Toric fan and Laurent potential data for the line-bundle case

A lattice here is a finite-rank free abelian group in the sense of “Free abelian group on a set”. For such a group $N$, put $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$, $M=\mathrm{Hom}(N,\mathbb{Z})$ and $M_{\mathbb{R}}=M{\otimes }_{\mathbb{Z}}\mathbb{R}$, with their evaluation pairing $⟨m,v⟩$. A strongly convex rational polyhedral cone is a set $\sigma ={\sum }_{i=1}^a{\mathbb{R}}_{\ge 0}{w}_i$ generated by finitely many $w_i\in N$ and satisfying $\sigma \cap (-\sigma )=\{0\}$. Write $n=\mathrm{rank}N$ and $\dim \sigma ={\dim }_{\mathbb{R}}{\mathrm{span}}_{\mathbb{R}}\sigma$. Its dual is ${\sigma }^{\vee }=\{m\in M_{\mathbb{R}}:⟨m,v⟩\ge 0\text{ for all }v\in \sigma \}$. A face is $\sigma \cap {\mathrm{\ell }}^{\perp }$ for a linear functional $\mathrm{\ell }\in {\sigma }^{\vee }$, including $\sigma$ itself by taking $\mathrm{\ell }=0$. For the smooth cones below, the zero face is obtained by a functional positive on every basis ray. A ray is a one-dimensional cone; $\mathrm{\Sigma }(1)$ denotes the set of rays in a fan $\mathrm{\Sigma }$. Its primitive generator $v_{\rho }$ is the unique nonzero vector with $\rho \cap N={\mathbb{Z}}_{\ge 0}{v}_{\rho }$.

A fan $\mathrm{\Sigma }$ is a nonempty finite collection of these cones that contains every face of each member and for which $\sigma \cap \tau$ is a face of both $\sigma$ and $\tau$ for all members. It is complete when ${\bigcup }_{\sigma \in \mathrm{\Sigma }}\sigma =N_{\mathbb{R}}$ and smooth when every cone has a presentation $\sigma ={\sum }_{i=1}^d{\mathbb{R}}_{\ge 0}{v}_i$ for vectors $v_1,\dots ,v_d$ that form part of an integral basis of $N$. In these basis coordinates its rays are exactly the positive coordinate axes, so these vectors are precisely its primitive ray generators. Its coordinate description also shows it is closed: the first $d$ coordinates are nonnegative and all remaining coordinates vanish. In a complete smooth fan every cone maximal under inclusion has dimension $n$: otherwise, for such a cone $\sigma$, let $x$ be the sum of its primitive rays and take a sequence tending to $x$ off its linear span. Completeness and finiteness give another cone $\tau$ containing a subsequence, and closedness gives $x\in \tau$. A nonnegative supporting functional defining the face $\sigma \cap \tau$ vanishes on $x$, hence on each ray of $\sigma$, so $\sigma \subseteq \tau$. The off-span points make this inclusion strict, contradicting maximality. Thus the rays of a maximal cone form a full integral basis. The rank-zero case is $\mathrm{\Sigma }=\{\{0\}\}$.

For a smooth fan $\mathrm{\Sigma }$, the associated toric scheme $X_{\mathrm{\Sigma }}$ uses the following explicit affine charts and gluing. For an additive submonoid $S\subset M$, $\mathbb{C}[S]$ is the algebra with basis symbols ${\chi }^m$ for $m\in S$, finite linear combinations as elements, and multiplication ${\chi }^m{\chi }^{m^{\mathrm{\prime }}}={\chi }^{m+m^{\mathrm{\prime }}}$. Set $S_{\sigma }={\sigma }^{\vee }\cap M$ and $U_{\sigma }=\mathrm{Spec}\mathbb{C}[S_{\sigma }]$, with affine scheme meaning “Affine schemes and their coordinate rings”. For a smooth cone choose an integral basis $v_1,\dots ,v_n$ whose first $d$ vectors are its primitive rays, and let $m_1,\dots ,m_n$ be the dual basis. Then $S_{\sigma }={\sum }_{i\le d}{\mathbb{Z}}_{\ge 0}{m}_i+{\sum }_{i>d}\mathbb{Z}{m}_i$, so its chart algebra is exactly $\mathbb{C}[x_1,\dots ,x_d,x_{d+1}^{\pm 1},\dots ,x_n^{\pm 1}]$, where $x_i={\chi }^{m_i}$. This coordinate calculation also gives finite generation directly in the smooth case; no separate general semigroup theorem is assumed.

In these smooth coordinates a face is generated by a subset of the rays: its nonnegative supporting functional vanishes on precisely that subset, and the nonnegative sum can vanish only on those generators. If a face $\tau ⪯\sigma$ is generated by the subset indexed by $I\subset \{1,\dots ,d\}$, its ring is $\mathbb{C}[S_{\tau }]=\mathbb{C}[S_{\sigma }][({\prod }_{j\in \{1,\dots ,d\}\setminus I}{x}_j{)}^{-1}]$. Indeed the inequalities removed from the dual cone allow precisely those formerly nonnegative coordinates to have arbitrary integer exponent. The empty product is $1$. By “A principal localization identifies its spectrum with a distinguished open” this identifies $U_{\tau }$ with the indicated distinguished open in $U_{\sigma }$, including its structure sheaf. These maps are intrinsic inclusions of the monomial algebras in $\mathbb{C}[M]$; a different integral basis gives the same map, not additional gluing data.

For $\sigma ,\tau \in \mathrm{\Sigma }$ glue their common open $U_{\sigma \cap \tau }$ by these identity-on-character identifications. Within any one smooth chart, intersecting two face opens inverts the union of their removed coordinates and gives the open for the intersection of the faces. Hence on a triple overlap both composites are the same inclusion into $\mathbb{C}[S_{\sigma \cap \tau \cap \eta }]$. The identity and cocycle conditions hold, and “Gluing affine schemes along compatible open isomorphisms” constructs $X_{\mathrm{\Sigma }}$ with these open charts. No additional pairwise identifications arise from chains: two consecutive identifications meet in precisely that triple overlap and compose to the stated direct one. Consequently, inside $X_{\mathrm{\Sigma }}$, the intersection $U_{\sigma }\cap U_{\tau }$ is exactly $U_{\sigma \cap \tau }$, and the same common-cone rule holds for every finite intersection. The chart for $\{0\}$ is the common dense torus $\mathrm{Spec}\mathbb{C}[M]$, whose complex points are $\mathrm{Hom}(M,{\mathbb{C}}^{\ast })$. In each chart this torus is the dense distinguished open obtained by inverting $x_1\cdots x_d$; the chart ring is a domain, so its zero prime belongs to this open and its closure is the whole chart. Its algebraic action is specified by the coaction ${\chi }^m\mapsto {\chi }^m\otimes {\chi }^m$ into $\mathbb{C}[M]{\otimes }_{\mathbb{C}}\mathbb{C}[S_{\sigma }]$. These coactions respect the displayed localizations and glue to a torus action on $X_{\mathrm{\Sigma }}$. This is the original toric torus, with character lattice $M$; the mirror torus below has character lattice $N$.

In this definition projective retains the additional condition that $X_{\mathrm{\Sigma }}\to \mathrm{Spec}\mathbb{C}$ admits a closed immersion into