematic showing the unit square polytope in $M_\mathbb{R}$, the four-quadrant normal fan in $N_\mathbb{R}$, and the resulting projective toric variety $\mathbb{P}^1 \times \mathbb{P}^1$ with four toric divisors.$

The picture captures the dictionary as a single combinatorial pipeline: vertices of $P$ index maximal cones of $Σ_{P}$ , facets of $P$ index rays of $Σ_{P}$ (hence toric divisors of $X_{P}$ ), and lattice points of $P$ index global sections of $L_{P}$ . Reading the picture left to right corresponds to applying the functor $P \mapsto (X_{P}, L_{P})$ ; reading right to left recovers $P$ as the Newton polytope of the ample line bundle $L$ on the projective toric variety.

Worked example [Beginner]

Build the projective plane $P^{2}$ as the toric variety of the unit simplex. Take $P = conv ((0, 0), (1, 0), (0, 1)) \subseteq M_{R} = R^{2}$ , the standard unit triangle with vertices $v_{0} = (0, 0)$ , $v_{1} = (1, 0)$ , $v_{2} = (0, 1)$ .

Step 1. The lattice points. The lattice points of $P$ are exactly the three vertices: $P \cap M = {(0, 0), (1, 0), (0, 1)}$ , a total of three points. The interior of $P$ contains no lattice points (it is the open triangle of area $1/2$ ), and the edges contain only the endpoints.

Step 2. The normal fan. At each vertex $v_{i}$ , compute the cone of inward-pointing directions. At $v_{0} = (0, 0)$ : the inward directions are the first quadrant in $N_{R}$ , the cone $σ_{0} = Cone (e_{1}, e_{2})$ .

At $v_{1} = (1, 0)$ : the inward directions form the wedge pointing back into the triangle from $v_{1}$ 's perspective, the cone $σ_{1} = Cone (- e_{1}, - e_{1} - e_{2})$ . By symmetry at $v_{2} = (0, 1)$ : $σ_{2} = Cone (- e_{2}, - e_{1} - e_{2})$ . The three maximal cones together with their faces (three rays, the zero cone) form the normal fan $Σ_{P}$ .

Step 3. The toric variety. The normal fan $Σ_{P}$ has three maximal cones in $R^{2}$ and three rays in the directions $u_{0} = e_{1} + e_{2}$ (after taking primitive generators correctly), and the standard basis vectors. Comparing to the fan of $P^{2}$ from [04.11.04] (three rays at $e_{1}, e_{2}, - e_{1} - e_{2}$ ), one recovers $X_{P} = P^{2}$ after the appropriate lattice identification. The line bundle $L_{P}$ is the ample line bundle $O_{P^{2}} (1)$ — the hyperplane bundle.

What this tells us. The three lattice points of $P$ are exactly the three coordinate basis elements ${X_{0}, X_{1}, X_{2}}$ of $H^{0} (P^{2}, O (1))$ — a dimension of $3$ , matching the lattice-point count. The Demazure character formula in action: counting lattice points of a polytope and computing global sections of a line bundle on the associated toric variety give the same answer. This pattern works in every dimension and for every lattice polytope, turning the algebraic-geometric question "how many sections does this ample line bundle have?" into the combinatorial question "how many integer points does this polytope contain?"

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The polytope-fan dictionary sends a lattice polytope $P \subseteq M_{R}$ to:

A. An affine variety $Spec C [P \cap M]$ without any line bundle. B. A projective toric variety $X_{P}$ together with an ample line bundle $L_{P}$ . C. A complex manifold with no algebraic structure. D. A finite group acting on the lattice $M$ .

Hint

The construction produces both a variety and a polarisation simultaneously: the variety from the normal fan, and the line bundle from how the polytope sits in $M_{R}$ .

Answer

B. The polytope-fan dictionary produces a projective toric variety $X_{P} = X_{Σ_{P}}$ (where $Σ_{P}$ is the normal fan of $P$ ) together with an ample line bundle $L_{P}$ on $X_{P}$ . The polytope encodes both pieces of data simultaneously — the fan from the inward-normal cones at vertices, and the line bundle from the location of $P$ inside $M_{R}$ . Option A is what an affine semigroup algebra construction would produce, missing the projective and polarisation content. Option C ignores the fact that $X_{P}$ is genuinely algebraic. Option D confuses the polytope-fan dictionary with the action of $T$ on $X_{P}$ .

Exercise (easy, true-false).

True or false: the projective toric variety $X_{P}$ depends only on the normal fan $Σ_{P}$ of the polytope, but the line bundle $L_{P}$ depends on the polytope $P$ itself, not just on its normal fan.

Hint

Many different polytopes can share the same normal fan (think of dilations or translations). Such polytopes share $X_{P}$ but differ in their polarisations.

Answer

True. The normal fan $Σ_{P}$ depends only on the directions of the facets of $P$ (the inward normals), and these directions are unchanged by scaling or translating $P$ . The variety $X_{P} = X_{Σ_{P}}$ therefore depends only on the shape (combinatorial type) of $P$ , not on its size or position. The line bundle $L_{P}$ does record the size: the dilation $k P$ produces the $k$ -th tensor power of $L_{P}$ , a different line bundle on the same toric variety. The lattice-point count of $k P$ is the Ehrhart polynomial of $P$ evaluated at $k$ , growing polynomially in $k$ with leading coefficient the volume of $P$ .

Formal definition [Intermediate+]

Let $N$ be a lattice of rank $n$ with dual lattice $M = Hom_{Z} (N, Z)$ , as in [04.11.01]. Let $M_{R} = M \otimes_{Z} R$ and $N_{R} = N \otimes_{Z} R$ .

Definition (lattice polytope). A lattice polytope $P \subseteq M_{R}$ is the convex hull of a finite subset of $M$ . The polytope is full-dimensional if its affine span equals $M_{R}$ . A face of $P$ is the intersection of $P$ with a supporting hyperplane; faces of dimension $0$ are vertices, of dimension $1$ are edges, of dimension $n - 1$ are facets.

Definition (normal fan). Given a full-dimensional lattice polytope $P \subseteq M_{R}$ , the normal fan $Σ_{P}$ in $N_{R}$ is constructed as follows. For each face $F \subseteq P$ , the inward-normal cone of $P$ at $F$ is $$ \sigma_F := {u \in N_\mathbb{R} : \langle m - m_F, u\rangle \geq 0 \text{ for all } m \in P, m_F \in F}, $$ the closed cone of inward-pointing directions at $F$ . The collection $Σ_{P} = {σ_{F} : F a face of P}$ is a complete fan in $N_{R}$ . Maximal cones $σ_{v}$ are indexed by vertices $v$ of $P$ ; rays $σ_{F}$ by facets $F$ ; the zero cone $σ_{P} = {0}$ by the full polytope $P$ .

Definition (polarised toric variety from $P$ ). Given a full-dimensional lattice polytope $P$ , the polarised toric variety from $P$ is the pair $(X_{P}, L_{P})$ where:

(i) $X_{P} = X_{Σ_{P}}$ is the toric variety of the normal fan, constructed by the fan-gluing of [04.11.04].

(ii) $L_{P}$ is the polarisation line bundle on $X_{P}$ , defined locally on each affine chart $U_{σ_{v}}$ by the section $χ^{v} \in C [S_{σ_{v}}]$ , with transition $χ^{v - v^{'}}$ from $U_{σ_{v}}$ to $U_{σ_{v^{'}}}$ on the overlap $U_{σ_{v} \cap σ_{v^{'}}}$ . The cocycle condition $χ^{v - v^{'}} \cdot χ^{v^{'} - v^{''}} = χ^{v - v^{''}}$ holds tautologically.

Definition (support function). The line bundle $L_{P}$ is encoded equivalently by the support function $ψ_{P} : N_{R} \to R$ defined by $$ \psi_P(u) := \min_{m \in P} \langle m, u\rangle = \min_{v \text{ vertex of } P} \langle v, u\rangle, $$ the lower-supporting function of $P$ . The function $ψ_{P}$ is piecewise linear on the cones of $Σ_{P}$ : on each maximal cone $σ_{v}$ , $ψ_{P} ∣_{σ_{v}} (u) = ⟨ v, u ⟩$ . The line bundle $L_{P}$ is strictly convex (equivalently $ψ_{P}$ is strictly convex on the fan) iff different maximal cones give different linear functions, iff different vertices $v$ of $P$ produce different elements of $M$ .

Definition (Demazure character). The Demazure character formula identifies the global sections of $L_{P}$ : $$ H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m, $$ with $T$ -weight decomposition under the dense-torus action. In particular, $$ \dim_\mathbb{C} H^0(X_P, L_P) = |P \cap M|. $$

Counterexamples to common slips

"The polytope $P$ is recovered from the fan $Σ_{P}$ alone." It is not. The polytope records both the shape (via the inward-normal directions, which determine $Σ_{P}$ ) and the size (via the actual lattice points of $P$ , which determine the polarisation $L_{P}$ ). Different lattice polytopes with the same normal fan — for example $P$ and $k P$ — share $X_{P}$ but have distinct polarisations $L_{P}, L_{P}^{\otimes k}$ . To recover $P$ from $(X_{P}, L_{P})$ one needs both pieces of data: the fan recovers the variety, and the support function $ψ_{P}$ recovers the polytope.
"Every projective toric variety has a unique polarisation." It does not. The Picard group $Pic (X_{Σ})$ of a smooth complete toric variety is the abelian group of line bundles, with finite rank equal to $∣Σ (1) ∣ - n$ ; the ample cone is a full-dimensional open cone in $Pic (X_{Σ}) \otimes R$ . Each ample line bundle $L$ corresponds to a different polytope $P_{L}$ , the Newton polytope. The polytope-fan dictionary applies to choices of polarisation, not to varieties alone.
"Lattice points of $P$ all give linearly independent sections." They do, but only because $L_{P}$ is ample (and in particular the polytope is full-dimensional with strictly convex support function). For non-ample line bundles on a toric variety, the lattice-point count of the associated polytope can fail to compute $h^{0}$ of the line bundle — the relevant invariant is the lattice-point count of the lower convex hull of the relevant lattice points, not of the polytope itself. The Demazure character formula in the form stated requires ampleness of $L_{P}$ .

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the polytope-fan equivalence: a bijection between full-dimensional lattice polytopes $P$ (up to lattice translation) and pairs $(X, L)$ of projective normal toric varieties with ample $T$ -equivariant line bundles (up to isomorphism). The Demazure character formula provides the explicit identification of global sections with lattice points.

Theorem (Polytope-fan correspondence; Demazure 1970, Fulton 1993 §3.4). Let $N$ be a free $Z$ -module of finite rank.

(a) From polytope to projective toric. Every full-dimensional lattice polytope $P \subseteq M_{R}$ produces a pair $(X_{P}, L_{P})$ consisting of a projective normal toric variety $X_{P} = X_{Σ_{P}}$ and an ample $T$ -equivariant line bundle $L_{P}$ on $X_{P}$ .

(b) Demazure character formula. The global sections decompose as $$ H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m, $$ as a $T$ -representation, where the right-hand side has $T$ -weight $m$ on each summand. In particular $dim H^{0} (X_{P}, L_{P}) = ∣ P \cap M ∣$ .

(c) From projective toric to polytope. Conversely, every pair $(X, L)$ of a projective normal toric variety with an ample $T$ -equivariant line bundle is of the form $(X_{P}, L_{P})$ for a unique lattice polytope $P$ up to lattice translation.

(d) Functoriality. The assignments $P \mapsto (X_{P}, L_{P})$ and $(X, L) \mapsto P_{L}$ are functorial in lattice maps respecting cone structure and ample-bundle morphisms respectively, and they are mutually inverse, giving an equivalence of categories.

Proof.

(a) Construction of $(X_{P}, L_{P})$ . The normal fan $Σ_{P}$ in $N_{R}$ is a complete fan by the standard convex-geometric argument: every direction $u \in N_{R}$ is the inward normal of $P$ at some face, hence lies in some $σ_{F}$ , so $∣ Σ_{P} ∣ = N_{R}$ . By the fan-to-toric construction of [04.11.04], $X_{P} = X_{Σ_{P}}$ exists as a normal separated complete scheme of finite type over $C$ .

To construct $L_{P}$ , work on each maximal cone $σ_{v}$ (for $v$ a vertex of $P$ ). On the affine chart $U_{σ_{v}} = Spec C [S_{σ_{v}}]$ , define the local section $s_{v} = χ^{v} \in C [S_{σ_{v}}]$ . (We have $v \in S_{σ_{v}}$ because $σ_{v}$ is the inward-normal cone at $v$ , so $⟨ v, u ⟩ \geq ⟨ v, u ⟩$ by definition, and shifting by the appropriate translation, $v \in (σ_{v})^{\lor} + v$ . The full check uses that $P - v$ lies in $(σ_{v})^{\lor}$ as a polytope of $M_{R}$ .)

On the overlap $U_{σ_{v}} \cap U_{σ_{v^{'}}} = U_{σ_{v} \cap σ_{v^{'}}}$ , the transition $s_{v} / s_{v^{'}} = χ^{v - v^{'}}$ is a regular invertible function. The element $v - v^{'}$ lies in $S_{σ_{v} \cap σ_{v^{'}}}^{\pm} := (σ_{v} \cap σ_{v^{'}})^{\lor} \cap M \oplus - (σ_{v} \cap σ_{v^{'}})^{\lor} \cap M$ — the lattice points in the linear span of the dual to the common face. The cocycle condition $$ \chi^{v - v'} \cdot \chi^{v' - v''} = \chi^{(v - v') + (v' - v'')} = \chi^{v - v''} $$ holds tautologically, so the local sections ${s_{v}}_{v}$ define a line bundle $L_{P}$ on $X_{P}$ .

Ampleness of $L_{P}$ is the strict-convexity of the support function $ψ_{P}$ . By construction $ψ_{P} ∣_{σ_{v}} (u) = ⟨ v, u ⟩$ , and strict convexity says different maximal cones give different linear functions: $v \neq = v^{'} \Rightarrow ψ_{P} ∣_{σ_{v}} \neq = ψ_{P} ∣_{σ_{v^{'}}}$ , equivalent to the polytope $P$ having distinct vertices. By the criterion for ampleness of toric line bundles (Cox-Little-Schenck Theorem 6.1.14; Fulton §3.4 Proposition), strict convexity of the support function is equivalent to $L_{P}$ being ample.

(b) Demazure character formula. The global sections of $L_{P}$ are functions on $X_{P}$ that, restricted to each affine chart $U_{σ_{v}}$ , give regular sections of $L_{P} ∣_{U_{σ_{v}}}$ . Since $L_{P} ∣_{U_{σ_{v}}}$ is the product line bundle on the affine chart (with local generator $s_{v} = χ^{v}$ ), a section over $U_{σ_{v}}$ is a function $f_{v} \in C [S_{σ_{v}}]$ — equivalently a polynomial in the characters $χ^{m}$ for $m \in S_{σ_{v}}$ .

A global section is a compatible system ${f_{v}}_{v}$ : on each overlap, $f_{v} / s_{v} = f_{v^{'}} / s_{v^{'}}$ (since both equal the global section value), equivalently $f_{v} \cdot χ^{- v} = f_{v^{'}} \cdot χ^{- v^{'}}$ . Writing $f_{v} = \sum_{m} a_{v, m} χ^{m}$ and matching coefficients of $χ^{m + (- v^{'})}$ on each side, the compatibility condition forces $a_{v, m} = a_{v^{'}, m^{'}}$ for $m - v = m^{'} - v^{'}$ , i.e., $m, m^{'}$ correspond to the same "absolute" lattice point. Setting $m = m - v \in M$ (translated to be measured from the polytope vertex), a global section is parametrised by an assignment of coefficients $a_{m}$ for $m \in M$ , with the constraint that on each chart $m + v \in S_{σ_{v}}$ , i.e., $m \in (S_{σ_{v}} - v) = P - v$ (translated).

For the global section to be regular on every chart, we need $m + v \in S_{σ_{v}}$ for every maximal cone $σ_{v}$ . Taking $m = m$ (un-translating), this is $m \in (σ_{v})^{\lor} + v$ for every $v$ , equivalent to $⟨ m - v, u ⟩ \geq 0$ for $u \in σ_{v}$ — equivalent to $m \in P$ (since $P$ is the intersection of the half-spaces ${m : ⟨ m - v_{i}, u_{i} ⟩ \geq 0}$ for the facet inward normals $u_{i}$ at the vertices $v_{i}$ ).

Hence the global sections are parametrised exactly by lattice points $m \in P \cap M$ , with the section $$ s_m|{U{\sigma_v}} = \chi^{m - v} \cdot s_v = \chi^{m - v} \cdot \chi^v = \chi^m, $$ a $T$ -semi-invariant of weight $m$ . The $T$ -weight decomposition is $$ H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot s_m, \qquad T \text{-weight}(s_m) = m, $$ which is the Demazure character formula. The dimension is the cardinality $∣ P \cap M ∣$ .

(c) Converse: from projective toric to polytope. Suppose $X$ is a projective normal toric variety with ample $T$ -equivariant line bundle $L$ . The global sections $H^{0} (X, L)$ form a finite-dimensional $T$ -representation. Since $T$ acts on the open dense torus $T \subseteq X$ by translation and the action lifts to $L$ (assumed $T$ -equivariant), the $T$ -action on $H^{0} (X, L)$ decomposes the space into $T$ -character spaces: $$ H^0(X, L) = \bigoplus_{m \in M} H^0(X, L)_m, $$ each $H^{0} (X, L)_{m}$ being the subspace of $T$ -weight $m$ . The set $supp (L) := {m \in M : H^{0} (X, L)_{m} \neq = 0}$ is finite.

Define $P_{L} := conv (supp (L)) \subseteq M_{R}$ , the convex hull. By the theory of toric line bundles (Cox-Little-Schenck Theorem 6.2.1), the ample condition implies that $supp (L) = P_{L} \cap M$ (every interior lattice point appears with multiplicity one) and that $supp (L)$ is exactly the set of lattice points of $P_{L}$ . Hence $P_{L}$ is a lattice polytope.

Moreover, the vertex cones of $P_{L}$ are exactly the maximal cones of the fan $Σ$ of $X$ . To see this: the support function of $L$ on $Σ$ is the function $ψ_{L} (u) = min_{m \in supp (L)} ⟨ m, u ⟩$ , the lower convex envelope of $supp (L)$ . Ampleness of $L$ is equivalent to strict convexity of $ψ_{L}$ on $Σ$ (Cox-Little-Schenck Theorem 6.1.14), and strict convexity says different maximal cones produce different linear functions $u \mapsto ⟨ v, u ⟩$ . The vertices $v$ realising the minimum on each maximal cone $σ$ are exactly the vertices of $P_{L}$ , and the inward-normal cone of $P_{L}$ at $v$ equals $σ$ . Hence $Σ$ is the normal fan of $P_{L}$ , and $X = X_{P_{L}}$ , $L = L_{P_{L}}$ .

Uniqueness of $P_{L}$ up to translation: any global section $s$ of $L$ that does not vanish on the open torus determines a translation of $P_{L}$ (different choices of nowhere-vanishing local generator on the torus shift $P_{L}$ by a lattice element of $M$ ). The polytope $P_{L}$ is well-defined modulo this lattice translation, and the line bundle is unchanged under translation.

(d) Functoriality. A morphism of lattice polytopes $ϕ : (M, P) \to (M^{'}, P^{'})$ (a $Z$ -linear map $M \to M^{'}$ sending lattice points of $P$ into lattice points of $P^{'}$ ) induces:

(i) A morphism of normal fans $ϕ^{*} : (N^{'}, Σ_{P^{'}}) \to (N, Σ_{P})$ on the dual lattices, with $ϕ^{*}$ the dual of $ϕ$ .

(ii) A toric morphism $X_{ϕ^{*}} : X_{P^{'}} \to X_{P}$ via the fan-functoriality of [04.11.04].

(iii) An equivariant morphism of line bundles $X_{ϕ^{*}}^{*} L_{P} \to L_{P^{'}}$ (pulled-back line bundle equals the polarisation from the pulled-back polytope, by construction).

The functor laws (identity, composition) follow from those of the underlying lattice-map operations. The inverse functor $(X, L) \mapsto P_{L}$ is functorial in toric morphisms via the support-function pullback. The mutual inversion of these functors — up to lattice translation of the polytope — is the assertion (c) above, completing the equivalence of categories.

$□$

Bridge. The polytope-fan correspondence builds toward [04.11.05] (orbit-cone correspondence — orbits of $X_{P}$ are stratified by faces of $P$ ) and [04.11.08] (toric divisors and support functions — the line bundle $L_{P}$ decomposes as $L_{P} = \sum_{F} a_{F} D_{F}$ over the facets $F$ of $P$ with toric divisors $D_{F}$ ), and the central insight is that the lattice polytope $P$ records both the projective toric variety and a polarisation simultaneously. The foundational reason this dual record is possible is that the normal-fan-construction is shape-only, depending only on the directions of facets, while the polarisation is size-also, depending on the actual lattice points of $P$ . The bridge is the support function $ψ_{P}$ : a single piecewise-linear convex function on $N_{R}$ that encodes both the fan (via its domains of linearity) and the line bundle (via its slope on each domain).

Putting these together, the polytope-fan dictionary identifies the category of lattice polytopes (with translation-equivariant lattice morphisms) with the category of polarised projective toric varieties (with $T$ -equivariant ample-bundle morphisms). This is exactly the algebraic-geometric counterpart of the Delzant correspondence in symplectic geometry — Delzant 1988 showed that simple smooth rational polytopes classify symplectic toric manifolds, and the projective-toric overlap of the two correspondences identifies polytopes with both algebraic and symplectic structures. The pattern appears again in [04.11.11] (algebraic moment map — the moment polytope is exactly $P$ for the compact-torus action on $X_{P}$ ), generalises in [04.11.13] (toric intersection theory — degrees and intersection numbers compute as lattice-volume invariants of $P$ ), and is dual to the orbit-stratification of $X_{P}$ by faces of $P$ — orbit closures of dimension $k$ are in bijection with faces of $P$ of dimension $k$ . The bridge is the Brion-Vergne identification $χ (X_{P}, L_{P}) = \sum_{m \in P \cap M} 1 = ∣ P \cap M ∣$ via Ehrhart polynomiality.

Exercises [Intermediate+]

Exercise 1 (easy, symbolic).

Compute the normal fan and lattice-point count for the unit simplex $P = conv ((0, 0), (1, 0), (0, 1)) \subseteq R^{2}$ . Identify the resulting toric variety and line bundle.

Hint

The three vertices give three maximal cones. Compute the inward-normal cone at each vertex. The result should be the fan of $P^{2}$ from [04.11.04].

Answer

Vertices. $v_{0} = (0, 0)$ , $v_{1} = (1, 0)$ , $v_{2} = (0, 1)$ .

Lattice points. $P \cap M = {v_{0}, v_{1}, v_{2}}$ (three points: only the vertices; the interior triangle and the edges contain no other lattice points).

Normal cones. At each vertex $v_{i}$ , compute the cone of inward directions ${u \in N_{R} : ⟨ m - v_{i}, u ⟩ \geq 0 for m \in P}$ .

At $v_{0} = (0, 0)$ : the cone is generated by inward normals to the two facets at $v_{0}$ , namely the facets along ${m_{1} = 0}$ and ${m_{2} = 0}$ (the two coordinate axes restricted to $P$ ). The inward normals are $e_{1}, e_{2}$ , so $σ_{0} = Cone (e_{1}, e_{2})$ , the first quadrant.
At $v_{1} = (1, 0)$ : the inward normals are to the facets along ${m_{1} = 1, m_{2} = 0}$ -bounding-direction (the edge from $v_{1}$ to $v_{2}$ , with inward normal $- e_{1} - e_{2}$ ) and along ${m_{2} = 0}$ (inward normal $e_{2}$ ). So $σ_{1} = Cone (- e_{1} - e_{2}, e_{2})$ .

Wait — the inward normal at $v_{1}$ to the facet ${m_{2} = 0}$ is $+ e_{2}$ (pointing into the polytope from that edge), and the inward normal to the edge from $v_{1}$ to $v_{2}$ (along direction $- e_{1} + e_{2}$ ) is the perpendicular $- e_{1} - e_{2}$ pointing into the polytope. So $σ_{1} = Cone (- e_{1} - e_{2}, e_{2})$ . The primitive ray generators are $- e_{1} - e_{2}$ and $e_{2}$ .
At $v_{2} = (0, 1)$ : by symmetry, $σ_{2} = Cone (- e_{1} - e_{2}, e_{1})$ .

Fan. Three rays in $R^{2}$ : primitive generators $u_{0} = e_{1}, u_{1} = e_{2}, u_{2} = - e_{1} - e_{2}$ (with $u_{0} + u_{1} + u_{2} = 0$ ). Three maximal cones (the three two-dimensional cones above) plus the zero cone. Comparing with [04.11.04] Exercise 1, this is exactly the fan of $P^{2}$ .

Toric variety and line bundle. $X_{P} = P^{2}$ , $L_{P} = O_{P^{2}} (1)$ (the hyperplane bundle). The Demazure formula gives $dim H^{0} (P^{2}, O (1)) = ∣ P \cap M ∣ = 3$ , matching the standard basis ${X_{0}, X_{1}, X_{2}}$ of the hyperplane sections.

Rubric: full credit for the three vertices, the lattice-point count of $3$ , the three maximal cones, identification with the fan of $P^{2}$ , and the line-bundle identification $O (1)$ .

Exercise 2 (easy, symbolic).

Compute the normal fan and lattice-point count for the unit square $P = [0, 1]^{2} \subseteq R^{2}$ . Identify the resulting toric variety and line bundle.

Hint

The four vertices of the square give four maximal cones — the four quadrants. The lattice-point count is the four corners (no interior or edge lattice points beyond corners). The result should be $P^{1} \times P^{1}$ with $O (1, 1)$ .

Answer

Vertices. $v_{1} = (0, 0), v_{2} = (1, 0), v_{3} = (1, 1), v_{4} = (0, 1)$ .

Lattice points. $P \cap M = {v_{1}, v_{2}, v_{3}, v_{4}}$ , total of $4$ lattice points (the four corners; no interior points).

Normal cones. By direct computation at each vertex:

$σ_{1} = Cone (e_{1}, e_{2})$ — first quadrant (inward from $v_{1} = (0, 0)$ ).
$σ_{2} = Cone (- e_{1}, e_{2})$ — second quadrant.
$σ_{3} = Cone (- e_{1}, - e_{2})$ — third quadrant.
$σ_{4} = Cone (e_{1}, - e_{2})$ — fourth quadrant.

Fan. Four rays $\pm e_{1}, \pm e_{2}$ and four maximal cones (the quadrants). This is the fan of $P^{1} \times P^{1}$ from [04.11.04] Exercise 2.

Toric variety and line bundle. $X_{P} = P^{1} \times P^{1}$ , $L_{P} = O (1, 1) = pr_{1}^{*} O (1) \otimes pr_{2}^{*} O (1)$ . The Demazure formula gives $$ \dim H^0(\mathbb{P}^1 \times \mathbb{P}^1, \mathcal{O}(1, 1)) = |P \cap M| = 4, $$ matching the bilinear-form basis ${X_{0} Y_{0}, X_{0} Y_{1}, X_{1} Y_{0}, X_{1} Y_{1}}$ for $O (1) ⊠ O (1)$ .

Rubric: full credit for the four vertices, four lattice points, four quadrant cones, identification with the fan of $P^{1} \times P^{1}$ , and the line bundle $O (1, 1)$ .

Exercise 3 (medium, numeric).

Compute the number of lattice points of the dilation $k P$ of the unit simplex $P$ in $R^{2}$ for $k = 1, 2, 3, 4$ . Find the polynomial $L (P, k) = ∣ k P \cap M ∣$ as a function of $k$ .

Hint

The dilation $k P$ is the triangle with vertices $(0, 0), (k, 0), (0, k)$ . Count integer points by triangular-number formulas. The Ehrhart polynomial of the unit simplex in dimension $n$ is $L (P, k) = (n k + n)$ .

Answer

$k = 1$ . $∣ P \cap M ∣ = 3$ (the three vertices of the simplex).

$k = 2$ . $2 P$ is the triangle with vertices $(0, 0), (2, 0), (0, 2)$ . Lattice points: $(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (0, 2)$ , total $6$ .

$k = 3$ . $3 P$ is the triangle with vertices $(0, 0), (3, 0), (0, 3)$ . Lattice points satisfy $m_{1}, m_{2} \geq 0$ and $m_{1} + m_{2} \leq 3$ . Count: $(m_{1} + m_{2} = 0)$ : $1$ ; $(= 1)$ : $2$ ; $(= 2)$ : $3$ ; $(= 3)$ : $4$ ; total $1 + 2 + 3 + 4 = 10$ .

$k = 4$ . $4 P$ : lattice points with $m_{1} + m_{2} \leq 4$ , all non-negative. Count: $1 + 2 + 3 + 4 + 5 = 15$ .

Polynomial. The sequence $3, 6, 10, 15$ is the triangular-number sequence $T_{k + 1} = (2 k + 2)$ . The Ehrhart polynomial of the unit 2-simplex is $$ L(P, k) = \binom{k + 2}{2} = \frac{(k + 1)(k + 2)}{2}, $$ a polynomial of degree $2$ in $k$ with leading coefficient $1/2 = vol (P)$ (the Euclidean area of the unit triangle). The general formula for the unit $n$ -simplex is $L (P, k) = (n k + n)$ .

Interpretation. The lattice-point count $∣ k P \cap M ∣$ is the Hilbert function $dim H^{0} (P^{n}, O (k))$ , the dimension of degree- $k$ homogeneous polynomials in $n + 1$ variables. For $n = 2$ : $dim H^{0} (P^{2}, O (k)) = (2 k + 2)$ , matching the count.

Rubric: full credit for the four count values, the polynomial $(2 k + 2)$ , and the identification with the Hilbert function of $O (k)$ on $P^{2}$ .

Exercise 4 (medium, symbolic).

Identify the projective toric variety $X_{P}$ and line bundle $L_{P}$ for the polytope $P = conv ((0, 0), (a, 0), (a, b), (0, b)) \subseteq R^{2}$ (an $a \times b$ rectangle). Verify the Demazure formula by computing both $∣ P \cap M ∣$ and $dim H^{0} (X_{P}, L_{P})$ .

Hint

The rectangle has the same normal fan as the unit square — it's a dilation in each factor independently. The line bundle changes with the dimensions.

Answer

Normal fan. The $a \times b$ rectangle has facets in the same four directions as the unit square: ${m_{1} = 0}, {m_{1} = a}, {m_{2} = 0}, {m_{2} = b}$ . The inward normals at the vertices are $\pm e_{1}, \pm e_{2}$ in the same combinatorial pattern as for the unit square. The normal fan is identical to that of the unit square: four quadrants, four rays $\pm e_{1}, \pm e_{2}$ .

Toric variety. $X_{P} = P^{1} \times P^{1}$ (same as for the unit square).

Line bundle. The polarisation $L_{P}$ is determined by the polytope's size, not its fan. By the construction: the local section on $U_{σ_{v}}$ for vertex $v$ is $χ^{v}$ , and the local generator changes with the dimensions. Concretely, $L_{P} = O (a, b) = pr_{1}^{*} O (a) \otimes pr_{2}^{*} O (b)$ .

Demazure formula check. The lattice-point count of the $a \times b$ rectangle (for $a, b \geq 1$ integers) is $$ |P \cap M| = (a + 1)(b + 1), $$ the product of the lattice-point counts in each factor. On the other side: $$ \dim H^0(\mathbb{P}^1 \times \mathbb{P}^1, \mathcal{O}(a, b)) = \dim H^0(\mathbb{P}^1, \mathcal{O}(a)) \cdot \dim H^0(\mathbb{P}^1, \mathcal{O}(b)) = (a + 1)(b + 1), $$ matching the lattice-point count by the Künneth formula. The Demazure formula is verified.

Rubric: full credit for the normal-fan identification, the identification $X_{P} = P^{1} \times P^{1}$ , the line bundle $O (a, b)$ , and the lattice-point count $(a + 1) (b + 1)$ matching the section count.

Exercise 5 (medium, symbolic).

The polytope $P = conv ((0, 0), (1, 0), (1, 1), (0, 2)) \subseteq R^{2}$ (a trapezoid) has normal fan equal to the fan of the Hirzebruch surface $F_{1}$ . Compute the line bundle $L_{P}$ in terms of the standard generators of $Pic (F_{1})$ .

Hint

The Hirzebruch surface $F_{1} = P (O_{P^{1}} \oplus O_{P^{1}} (1))$ has Picard group $Pic (F_{1}) = Z^{2}$ generated by the fiber class $F$ and a section class $S$ (or equivalently $H$ and $E$ for hyperplane and exceptional). The line bundle $L_{P}$ is determined by the height differences of the polytope's vertices.

Answer

Normal fan. The trapezoid has four facets: two vertical (along $m_{1} = 0$ and $m_{1} = 1$ ), one horizontal bottom ( $m_{2} = 0$ ), and one diagonal top (along $m_{1} + m_{2} = 2$ , from $(1, 1)$ to $(0, 2)$ ). The inward normals are $e_{1}, - e_{1}, e_{2}$ , and $- e_{1} - e_{2}$ respectively. These four rays in $R^{2}$ together with the four maximal cones spanned by adjacent rays form the fan of the Hirzebruch surface $F_{1}$ (compare [04.11.04] Exercise 3 with parameter $a = 1$ ).

Toric variety. $X_{P} = F_{1}$ , the first Hirzebruch surface.

Line bundle. The Picard group $Pic (F_{1}) = Z ⟨ F, S ⟩$ has two generators: $F$ (the fiber class of the $P^{1}$ -bundle structure) and $S$ (a section class — specifically $S = D_{4}$ , the toric divisor corresponding to the ray $ρ_{4} = - e_{2}$ , or equivalently the negative-curve section).

The line bundle $L_{P}$ on $F_{1}$ corresponds to the toric divisor $D = a_{1} D_{ρ_{1}} + a_{2} D_{ρ_{2}} + a_{3} D_{ρ_{3}} + a_{4} D_{ρ_{4}}$ , where $a_{i} = - ψ_{P} (u_{i})$ are the negatives of the support-function values on the ray generators. Reading off the trapezoid:

$ψ_{P} (e_{1}) = 0$ (since the inward-normal to ${m_{1} = 0}$ from the vertex $(0, 0)$ gives the linear function $⟨(0, 0), e_{1} ⟩ = 0$ ).
$ψ_{P} (- e_{1}) = - 1$ (from the vertex $(1, *)$ , value $⟨(1, *), - e_{1} ⟩ = - 1$ ).
$ψ_{P} (e_{2}) = 0$ .
$ψ_{P} (- e_{1} - e_{2}) = - 2$ (from the vertex $(0, 2)$ : $⟨(0, 2), - e_{1} - e_{2} ⟩ = - 2$ ).

Hence $a_{1} = 0, a_{2} = 1, a_{3} = 0, a_{4} = 2$ , and $$ L_P = D_2 + 2 D_4 = (\text{section class}) + 2 \cdot (\text{negative-curve class}) = S + 2 \cdot S = 3 S \quad \text{or, in $(F, S)$ basis: } L_P = F + 2 S. $$

(Choice of generators varies by convention; the canonical reading is $L_{P} = (fiber) + 2 \cdot (negative section) = F + 2 σ$ in CLS conventions for $F_{1}$ , with $σ$ the negative section.)

Demazure check. Lattice-point count: the trapezoid has $5$ lattice points: $(0, 0), (1, 0), (1, 1), (0, 1), (0, 2)$ (note: $(0, 1)$ is on the boundary edge of the trapezoid). The line bundle $L_{P} = F + 2 σ$ on $F_{1}$ has $dim H^{0} (F_{1}, L_{P}) = 5$ , matching the count.

Rubric: full credit for the normal-fan identification with $F_{1}$ , the line-bundle decomposition $L_{P} = F + 2 σ$ (or equivalent), and the lattice-point count of $5$ .

Exercise 6 (hard, short-answer).

Prove that the support function $ψ_{P} : N_{R} \to R$ defined by $ψ_{P} (u) = min_{m \in P} ⟨ m, u ⟩$ is strictly convex on the normal fan $Σ_{P}$ , in the sense that for any two adjacent maximal cones $σ_{v}, σ_{v^{'}}$ with common facet $τ = σ_{v} \cap σ_{v^{'}}$ , the linear functions $ψ_{P} ∣_{σ_{v}}$ and $ψ_{P} ∣_{σ_{v^{'}}}$ disagree along the dimension transverse to $τ$ .

Hint

On the maximal cone $σ_{v}$ , $ψ_{P} (u) = ⟨ v, u ⟩$ (the minimum is attained at the vertex $v$ ). Strict convexity at the wall says the function bends upward across the wall, equivalently the vertex $v$ is genuinely different from $v^{'}$ .

Answer

By construction, on the maximal cone $σ_{v}$ (the inward-normal cone at the vertex $v$ ), the support function $ψ_{P} ∣_{σ_{v}}$ is the linear function $u \mapsto ⟨ v, u ⟩$ : for any $u \in σ_{v}$ , the minimum $min_{m \in P} ⟨ m, u ⟩$ is attained at $m = v$ (this is exactly the defining property of $σ_{v}$ ).

For two adjacent maximal cones $σ_{v}, σ_{v^{'}}$ sharing a facet $τ$ , the cones $σ_{v}$ and $σ_{v^{'}}$ correspond to two adjacent vertices $v, v^{'}$ of $P$ (sharing an edge of $P$ — the edge being the face dual to the facet $τ$ ). Since $v \neq = v^{'}$ as elements of $M$ (different vertices of a polytope are different lattice points), the linear functions $u \mapsto ⟨ v, u ⟩$ and $u \mapsto ⟨ v^{'}, u ⟩$ are different elements of $Hom (N, R) = M_{R}$ .

These two linear functions agree on the facet $τ$ (the common boundary of $σ_{v}$ and $σ_{v^{'}}$ ): for $u \in τ$ , both $v$ and $v^{'}$ achieve the minimum $⟨ m, u ⟩$ over $m \in P$ — because the facet $τ$ is exactly the locus of $u$ for which the minimum is achieved at multiple vertices, namely both endpoints of the edge $v v^{'}$ of $P$ . Hence $⟨ v, u ⟩ = ⟨ v^{'}, u ⟩$ for $u \in τ$ , equivalent to $v - v^{'} \in τ^{⊥} = span_{R} (τ)^{⊥}$ . The vector $v - v^{'}$ is non-zero (different vertices) and lies in $τ^{⊥}$ , so it represents the "jump direction" of the support function across $τ$ .

The strict convexity assertion is then: $$ \psi_P(u_+ + u_-) < \psi_P|{\sigma_v}(u_+ + u_-) + \psi_P|{\sigma{v'}}(u_+ + u_-) $$ for $u_{+} \in σ_{v} ∖ τ$ and $u_{-} \in σ_{v^{'}} ∖ τ$ such that $u_{+} + u_{-}$ is in some larger ambient region. The relevant statement is more directly: $$ \psi_P(u) > \langle v, u\rangle \quad \text{for } u \in \sigma{v'} \setminus \tau, $$ which says the linear extension of $ψ_{P} ∣_{σ_{v}}$ to the neighbouring cone $σ_{v^{'}}$ is strictly smaller than the actual value of $ψ_{P}$ there — equivalently, $ψ_{P}$ is strictly upper-bounded by its linear extension from $σ_{v}$ across the wall $τ$ . The strict upper-bound is equivalent to $⟨ v^{'}, u ⟩ < ⟨ v, u ⟩$ for $u \in σ_{v^{'}} ∖ τ$ , which holds because $v$ achieves the minimum $⟨ m, u ⟩ = ⟨ v^{'}, u ⟩$ at $m = v^{'}$ on $σ_{v^{'}}$ , so by definition $⟨ v^{'}, u ⟩ \leq ⟨ v, u ⟩$ with equality only on $τ$ .

Hence strict convexity holds at every wall $τ$ . This is exactly the ampleness criterion for $L_{P}$ : by the toric-line-bundle ampleness criterion (Cox-Little-Schenck Theorem 6.1.14; Fulton §3.4), a $T$ -equivariant line bundle on a complete normal toric variety $X_{Σ}$ is ample iff its support function is strictly convex on $Σ$ , in exactly this sense. Therefore $L_{P}$ is ample. $□$

Rubric: full credit for the identification of $ψ_{P} ∣_{σ_{v}} (u) = ⟨ v, u ⟩$ , the agreement-on-wall analysis, the strict-inequality argument off the wall, and the linkage to ampleness of $L_{P}$ .

Exercise 7 (medium, multiple choice).

For a full-dimensional lattice polytope $P \subseteq M_{R}$ , the Ehrhart polynomial $L (P, k) = ∣ k P \cap M ∣$ has degree:

A. $0$ — it is constant. B. $1$ — it is linear. C. $n = dim M_{R}$ — it is a polynomial of degree equal to the dimension of $P$ . D. $∣ P \cap M ∣ - 1$ .

Hint

The leading term of $L (P, k)$ is $vol (P) \cdot k^{n}$ , where $vol (P)$ is the Euclidean volume of $P$ . This forces the polynomial to have degree exactly $n$ for a full-dimensional polytope.

Answer

C. Ehrhart's theorem (Ehrhart 1962, C. R. Acad. Sci.) states that for a full-dimensional lattice polytope $P \subseteq M_{R}$ of dimension $n$ , the function $L (P, k) = ∣ k P \cap M ∣$ extends from $k \in Z_{> 0}$ to a polynomial in $k$ of degree exactly $n$ , with:

Leading coefficient $vol (P)$ (the Euclidean volume of $P$ ).
Second-highest coefficient $vol_{n - 1} (\partial P) /2$ (half the surface area).
Constant term $L (P, 0) = 1$ (Ehrhart's constant-term theorem).

For the unit 2-simplex (Exercise 3), $L (P, k) = (k + 1) (k + 2) /2$ has degree $2 = dim M_{R}$ with leading coefficient $1/2 = vol (unit triangle)$ . For the unit square, $L (P, k) = (k + 1)^{2}$ has degree $2$ with leading coefficient $1$ . For the standard simplex in dimension $n$ , $L (P, k) = (n k + n)$ has degree $n$ .

The algebraic-geometric interpretation: the Ehrhart polynomial is exactly the Hilbert polynomial of the projective toric variety $X_{P}$ with respect to the ample line bundle $L_{P}$ : $$ L(P, k) = \dim H^0(X_P, L_P^{\otimes k}), $$ by the Demazure character formula applied to $k P$ . The polynomial degree $n$ matches the dimension of $X_{P}$ (equal to $dim M_{R} = n$ for a full-dimensional polytope), and the leading coefficient $vol (P)$ matches the degree of $X_{P} ↪ P^{∣ P \cap M ∣ - 1}$ under the polarisation. Options A, B are wrong for any full-dimensional polytope; option D confuses the polynomial degree with the section dimension at $k = 1$ .

Exercise 8 (hard, short-answer).

Prove the Demazure character formula $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ by working through the $T$ -equivariant local-to-global computation on the fan of $X_{P}$ . Explain why ampleness of $L_{P}$ is needed.

Hint

Compute $H^{0}$ via the Čech complex on the open cover ${U_{σ_{v}}}$ for vertices $v$ of $P$ . Each local section is a $T$ -weight space; gluing conditions on overlaps cut out exactly the lattice-point condition $m \in P$ .

Answer

Setup. Let $P$ be a full-dimensional lattice polytope with vertices $v_{1}, \dots, v_{N}$ and normal fan $Σ_{P}$ with maximal cones $σ_{1}, \dots, σ_{N}$ (cone $σ_{i}$ is the inward-normal cone at vertex $v_{i}$ ). The toric variety $X_{P} = X_{Σ_{P}}$ has affine cover ${U_{σ_{i}}}_{i = 1}^{N}$ , and the line bundle $L_{P}$ has local generator $s_{i} = χ^{v_{i}}$ on $U_{σ_{i}}$ , with transitions $s_{i} / s_{j} = χ^{v_{i} - v_{j}}$ on the overlaps.

Step 1: $T$ -weight decomposition. Each $H^{0} (U_{σ_{i}}, L_{P} ∣_{U_{σ_{i}}})$ is a free rank-one $C [S_{σ_{i}}]$ -module (with generator $s_{i}$ ), so as a $T$ -representation: $$ H^0(U_{\sigma_i}, L_P|{U{\sigma_i}}) = \bigoplus_{m \in S_{\sigma_i} + v_i} \mathbb{C} \cdot \chi^m, $$ where the shift by $v_{i}$ comes from the local generator $s_{i} = χ^{v_{i}}$ : a local section is $f \cdot s_{i}$ for $f \in C [S_{σ_{i}}]$ , with $T$ -weight of $f \cdot s_{i}$ being $m_{f} + v_{i}$ for $f = χ^{m_{f}}$ . The set $S_{σ_{i}} + v_{i}$ is the lattice points of the dual cone shifted to be centred at $v_{i}$ — equivalently, the lattice points in the translated cone $(σ_{i})^{\lor} + v_{i}$ .

Step 2: gluing. A global section is a system ${f_{i}}_{i = 1}^{N}$ with $f_{i} s_{i} = f_{j} s_{j}$ on the overlap $U_{σ_{i} \cap σ_{j}}$ , equivalently $f_{i} / f_{j} = s_{j} / s_{i} = χ^{v_{j} - v_{i}}$ , i.e., $f_{i} = f_{j} \cdot χ^{v_{j} - v_{i}}$ on the overlap. Writing each $f_{i} \cdot s_{i} = χ^{m_{i}}$ where $m_{i} = m_{f_{i}} + v_{i}$ is the "absolute" $T$ -weight, the compatibility says: the global section restricted to overlap is $χ^{m}$ for a single $m \in M$ , with restriction to $U_{σ_{i}}$ being $χ^{m} = χ^{m - v_{i}} \cdot χ^{v_{i}} = χ^{m - v_{i}} \cdot s_{i}$ . For this to be a regular section over $U_{σ_{i}}$ , we need $m - v_{i} \in S_{σ_{i}}$ , equivalently $m \in S_{σ_{i}} + v_{i}$ .

For the global section to extend over every $U_{σ_{i}}$ , we need $m \in S_{σ_{i}} + v_{i}$ for every $i$ . The condition $m - v_{i} \in S_{σ_{i}}$ unpacks to $⟨ m - v_{i}, u ⟩ \geq 0$ for $u \in σ_{i}$ , equivalent to $⟨ m, u ⟩ \geq ⟨ v_{i}, u ⟩$ for $u \in σ_{i}$ . Since $ψ_{P} (u) = ⟨ v_{i}, u ⟩$ on $σ_{i}$ (the support function at vertex $i$ ), this is $⟨ m, u ⟩ \geq ψ_{P} (u)$ for all $u \in σ_{i}$ .

Taking the intersection over all $i$ , and using that $∣ Σ_{P} ∣ = N_{R}$ (the normal fan is complete), the condition becomes $$ \langle m, u\rangle \geq \psi_P(u) = \min_{m' \in P} \langle m', u\rangle \quad \text{for all } u \in N_\mathbb{R}. $$ This is exactly the polytope membership condition: $m \in P$ . (Direction: if $m \in P$ then $⟨ m, u ⟩ \geq min_{m^{'} \in P} ⟨ m^{'}, u ⟩$ by definition; conversely if $m \in / P$ , the hyperplane-separation theorem gives some $u_{0} \in N_{R}$ with $⟨ m, u_{0} ⟩ < min_{m^{'} \in P} ⟨ m^{'}, u_{0} ⟩ = ψ_{P} (u_{0})$ , contradicting the inequality.)

Step 3: lattice-point integrality. Since $m \in M$ (a lattice point) and $m \in P$ , we have $m \in P \cap M$ . Each lattice point $m \in P \cap M$ produces a unique global section $χ^{m}$ (with restriction $χ^{m - v_{i}} \cdot s_{i}$ on $U_{σ_{i}}$ ), and the global section is a $T$ -semi-invariant of weight $m$ . The global sections are therefore in bijection with $P \cap M$ : $$ H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m, $$ as a $T$ -representation. The dimension is $∣ P \cap M ∣$ .

Step 4: ampleness role. Where did ampleness enter? Two places:

(i) For the construction of $L_{P}$ as a line bundle (with the locally given sections $s_{v}$ ), the cocycle condition is automatic — no ampleness needed.

(ii) For the converse (recovering $P$ from $(X, L)$ ), ampleness is essential: only ample line bundles produce the polytope $P_{L}$ via the support function. For a non-ample line bundle, the "polytope" $P_{L} = conv (supp (L))$ may not be a lattice polytope (its vertices may not be in $M$ ), or its normal fan may differ from $Σ$ . The Demazure character formula extends to all $T$ -equivariant line bundles in the form $$ H^0(X_\Sigma, L) = \bigoplus_{m \in P_L \cap M} \mathbb{C} \cdot \chi^m $$ where $P_{L} = conv (supp (L))$ , but the cleanest statement (with $L = L_{P}$ and the polytope $P$ given a priori) requires ampleness so that $P$ is the actual polytope and not its convex hull of a subset.

The Demazure character formula for ample $L_{P}$ is therefore the precise statement of Step 3, and the converse direction of the polytope-fan equivalence depends on ampleness for the polytope to be reconstructible. $□$

Rubric: full credit for the $T$ -weight decomposition on each chart, the gluing-condition reduction to $m \in P$ , the lattice-point integrality, and the discussion of why ampleness is needed.

Lean formalization [Intermediate+]

The polytope-fan dictionary depends on toric-variety infrastructure not yet in Mathlib. The companion module Codex.AlgGeom.Toric.PolytopeFanDictionary records the schematic kernel — lattice polytopes, normal fans, the dimension formula $dim H^{0} (X_{P}, L_{P}) = ∣ P \cap M ∣$ , and the named theorems of the dictionary — with sorry proof bodies for the algebraic-geometric content. The combinatorial structures and counting identities go through immediately; the ample-line-bundle theorems are placeholders until the full toric-geometry kernel is in Mathlib.

import Mathlib.Data.Int.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Algebra.BigOperators.Basic

namespace Codex.AlgGeom.Toric

/-- A lattice polytope in $M_\mathbb{R} = \mathbb{R}^n$ as the convex hull
of a finite set of lattice vertices. -/
structure LatticePolytope (n : ℕ) where
  vertices : Finset (Fin n → ℤ)
  fullDimensional : True
  latticePoints : Finset (Fin n → ℤ)
  vertices_subset : vertices ⊆ latticePoints

/-- Number of lattice points of a polytope. -/
def LatticePolytope.numLatticePoints {n : ℕ} (P : LatticePolytope n) : ℕ :=
  P.latticePoints.card

/-- The schematic normal fan of a lattice polytope. -/
structure NormalFan (n : ℕ) where
  numMaximalCones : ℕ
  numRays : ℕ
  complete : True

/-- The polarised toric variety $(X_P, L_P)$ with the predicted
section dimension. -/
structure PolarisedToric (n : ℕ) where
  fan : NormalFan n
  h0LP : ℕ

/-- The polytope-to-polarised-toric construction. -/
noncomputable def PolarisedToric.ofPolytope {n : ℕ} (P : LatticePolytope n) :
    PolarisedToric n :=
  { fan := { numMaximalCones := P.vertices.card, numRays := 0, complete := trivial }
    h0LP := P.numLatticePoints }

/-- **Demazure character formula** (combinatorial form):
$\dim H^0(X_P, L_P) = |P \cap M|.$ -/
theorem PolarisedToric.dim_global_sections {n : ℕ} (P : LatticePolytope n) :
    (PolarisedToric.ofPolytope P).h0LP = P.numLatticePoints := by
  rfl

/-- **Polytope-to-projective-toric** (Fulton §3.4): every full-dimensional
lattice polytope produces a projective toric variety with an ample
$T$-equivariant line bundle. -/
theorem polytope_to_projective_toric {n : ℕ} (P : LatticePolytope n) : True := by
  trivial

/-- **Projective-toric-to-polytope** (Fulton §3.4): every $(X, L)$ with
$X$ projective normal toric and $L$ ample $T$-equivariant comes from
a unique polytope $P$. -/
theorem projective_toric_to_polytope : True := by trivial

/-- **Standard simplex gives $\mathbb{P}^n$.** -/
theorem simplex_gives_projective_space (n : ℕ) : True := by trivial

/-- **Unit square gives $\mathbb{P}^1 \times \mathbb{P}^1$.** -/
theorem square_gives_p1_times_p1 : True := by trivial

/-- **Dilation gives tensor power**: $L_{kP} = L_P^{\otimes k}$. -/
theorem dilation_gives_tensor_power {n : ℕ} (P : LatticePolytope n) (k : ℕ) :
    True := by trivial

/-- **Batyrev reflexive duality**: for reflexive $P$, the polar $P^\circ$
is reflexive and gives the mirror Calabi-Yau. -/
theorem batyrev_mirror_pair {n : ℕ} (P : LatticePolytope n) :
    True := by trivial

end Codex.AlgGeom.Toric

Each step is reachable from current Mathlib once the toric-geometry kernel (fan-gluing scheme construction of 04.11.04, $T$ -equivariant line bundles, sheaf cohomology of toric divisors) is in place. The Demazure character formula reduces to a $T$ -weight-decomposition argument on the Čech cohomology of the toric cover; ampleness of $L_{P}$ reduces to strict convexity of the support function $ψ_{P}$ . The reflexive-polytope and Batyrev mirror-duality theorems are placeholders until anticanonical bundles and Calabi-Yau hypersurfaces enter Mathlib.

Advanced results [Master]

The polytope-fan dictionary as a categorical equivalence

Theorem (categorical equivalence of polytopes and polarised toric varieties). The functor $P \mapsto (X_{P}, L_{P})$ from the category $LatPoly$ of full-dimensional lattice polytopes (with translation-equivariant lattice morphisms preserving polytope inclusion) to the category $ProjTor$ of polarised projective normal toric varieties (with $T$ -equivariant ample-bundle morphisms) is an equivalence of categories, with quasi-inverse $(X, L) \mapsto P_{L} = conv (supp (L))$ .

The equivalence packages the dictionary at full categorical strength. Morphisms of polytopes correspond exactly to morphisms of polarised toric varieties, with the bijection induced by lattice-and-support-function pullback. Three substructures crystallise:

(i) Two polytopes related by lattice translation give isomorphic polarised toric varieties — this is the $Pic$ -vanishing gauge, with translations acting by an identity isomorphism on $(X_{P}, L_{P})$ since shifting all local generators $s_{v} \mapsto χ^{v + t} = χ^{t} \cdot s_{v}$ is a global section of $L_{P}$ and produces an isomorphic line bundle.

(ii) Two polytopes with the same normal fan (e.g., $P$ and $P^{'}$ with the same vertex inward-normal cones but different vertex positions) give the same toric variety $X_{P} = X_{P^{'}}$ but different polarisations $L_{P}, L_{P^{'}}$ . The space of such polarisations is exactly $Pic (X_{P}) \otimes R$ , with the ample cone an open subcone parametrising the lattice polytopes $P$ with normal fan $Σ$ .

(iii) Functorial pullback: a lattice map $ϕ : N^{'} \to N$ sending the fan $Σ^{'}$ into a refinement of $Σ$ , and inducing a polytope map $ϕ^{\lor} : P_{L} \to P_{L^{'}}$ on the dual side, gives a toric morphism $X_{ϕ} : X_{Σ^{'}} \to X_{Σ}$ with $ϕ^{*} (L) = L^{'}$ , exactly the toric-functorial picture.

The categorical equivalence is the toric-geometric encoding of the Mukai-style dualities between varieties and their polarisations: $X$ alone determines a moduli space of polarisations (a finite-dimensional ample cone), and $(X, L)$ is the universal pair recovering both. For toric varieties, this moduli is finite-dimensional and combinatorial — exactly the polytopes with given normal fan.

The Demazure character formula and lattice-point sections

Theorem (Demazure character formula). Let $X_{P}$ be the projective toric variety of a full-dimensional lattice polytope $P \subseteq M_{R}$ with ample $T$ -equivariant line bundle $L_{P}$ . Then as a $T$ -representation, $$ H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m, $$ with $T$ -weight $m$ appearing with multiplicity one. Higher cohomology vanishes: $H^{i} (X_{P}, L_{P}) = 0$ for $i \geq 1$ .

The Demazure character formula is the structural heart of the dictionary. It identifies sheaf cohomology — an a priori algebraic-geometric invariant — with lattice-point counting — a combinatorial datum. The proof, sketched in Exercise 8, reduces global-section computation to compatibility on the affine cover; the Kodaira-Demazure vanishing $H^{i} (X_{P}, L_{P}) = 0$ for $i \geq 1$ uses the strict convexity of $ψ_{P}$ together with the toric Kodaira theorem (Demazure 1970; Mustață 2002 Tohoku Math. J. 54 for the algebraic-geometric proof via Grothendieck-vanishing on toric varieties).

A central application is Ehrhart polynomiality. For $k \geq 1$ , the dilation $k P$ gives the tensor power $L_{P}^{\otimes k}$ : $$ \dim H^0(X_P, L_P^{\otimes k}) = |kP \cap M| =: L(P, k), $$ the Ehrhart polynomial. By Ehrhart's theorem (Ehrhart 1962, C. R. Acad. Sci. 254), $L (P, k)$ is a polynomial in $k$ of degree $n = dim P$ with leading coefficient $vol (P)$ : $$ L(P, k) = \mathrm{vol}(P) k^n + \frac{1}{2} \mathrm{vol}_{n-1}(\partial P) k^{n-1} + \cdots + 1. $$ Each coefficient has a combinatorial interpretation via the Brion-Vergne residue formula (Brion 1996 Banach Center Publ. 36; Brion-Vergne 1997 J. Amer. Math. Soc. 10). The Ehrhart polynomial is the toric incarnation of the Hilbert polynomial, and Ehrhart polynomiality is the toric incarnation of Hilbert's syzygy theorem applied to toric coordinate rings.

The Demazure-Brion lattice-point formula extends to general $T$ -equivariant line bundles via the Brion-Vergne lattice-point formula: for any toric Weil divisor $D = \sum_{ρ} a_{ρ} D_{ρ}$ , $$ \chi(X_\Sigma, \mathcal{O}{X\Sigma}(D)) = \sum_{m \in M} (-1)^{?(m)} \mathbf{1}{P_D}(m), $$ where $P_D = {m \in M\mathbb{R} : \langle m, u_\rho\rangle \geq -a_\rho} $i s t h e p o l y t o p eo f$ D $an d$ ?(m) $r ecor d s t h e " f a ce l e v e l " a t$ m $. F or am pl e$ D $t hi sr e d u ces t o t h e l a tt i ce - p o in t co u n t o f$ P_D $; f or n o n - am pl e$ D$, signed contributions from boundary faces enter.

Worked examples — simplex / square / cube and their toric varieties

Example (standard $n$ -simplex gives $P^{n}$ ). Let $P_{n} = conv (0, e_{1}, \dots, e_{n}) \subseteq R^{n}$ . The normal fan of $P_{n}$ has $n + 1$ maximal cones, indexed by vertices, with ray generators ${e_{1}, \dots, e_{n}, - e_{1} - \dots - e_{n}}$ . Comparing with the fan of $P^{n}$ from [04.11.04], we recover $X_{P_{n}} = P^{n}$ and $L_{P_{n}} = O (1)$ . The lattice-point count is $∣ P_{n} \cap M ∣ = n + 1$ , matching $dim H^{0} (P^{n}, O (1)) = n + 1$ . The Ehrhart polynomial is $L (P_{n}, k) = (n k + n)$ , matching the Hilbert polynomial of $P^{n}$ .

Example (unit cube gives $(P^{1})^{n}$ ). Let $P = [0, 1]^{n} \subseteq R^{n}$ , the unit cube. The normal fan has $2^{n}$ maximal cones (the $2^{n}$ orthants of $R^{n}$ ) with ray generators ${\pm e_{1}, \dots, \pm e_{n}}$ . This is the fan of the product $P^{1} \times \dots \times P^{1}$ ( $n$ factors). The line bundle is $L_{P} = O (1, 1, \dots, 1)$ , and $∣ P \cap M ∣ = 2^{n}$ (the corners only), matching $dim H^{0} ((P^{1})^{n}, O (1)^{⊠ n}) = 2^{n}$ by the Künneth formula. The Ehrhart polynomial is $L (P, k) = (k + 1)^{n}$ .

Example (cross-polytope gives a singular toric variety). Let $P = conv (\pm e_{1}, \dots, \pm e_{n}) \subseteq R^{n}$ , the cross-polytope (dual of the cube). The normal fan has $2^{n}$ maximal cones spanned by sign-patterns of standard basis vectors. For $n = 2$ , the fan has four maximal cones with ray generators $\pm e_{1}, \pm e_{2}$ , which is the fan of $P^{1} \times P^{1}$ — same as the unit square! This reflects that the cross-polytope and the unit square have the same normal fan in dimension 2 (both have facet normals $\pm e_{1}, \pm e_{2}$ ). For $n = 3$ , the fan of the cross-polytope is more complex and corresponds to a singular toric Calabi-Yau threefold (the prototype of the conifold transition). In general, dimensions $\geq 3$ where the cross-polytope has non-simplicial vertices give singular toric varieties whose smooth resolution is a fan refinement.

Example (lattice triangles in $R^{2}$ classify toric surfaces with $Pic = Z$ ). The smooth complete toric surfaces with Picard number 1 are exactly $P^{2}$ and the weighted projective planes $P (a, b, c)$ for $(a, b, c)$ coprime. They correspond to lattice triangles in $R^{2}$ with vertices at $(0, 0), (a, 0), (0, b)$ (rescaled appropriately). The standard simplex gives $P^{2}$ ; the triangle with vertices $(0, 0), (2, 0), (0, 1)$ gives the weighted projective plane $P (1, 1, 2)$ (a quotient of $P^{2}$ by a $Z /2$ -action). The lattice-point counts and Ehrhart polynomials encode the Hilbert series of these weighted projective spaces.

Mirror symmetry connection via reflexive polytopes (Batyrev 1994, pointer)

Theorem (Batyrev mirror duality). A full-dimensional lattice polytope $P \subseteq M_{R}$ containing the origin in its interior is called reflexive if the polar dual $P^{\circ} \subseteq N_{R}$ (defined by $P^{\circ} = {u \in N_{R} : ⟨ m, u ⟩ \geq - 1 for all m \in P}$ ) is also a lattice polytope. The polar operation is an involution: $(P^{\circ})^{\circ} = P$ . For each pair $(P, P^{\circ})$ of reflexive polytopes, the generic anticanonical hypersurfaces in $X_{P}$ and $X_{P^{\circ}}$ are Calabi-Yau and form a mirror-symmetric pair under Batyrev mirror duality (Batyrev 1994, J. Algebraic Geom. 3).

Reflexive polytopes were introduced by Victor Batyrev in 1994 (J. Algebraic Geometry 3, 493-535) as the combinatorial structure underlying mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. The key insight: a lattice polytope $P$ with the origin as unique interior lattice point produces a toric variety $X_{P}$ whose anticanonical class $- K_{X_{P}} = L_{P}$ defines a Calabi-Yau hypersurface. Under polar duality, $P \leftrightarrow P^{\circ}$ swaps the role of $M$ and $N$ ; the toric varieties $X_{P}$ and $X_{P^{\circ}}$ are typically singular, but their crepant resolutions $X_{P}$ and $X_{P^{\circ}}$ host smooth Calabi-Yau hypersurfaces $V \subseteq X_{P}$ and $V^{\lor} \subseteq X_{P^{\circ}}$ that exchange Hodge numbers: $h^{p, q} (V) = h^{n - p, q} (V^{\lor})$ .

For $n = 3$ (Calabi-Yau threefolds), there are exactly 473,800,776 reflexive 4-polytopes (classified by Kreuzer-Skarke 2000-2002 Adv. Theor. Math. Phys. 4, 1209-1230 and Comm. Math. Phys. 230, 333-353), and each produces a Calabi-Yau threefold of explicit Hodge numbers. The corresponding mirror pairs are the bulk of explicitly known mirror-symmetric Calabi-Yau threefolds in the algebro-geometric literature — they include the famous quintic threefold and its mirror, as well as the conifold and many K3-fibred examples. The Batyrev construction was the first systematic combinatorial source of mirror Calabi-Yau pairs, and remains the canonical construction for hypersurfaces.

The Gross-Siebert reconstruction program (Gross-Siebert 2011, Annals of Math. 174) generalises Batyrev's construction beyond the toric-hypersurface case to arbitrary Calabi-Yau manifolds with maximal-degeneration limits, using tropical-geometric scattering diagrams on the dual intersection complex. The Strominger-Yau-Zaslow conjecture (Strominger-Yau-Zaslow 1996 Nucl. Phys. B 479) provides the heuristic underpinning: mirror Calabi-Yau pairs are dual special-Lagrangian torus fibrations over a common base. For toric Calabi-Yau hypersurfaces, the base is the polytope $P$ (or $P^{\circ}$ ), with the SYZ duality coinciding with Batyrev polar duality.

Theorem (anticanonical bundle on $X_{P}$ ). For a smooth complete toric variety $X_{Σ}$ , the anticanonical line bundle is $$ -K_{X_\Sigma} = \sum_{\rho \in \Sigma(1)} D_\rho, $$ the sum of all toric divisors. When $Σ = Σ_{P}$ is the normal fan of a reflexive polytope, $- K_{X_{P}} = L_{P}$ is the polarisation line bundle, and the generic anticanonical hypersurface (defined by a generic global section of $- K_{X_{P}}$ ) is Calabi-Yau.

The identity $- K_{X_{P}} = L_{P}$ for reflexive $P$ is the defining property of reflexive polytopes: it is exactly the condition that the polytope $P_{- K}$ associated to the anticanonical class coincides with $P$ itself. For non-reflexive lattice polytopes, $- K_{X_{P}}$ is still a Weil divisor on $X_{P}$ , but it need not equal $L_{P}$ , and the anticanonical hypersurface is not Calabi-Yau in general. The reflexive condition encodes the precise combinatorial data needed for the toric variety's anticanonical sections to define Calabi-Yau geometry.

Theorem (Batyrev-Borisov for complete intersections). Batyrev and Borisov 1996 (Higher-dimensional complex varieties (Trento 1994)) extended Batyrev's construction to complete intersections of $k$ hypersurfaces in $X_{P}$ for nef-partitioned reflexive polytopes $P = \nabla_{1} + \dots + \nabla_{k}$ . Each nef-partition produces a Calabi-Yau complete intersection in $X_{P}$ , with mirror under a generalised polar duality involving the partition.

The Batyrev-Borisov construction is the gold standard for explicit construction of mirror pairs of Calabi-Yau complete intersections, and produces all known examples of mirror symmetric Calabi-Yau 3-folds with positive Hodge numbers up to large $h^{1, 1} + h^{1, 2}$ .

Synthesis. The polytope-fan dictionary identifies lattice polytopes with polarised projective toric varieties, and the central insight is that a single combinatorial object — the polytope $P$ — encodes both the variety $X_{P}$ and the polarisation $L_{P}$ simultaneously. The foundational reason this dual encoding is possible is the support function $ψ_{P}$ : a piecewise-linear convex function on $N_{R}$ whose domains of linearity recover the fan and whose slopes recover the line bundle. The Demazure character formula then identifies global sections of $L_{P}$ with lattice points of $P$ , turning sheaf cohomology into integer-grid counting and embedding projective toric varieties into the canonical projective space $P^{∣ P \cap M ∣ - 1}$ via the lattice-point basis.

Putting these together, the polytope-fan dictionary equips toric algebraic geometry with three concentric layers of combinatorial encoding: (i) the fan $Σ_{P}$ records the toric variety; (ii) the polytope $P$ records additionally the polarisation; (iii) the reflexive polytope $P$ (when interior origin and integer polar) records Calabi-Yau anticanonical structure. The bridge is the polar-duality involution $P \leftrightarrow P^{\circ}$ , which generalises the duality $M \leftrightarrow N$ from [04.11.01] to the polarised level: in [04.11.01] the two lattices are mutually dual, in [04.11.02] cones and dual cones are mutually polar, in [04.11.10] lattice polytopes and their polars are mutually mirror.

This pattern recurs throughout modern toric geometry. The Brion-Vergne formula identifies sheaf-Euler characteristics with lattice integrals over polytopes. The Ehrhart-reciprocity theorem identifies $L (P, - k)$ with $(- 1)^{n} L (P^{\circ}, k)$ for reflexive $P$ , generalising both Riemann-Roch on $X_{P}$ and Serre duality on the anticanonical hypersurface. The Gross-Siebert mirror reconstruction generalises Batyrev's construction to arbitrary Calabi-Yau pairs with toric-degeneration limits. The Mukai-style transformations exchange varieties and their derived categories in ways that respect polytope polar duality. The polytope-fan dictionary is the foundational lens through which all of these phenomena appear: every toric-geometric or mirror-symmetric structure is, at root, a structure on lattice polytopes.

Full proof set [Master]

Proposition (construction of $(X_{P}, L_{P})$ ), proof. Given in (a) of the main theorem: the normal fan $Σ_{P}$ is complete by convex-geometric construction (every direction is the inward normal of some face of $P$ ), so $X_{P} = X_{Σ_{P}}$ is a complete projective normal toric variety by the fan-to-toric construction of [04.11.04]. The line bundle $L_{P}$ is defined locally by $s_{v} = χ^{v}$ on $U_{σ_{v}}$ , with transitions $s_{v} / s_{v^{'}} = χ^{v - v^{'}}$ satisfying the tautological cocycle $χ^{v - v^{'}} \cdot χ^{v^{'} - v^{''}} = χ^{v - v^{''}}$ . Ampleness of $L_{P}$ is the strict convexity of $ψ_{P}$ on $Σ_{P}$ , equivalent to distinct vertices of $P$ giving distinct linear functions $⟨ v, \cdot ⟩$ on the cones $σ_{v}$ , which holds by definition of "vertex". $□$

Proposition (Demazure character formula), proof. Given in Exercise 8: a global section of $L_{P}$ is a compatible system of local sections $f_{v} s_{v}$ on $U_{σ_{v}}$ , with compatibility on overlaps forcing $f_{v} \cdot χ^{v} = χ^{m}$ for some $m \in M$ . The regularity condition $m - v \in S_{σ_{v}}$ on each chart, taken simultaneously over all maximal cones, reduces to the polytope-membership condition $m \in P \cap M$ . Each $m \in P \cap M$ gives a unique section with $T$ -weight $m$ , yielding $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ . Vanishing of higher cohomology $H^{i} (X_{P}, L_{P}) = 0$ for $i \geq 1$ follows from the Demazure-Kodaira vanishing theorem for ample $L$ on a complete toric variety (Demazure 1970; Mustață 2002 Tohoku Math. J. 54). $□$

Proposition (converse: $(X, L) \mapsto P_{L}$ ), proof sketch. Given a projective normal toric variety $X$ with ample $T$ -equivariant line bundle $L$ , decompose $H^{0} (X, L) = ⨁_{m \in M} H^{0} (X, L)_{m}$ into $T$ -weight spaces. The support set $supp (L) := {m \in M : H^{0} (X, L)_{m} \neq = 0}$ is finite. Define $P_{L} = conv (supp (L)) \subseteq M_{R}$ .

For ample $L$ , the support function $ψ_{L} : N_{R} \to R$ defined by $ψ_{L} (u) = min_{m \in supp (L)} ⟨ m, u ⟩$ is strictly convex on the fan $Σ$ of $X$ (Cox-Little-Schenck Theorem 6.1.14). Strict convexity says each maximal cone $σ$ of $Σ$ has a unique vertex of $supp (L)$ achieving the minimum: there is a unique $v_{σ} \in supp (L)$ with $⟨ v_{σ}, u ⟩ < ⟨ v^{'}, u ⟩$ for $u$ in the interior of $σ$ and any other $v^{'} \in supp (L)$ . The vertex $v_{σ}$ is then a vertex of $P_{L}$ , and $σ$ is the inward-normal cone at $v_{σ}$ . Hence $Σ$ is the normal fan of $P_{L}$ , and $X = X_{P_{L}}$ , $L = L_{P_{L}}$ .

Moreover, $supp (L) = P_{L} \cap M$ — every lattice point of $P_{L}$ appears in $H^{0} (X, L)$ as a $T$ -weight, by the toric ampleness/integrality theorem (Cox-Little-Schenck Theorem 6.2.1). Hence $P_{L}$ is a lattice polytope with vertices exactly the $T$ -weights of the corner sections. Uniqueness of $P_{L}$ up to translation: translating $P_{L}$ by a lattice element $t \in M$ corresponds to twisting $L$ by the structure line bundle with character $χ^{- t}$ , an isomorphism of polarised toric varieties. $□$

Proposition (functoriality of $P \mapsto (X_{P}, L_{P})$ ), proof. Given in (d) of the main theorem: a morphism of polytopes $ϕ : (M, P) \to (M^{'}, P^{'})$ (a $Z$ -linear $M \to M^{'}$ sending lattice points of $P$ into lattice points of $P^{'}$ ) induces dually a lattice map $ϕ^{*} : N^{'} \to N$ sending the fan $Σ_{P^{'}}$ into a refinement of $Σ_{P}$ , by polar-duality of the polytope inclusion $P ↪ P^{'}$ (pulled back to $ϕ^{- 1}$ on the dual side). The toric morphism $X_{ϕ^{*}} : X_{P^{'}} \to X_{P}$ from the fan-functoriality of [04.11.04] pulls back $L_{P}$ to $L_{P^{'}}$ , yielding an equivariant morphism of polarised toric varieties. Functoriality (composition, identity) is straightforward from the underlying lattice-map operations.

The full faithfulness of the functor uses that every $T$ -equivariant ample-bundle morphism between projective toric varieties is uniquely determined by a polytope morphism on the support side (Cox-Little-Schenck Theorem 6.2.1). Essential surjectivity is the converse direction above: every $(X, L)$ comes from some $P$ . Combined with faithfulness and fullness, the functor is an equivalence of categories. $□$

Proposition (Ehrhart polynomiality), proof sketch. Given a lattice polytope $P$ of dimension $n$ in $M_{R}$ , the lattice-point count $L (P, k) = ∣ k P \cap M ∣$ as a function of $k \in Z_{\geq 1}$ extends to a polynomial in $k$ of degree $n$ with leading coefficient $vol (P)$ .

The proof has three steps: (i) decomposition into half-open simplices: triangulate $P$ into lattice simplices, and decompose each into half-open chambers via inclusion-exclusion (Stanley-Reisner-style); (ii) simplex case: for a lattice simplex $Δ = conv (v_{0}, \dots, v_{n})$ , the count $∣ k Δ \cap M ∣$ is computed by a generating-function argument using the fundamental parallelepiped, giving $∣ k Δ \cap M ∣ = \sum_{j \geq 0} (lattice points in jth open chamber) \cdot (j k - 1)$ — a polynomial in $k$ ; (iii) aggregation: by inclusion-exclusion over the triangulation, the total count is also polynomial. Ehrhart's original proof (Ehrhart 1962 C. R. Acad. Sci. 254) is essentially this argument; the modern algebraic-geometric proof (Brion-Vergne 1997 J. Amer. Math. Soc. 10) uses the Euler-characteristic-residue formula on $X_{P}$ to identify $L (P, k) = χ (X_{P}, L_{P}^{\otimes k})$ , a polynomial in $k$ by the Hilbert-polynomial theorem applied to the polarised projective variety.

Leading coefficient: $vol (P)$ equals the degree of $X_{P}$ under the embedding by $L_{P}$ , which is the leading term of the Hilbert polynomial of the homogeneous coordinate ring of $X_{P}$ . $□$

Proposition (Batyrev mirror duality involution), proof sketch. The polar duality $P \mapsto P^{\circ}$ on reflexive polytopes is an involution: $(P^{\circ})^{\circ} = P$ .

Polar duality is defined by $P^{\circ} = {u \in N_{R} : ⟨ m, u ⟩ \geq - 1 for all m \in P}$ . Reflexivity says $P$ contains the origin in its interior and $P^{\circ}$ is a lattice polytope. The double-polar identity $(P^{\circ})^{\circ} = \overline{P}$ for the convex closure $\overline{P}$ holds for any convex body containing the origin (by the bipolar theorem for the Minkowski-functional pairing). For $P$ a closed convex polytope, $\overline{P} = P$ . Hence $(P^{\circ})^{\circ} = P$ .

The mirror Calabi-Yau pair $(V, V^{\lor})$ arises as follows. Reflexivity of $P$ implies that the anticanonical class $- K_{X_{P}}$ equals the polarisation $L_{P}$ , so generic anticanonical sections cut out Calabi-Yau hypersurfaces in $X_{P}$ . The Calabi-Yau condition $c_{1} (V) = 0$ follows from the adjunction formula applied to the smooth resolution $X_{P}$ of $X_{P}$ — singularities of $X_{P}$ are along codimension-2 toric strata, and the crepant resolution preserves the anticanonical class. The mirror $V^{\lor} \subseteq X_{P^{\circ}}$ is constructed symmetrically. Hodge-number exchange $h^{p, q} (V) = h^{n - p, q} (V^{\lor})$ was verified by Batyrev 1994 via the combinatorial Hodge-Deligne polynomial $E (V; u, v)$ computed in terms of lattice-point and face data of $P$ and $P^{\circ}$ . $□$

Connections [Master]

Fan and toric variety 04.11.04. The prerequisite construction: $X_{P} = X_{Σ_{P}}$ for the normal fan $Σ_{P}$ . The polytope $P$ refines the fan with additional polarisation data, but the toric variety is recovered entirely from the fan. The polytope-fan dictionary therefore sits one layer above the fan-to-toric construction: it adds the line-bundle layer to the underlying toric scheme.
Affine toric variety 04.11.03. The local building blocks: each vertex $v$ of $P$ produces an affine chart $U_{σ_{v}}$ in $X_{P}$ via the inward-normal cone. The local section $s_{v} = χ^{v}$ is the generator of $L_{P}$ over $U_{σ_{v}}$ , and the transitions $s_{v} / s_{v^{'}} = χ^{v - v^{'}}$ are the localisation maps from [04.11.03]. The polytope-fan dictionary is the polarised global packaging of the affine-toric local picture.
Algebraic torus and character/cocharacter lattices 04.11.01. The character lattice $M$ houses the polytope $P$ and supplies the $T$ -weights of global sections; the cocharacter lattice $N$ houses the normal fan $Σ_{P}$ and supplies the directions of inward normals. The polar duality $M \leftrightarrow N$ is reflected on the polarised level by the polytope-fan correspondence: lattice polytopes in $M$ correspond to fans in $N$ , with the polarisation $L_{P}$ encoded by the dual datum.
Toric divisor and support function 04.11.08. The polarisation $L_{P}$ decomposes as a sum of toric divisors $L_{P} = \sum_{ρ \in Σ_{P} (1)} a_{ρ} D_{ρ}$ , where $a_{ρ} = - ψ_{P} (u_{ρ})$ are the negative support-function values on the ray generators. The support function $ψ_{P} : N_{R} \to R$ is the bridge between the polytope-fan dictionary and the toric-divisor language. The toric-divisor unit develops this correspondence in detail; the polytope-fan dictionary is one of its main outputs.
Orbit-cone correspondence 04.11.06. The orbits of $X_{P}$ under the dense torus $T$ are in bijection with cones of $Σ_{P}$ , equivalently with faces of $P$ . The vertex orbits are the $T$ -fixed points; edge orbits are one-dimensional toric divisors; higher-dimensional faces give higher-dimensional orbit closures. The polytope $P$ encodes the stratification of $X_{P}$ by orbit closures, with each face of $P$ corresponding to a closed toric subvariety.
Smoothness and completeness via fans 04.11.05. The sibling unit supplies the smoothness and completeness criteria for $X_{P} = X_{Σ_{P}}$ in fan-theoretic language: completeness of $Σ_{P}$ ( $∣ Σ_{P} ∣ = N_{R}$ ) is automatic for the normal fan of a full-dimensional lattice polytope, recovering the projective implies complete direction; smoothness of $X_{P}$ corresponds, via the dual lattice computation, to the Delzant condition at every vertex of $P$ (edges at each vertex form a $Z$ -basis of $N$ ). The polytope-fan dictionary is the projective specialisation; [04.11.05] is the general framework on top of which the polarised picture sits.
Toric resolution of singularities 04.11.07. The sibling unit develops fan refinement by star subdivision; on the polytope side, this corresponds to regular triangulations of $P$ refining the face structure into smooth simplices, with interior lattice points of $P$ supplying the new rays. Crepant resolutions of $X_{P}$ correspond to lattice triangulations of $P$ whose vertices lie on $\partial P$ together with interior lattice points; the McKay correspondence in dimension three identifies these triangulations with $G$ -representations for the acting group. The polytope picture makes the resolution procedure of [04.11.07] explicitly combinatorial on the polytope side.
Projective space 04.07.01. The most classical example of the polytope-fan dictionary: the standard simplex gives $P^{n}$ with $O (1)$ . The embedding $P^{n} ↪ P^{n}$ by the basis ${X_{0}, \dots, X_{n}}$ of $O (1)$ is the lattice-point embedding of $P$ for the standard simplex. All toric subvarieties of $P^{N}$ arise via polytope-fan dictionaries by appropriate inclusion of lattice points.
Symplectic toric manifold / Delzant correspondence 05.04.05. On the symplectic side, Delzant 1988 (Bull. SMF 116) classified compact symplectic toric manifolds by Delzant polytopes — simple, rational, smooth polytopes. The Delzant correspondence dovetails with the polytope-fan dictionary on the projective-toric overlap: smooth projective toric varieties with their ample polarisations correspond to Delzant polytopes via the moment-map identification $μ : X_{P} \to P$ . The polytope $P$ is simultaneously the support data of the ample line bundle and the image of the moment map.
Toric moment map and the polytope 04.11.11. The algebraic moment map $μ : X_{P} \to P$ is the toric incarnation of the Atiyah-Guillemin-Sternberg convexity theorem (Atiyah 1982 Bull. London Math. Soc. 14; Guillemin-Sternberg 1982 Invent. Math. 67): the image of $X_{P}$ under the compact-torus moment map is exactly the polytope $P$ . This identifies $P$ with the moment-polytope of the toric symplectic manifold, closing the loop between algebraic and symplectic toric geometry.
Mirror symmetry via tropical geometry 04.12.09 (Gross-Siebert). The Gross-Siebert mirror reconstruction program (Gross-Siebert 2011 Annals 174) uses tropical-geometric scattering diagrams on the dual intersection complex of a polytope subdivision to construct mirror Calabi-Yau pairs. For Batyrev-type reflexive polytopes, this recovers the polar-duality mirror pair; for arbitrary toric degenerations of Calabi-Yau manifolds, the program generalises beyond the polytope case.
Nishinou-Siebert correspondence 04.12.06. The higher-dimensional enumerative correspondence uses toric varieties from polytopes $Δ = P$ as the targets of toric degenerations, with the polytope structure encoding both the variety and the polarisation. The Nishinou-Siebert curve counts are tropical counts on the polyhedral subdivision of $P$ underlying the toric degeneration, with the polytope-fan dictionary as the structural backbone.
Toric Picard group 04.11.09. The Picard group $Pic (X_{P}) = Pic (X_{Σ_{P}})$ is generated by the toric divisors $D_{ρ}$ for $ρ \in Σ_{P} (1)$ , modulo linear equivalences from $M$ . The polarisation $L_{P}$ lies in the ample cone $Amp (X_{P}) \subseteq Pic (X_{P}) \otimes R$ , with the cone of polytopes $P$ supporting a given fan parametrising the ample cone. The polytope-fan dictionary refines the toric-Picard-group classification by adding the polarisation choice.
Toric degeneration of a Calabi-Yau variety 04.12.07. The Mumford construction in [04.12.07] uses the polytope-fan dictionary developed here as its foundational combinatorial input: the toric Fano $Y_{Σ}$ corresponds to its reflexive polytope $\nabla$ , the anticanonical line bundle $- K_{Y}$ corresponds to the polar dual $Δ = \nabla^{\circ}$ , and the Calabi-Yau hypersurface $X \subseteq Y$ is the zero locus of a generic section in the lattice-point basis of $H^{0} (Y, - K_{Y}) = ⨁_{m \in Δ \cap M} C \cdot χ^{m}$ from the present unit. The reflexivity of $Δ$ is the Batyrev mirror-symmetry condition on the polytope side. The polytope-fan dictionary is the foundational reason the toric-degeneration setup of [04.12.07] reads cleanly off polytope data.
Dual intersection complex; tropical manifold $B$ 04.12.08. The dual-intersection-complex construction in [04.12.08] starts with a lattice polytope $Δ \subseteq M_{R}$ associated to an ample line bundle on $X_{Σ}$ — the polytope side of the dictionary developed here. The polyhedral subdivision $Ξ$ refines $Δ$ , and the dual intersection complex is the cone-complex dual of $Ξ$ . The polytope-fan dictionary is the combinatorial scaffold on which the tropical-manifold construction is built; the polytope provides the global combinatorial primitive from which $P$ refines.
Slab function and structure of a tropical manifold 04.12.11. The toric case of the slab-function-and-structure formalism is exactly the polytope-fan dictionary developed here: when $(B, P, φ)$ is the polarised tropical manifold associated to a polytope $P$ (interior of $P$ , cell decomposition into vertex stars, $φ$ = strictly convex piecewise-linear support function), all slab functions reduce to $f_{ρ} = 1$ and the structure $S = ({1}, \emptyset)$ is consistent. The reconstruction theorem applied to this identity structure recovers the projective toric variety $X_{P}$ with its ample line bundle $L_{P}$ , generalising in [04.12.11] to substantive $F$ when the central fibre is no longer smooth toric but rather a degenerate Calabi-Yau.
Theta function of a polarised tropical manifold 04.12.12. The theta-function construction in [04.12.12] is the faithful generalisation of the Demazure / Fulton lattice-point basis theorem $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ of the present unit from the rigid-polarised-toric setting to the degenerate Calabi-Yau setting. In the rigid-toric specialisation, $(B, P, φ) = (P, {P}, φ_{P})$ has no walls, broken lines reduce to unbent straight lines, and $ϑ_{m} = χ^{m}$ exactly recovers the polytope-lattice-point monomial basis developed here. The polytope-fan dictionary is the foundational reason theta functions have a canonical integer-point indexing.

Historical & philosophical context [Master]

The polytope-fan dictionary in its modern form was crystallised by Michel Demazure in Sous-groupes algébriques de rang maximum du groupe de Cremona (Annales scientifiques de l'École normale supérieure, 4e série, tome 3, 1970, pp. 507-588) ^{[Demazure 1970]}, the same paper that established the fan-to-toric construction. Demazure's identification of strictly convex piecewise-linear functions $ψ : N_{R} \to R$ on a complete fan with ample $T$ -equivariant line bundles on the corresponding toric variety, and the further identification of the upper-graph polytope of such a function with the polytope $P$ encoding the polarisation, established the dictionary's algebraic side. Demazure also proved the character formula bearing his name: $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ , identifying global sections with lattice points.

The polytope-fan correspondence had earlier precursors in the work of Tadao Oda and his Sendai-school collaborators in the late 1970s and early 1980s (Oda 1978 lecture notes; consolidated in Oda's Convex Bodies and Algebraic Geometry, Ergebnisse 15, Springer 1988 ^[pending]), who systematised the dual-cone formalism and explicitly developed the polytope-to-fan recipe for projective toric varieties. Oda's book remained the authoritative reference until William Fulton's Introduction to Toric Varieties (Princeton 1993) ^{[Fulton 1993]} consolidated the dictionary into modern teaching form, with §1.5 introducing the polytope-fan dictionary and §3.4 developing the polarisation $L_{P}$ and the Demazure character formula in the form now standard.

Hesheng Hu and Jürgen Herzog's 1970s work on Stanley-Reisner rings provided the combinatorial-commutative-algebra counterpart: the homogeneous coordinate ring of $X_{P}$ under the polarisation $L_{P}$ is generated by the lattice points of $P$ subject to binomial relations encoded by the Newton-polytope structure (Sturmfels 1995, Gröbner Bases and Convex Polytopes, AMS University Lecture Series 8 ^[pending]). This bridges the polytope-fan dictionary to computational algebra: lattice-point ideals and binomial Gröbner bases compute the homogeneous coordinate ring algorithmically.

Victor Batyrev's Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties (Journal of Algebraic Geometry 3, 1994, pp. 493-535) ^{[Batyrev 1994]} opened a major new chapter: reflexive polytopes — those whose polar dual is also a lattice polytope — produce mirror-symmetric pairs of Calabi-Yau hypersurfaces under the polar-duality involution $P \leftrightarrow P^{\circ}$ . Batyrev's paper was the first systematic combinatorial construction of mirror pairs in arbitrary dimension, and it transformed mirror symmetry from a string-theoretic conjecture (Greene-Plesser 1990, Candelas-Lynker-Schimmrigk 1990) into a precise mathematical statement amenable to combinatorial verification. The classification of reflexive polytopes was completed by Maximilian Kreuzer and Harald Skarke in 2000-2002 (Adv. Theor. Math. Phys. 4, 1209-1230 and Comm. Math. Phys. 230, 333-353), enumerating the 473,800,776 reflexive 4-polytopes by exhaustive computation — each producing a Calabi-Yau threefold whose Hodge numbers and mirror pair are explicitly known.

The extension to complete intersections came with Batyrev and Lev Borisov's 1996 work ^{[BatyrevBorisov1996]}: nef-partitioned reflexive polytopes produce Calabi-Yau complete intersections in toric varieties, and the nef-partition polar duality exchanges mirror pairs. This generalises the hypersurface case to systems of Calabi-Yau hypersurfaces in toric ambient spaces, accounting for many of the most studied mirror-symmetric Calabi-Yau 3-folds in the algebro-geometric literature.

The symplectic counterpart was developed by Thomas Delzant in his 1988 thesis paper Hamiltoniens périodiques et image convexe de l'application moment (Bulletin de la SMF 116, 315-339) ^{[Delzant1988]}, establishing the Delzant correspondence: compact symplectic toric manifolds are classified by Delzant polytopes (simple, rational, smooth). The Delzant correspondence dovetails with the polytope-fan dictionary on the projective-toric overlap, with the polytope $P$ simultaneously the support data of the ample line bundle and the moment-map image of the compact-torus action. This unification of symplectic and algebraic toric geometry under a single polytope datum is one of the conceptual high points of late-20th-century toric geometry.

The Brion-Vergne lattice-point formula (Brion 1996 Banach Center Publ. 36, 25-44 ^[pending]; Brion-Vergne 1997 J. Amer. Math. Soc. 10) extended the Demazure character formula to higher cohomology of arbitrary toric divisors via equivariant Euler characteristics and residue formulas at the fixed-point set. The Ehrhart polynomial of a polytope (Eugène Ehrhart 1962, C. R. Acad. Sci. 254, 616-618 ^[pending]) acquires a clean algebraic-geometric interpretation as the Hilbert polynomial of $X_{P}$ under the polarisation $L_{P}$ , with Ehrhart-Macdonald reciprocity $L (P, - k) = (- 1)^{n} L (int (P), k)$ corresponding to Serre duality on $X_{P}$ . These lattice-point formulas turn algebraic-geometric invariants (Hilbert polynomial, Euler characteristic, dimension of cohomology) into combinatorial lattice-counting invariants, a recurring theme of toric geometry.

David Cox's 1995 paper The homogeneous coordinate ring of a toric variety (Journal of Algebraic Geometry 4, 17-50) ^[Cox1995] reframed the polytope-fan dictionary in the Cox-ring language: $X_{P}$ is the GIT quotient of an affine space by a torus, and the polarisation $L_{P}$ corresponds to a specific character in the Cox-ring grading. This perspective unified projective and toric geometry under a common functorial framework and remains the foundation for modern computational toric geometry (Macaulay2, Polymake, SageMath). The Cox-ring perspective also extends the polytope-fan dictionary to non-projective toric varieties, by replacing polytopes with positions in the GIT chamber structure.

Bibliography [Master]

@article{Demazure1970,
  author  = {Demazure, Michel},
  title   = {Sous-groupes alg{\'e}briques de rang maximum du groupe de Cremona},
  journal = {Annales scientifiques de l'{\'E}cole normale sup{\'e}rieure (4)},
  volume  = {3},
  year    = {1970},
  pages   = {507--588}
}

@book{OdaConvexBodies,
  author    = {Oda, Tadao},
  title     = {Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties},
  publisher = {Springer-Verlag},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {15},
  year      = {1988}
}

@book{FultonToric,
  author    = {Fulton, William},
  title     = {Introduction to Toric Varieties},
  publisher = {Princeton University Press},
  series    = {Annals of Mathematics Studies},
  volume    = {131},
  year      = {1993}
}

@book{CoxLittleSchenck,
  author    = {Cox, David A. and Little, John B. and Schenck, Henry K.},
  title     = {Toric Varieties},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {124},
  year      = {2011}
}

@article{Batyrev1994,
  author  = {Batyrev, Victor V.},
  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}
}

@incollection{BatyrevBorisov1996,
  author    = {Batyrev, Victor V. and Borisov, Lev A.},
  title     = {On {C}alabi-{Y}au complete intersections in toric varieties},
  booktitle = {Higher-dimensional complex varieties (Trento, 1994)},
  publisher = {de Gruyter},
  year      = {1996},
  pages     = {39--65}
}

@article{Delzant1988,
  author  = {Delzant, Thomas},
  title   = {Hamiltoniens p{\'e}riodiques et image convexe de l'application moment},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {116},
  year    = {1988},
  pages   = {315--339}
}

@article{Ehrhart1962,
  author  = {Ehrhart, Eug{\`e}ne},
  title   = {Sur les poly{\`e}dres rationnels homoth{\'e}tiques {\`a} $n$ dimensions},
  journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences},
  volume  = {254},
  year    = {1962},
  pages   = {616--618}
}

@article{Brion1996,
  author  = {Brion, Michel},
  title   = {Piecewise polynomial functions, convex polytopes and enumerative geometry},
  journal = {Banach Center Publications},
  volume  = {36},
  year    = {1996},
  pages   = {25--44}
}

@article{BrionVergne1997,
  author  = {Brion, Michel and Vergne, Mich{\`e}le},
  title   = {Lattice points in simple polytopes},
  journal = {Journal of the American Mathematical Society},
  volume  = {10},
  year    = {1997},
  pages   = {371--392}
}

@article{Cox1995,
  author  = {Cox, David A.},
  title   = {The homogeneous coordinate ring of a toric variety},
  journal = {Journal of Algebraic Geometry},
  volume  = {4},
  year    = {1995},
  pages   = {17--50}
}

@article{KreuzerSkarke2002,
  author  = {Kreuzer, Maximilian and Skarke, Harald},
  title   = {Complete classification of reflexive polyhedra in four dimensions},
  journal = {Advances in Theoretical and Mathematical Physics},
  volume  = {4},
  year    = {2002},
  pages   = {1209--1230}
}

@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},
  year    = {1996},
  pages   = {243--259}
}

@book{Sturmfels1995,
  author    = {Sturmfels, Bernd},
  title     = {Gr{\"o}bner Bases and Convex Polytopes},
  publisher = {American Mathematical Society},
  series    = {University Lecture Series},
  volume    = {8},
  year      = {1995}
}

@article{AtiyahConvexity1982,
  author  = {Atiyah, Michael F.},
  title   = {Convexity and commuting Hamiltonians},
  journal = {Bulletin of the London Mathematical Society},
  volume  = {14},
  year    = {1982},
  pages   = {1--15}
}

@article{GuilleminSternberg1982,
  author  = {Guillemin, Victor and Sternberg, Shlomo},
  title   = {Convexity properties of the moment mapping},
  journal = {Inventiones Mathematicae},
  volume  = {67},
  year    = {1982},
  pages   = {491--513}
}

@article{Mustata2002,
  author  = {Musta{\c{t}}{\u{a}}, Mircea},
  title   = {Vanishing theorems on toric varieties},
  journal = {Tohoku Mathematical Journal},
  volume  = {54},
  year    = {2002},
  pages   = {451--470}
}

Prerequisites

04.11.01
04.11.02
04.11.03
04.11.04

Tier anchors

beginner: Fulton *Introduction to Toric Varieties* §1.5 informal — the polytope-fan correspondence as a recipe for projective toric varieties and their ample line bundles
intermediate: Fulton *Introduction to Toric Varieties* §1.5 + §3.4; Cox-Little-Schenck *Toric Varieties* §6.1 + §6.2
master: Fulton §1.5, §3.4; Cox-Little-Schenck §6.1, §6.2, §6.3; Oda *Convex Bodies and Algebraic Geometry* Ch. 1 §1.5, Ch. 2 §2.3; Demazure 1970 *Sous-groupes algébriques de rang maximum*; Kempf-Knudsen-Mumford-Saint-Donat *Toroidal Embeddings I* Ch. I §3; Mumford-Suominen 1972 *Introduction to the theory of moduli*; Brion 1996 *Piecewise polynomial functions, convex polytopes and enumerative geometry*; Batyrev 1994 *Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties*

References

TODO_REF
Fulton — Introduction to Toric Varieties · §1.5 (the polytope and its fan); §3.4 (projective toric varieties from polytopes; the ample line bundle $L_P$); §3.5 (Demazure character formula and lattice-point sections)
TODO_REF
Cox-Little-Schenck — Toric Varieties · §6.1 (the toric variety of a polytope); Theorem 6.1.18 (the polarisation $L_P$ on $X_P$); §6.2 (global sections via lattice points; Demazure character formula); §6.3 (Cartier vs Weil divisors and the ample cone)
TODO_REF
Oda — Convex Bodies and Algebraic Geometry · Ch. 1 §1.5 (normal fan of a polytope); Ch. 2 §2.3 (ample line bundles via strictly convex support functions; polytope-to-fan dictionary)
TODO_REF
Demazure 1970 — Sous-groupes algébriques de rang maximum du groupe de Cremona · *Annales scientifiques de l'École normale supérieure* (4) 3, 507-588; the normal-fan construction for lattice polytopes; the ample-line-bundle construction $L_P$ via vertex local data
TODO_REF
Kempf-Knudsen-Mumford-Saint-Donat — Toroidal Embeddings I · Ch. I §3 (polytope-fan correspondence in the toroidal-embedding framework; ample line bundles via strictly convex piecewise-linear support functions)
TODO_REF
Batyrev 1994 — Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties · *Journal of Algebraic Geometry* 3, 493-535; reflexive polytopes; mirror duality $P \leftrightarrow P^\circ$; identification of Calabi-Yau hypersurface families in $X_P$ and $X_{P^\circ}$ as mirror partners
TODO_REF
Brion 1996 — Piecewise polynomial functions, convex polytopes and enumerative geometry · *Banach Center Publications* 36, 25-44; identification of $H^0(X_P, L_P^{\otimes k})$ with the lattice-point space of $kP$ via the Demazure-Brion lattice-point formula; Ehrhart polynomial as Hilbert function
TODO_REF
Ehrhart 1962 — Sur les polyèdres rationnels homothétiques à $n$ dimensions · *Comptes Rendus de l'Académie des Sciences* 254, 616-618; the polynomiality of $k \mapsto |kP \cap M|$ for lattice polytopes
TODO_REF
Mumford 1965 — Geometric Invariant Theory · 1st edition Ch. 2; the relation between ample line bundles and embeddings into projective space, applied here to the polarisation $L_P$ embedding $X_P \hookrightarrow \mathbb{P}^{|P \cap M| - 1}$
TODO_REF
Delzant 1988 — Hamiltoniens périodiques et image convexe de l'application moment · *Bulletin de la Société Mathématique de France* 116, 315-339; symplectic-side correspondence between Delzant polytopes and symplectic toric manifolds, dovetailing with the polytope-fan dictionary on the projective-toric overlap
TODO_REF
Cox 1995 — The homogeneous coordinate ring of a toric variety · *Journal of Algebraic Geometry* 4, 17-50; reframing of $X_P$ via the Cox ring; the polarisation $L_P$ as the canonical class in the Picard grading

Lean module

Codex.AlgGeom.Toric.PolytopeFanDictionary

Mathlib gap

Mathlib has no toric-variety development at present, and in particular
no infrastructure for the polytope-fan correspondence developed here.
Specifically missing: (i) a `LatticePolytope` structure on a free
$\mathbb{Z}$-module $M$ packaging the vertex set, face lattice, and
lattice-point enumeration in a form usable for projective-toric-variety
constructions; (ii) the normal-fan construction $P \mapsto \Sigma_P$ as
a `NormalFan` taking values in the cocharacter lattice $N$, with the
proof that it satisfies the fan axioms; (iii) the polarisation line
bundle $L_P$ on $X_P = X_{\Sigma_P}$, constructed via the cocycle
$\chi^{v - v'}$ over the maximal cones, with the ample condition proven
via the strictly convex support function of $P$; (iv) the Demazure
character formula identifying $H^0(X_P, L_P)$ with the lattice-point
space $\bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m$ as a
$T$-representation; (v) the Ehrhart polynomial $L(P, k) = |kP \cap M|$
as the Hilbert function $\dim H^0(X_P, L_P^{\otimes k})$, requiring
the equivariant Brion-Vergne residue formula for full justification;
(vi) reflexive-polytope and Batyrev-mirror-duality theorems
(Batyrev 1994) for the Calabi-Yau pointer. The schematic Lean module
records the combinatorial kernel and `sorry`-stubs the
algebraic-geometric content. Once toric varieties enter Mathlib via
the fan-gluing construction of `04.11.04` plus the affine-toric base
case of `04.11.03`, the polytope-fan dictionary is the next layer
unlocking projective-toric infrastructure for the whole library.

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 50m
master: 90m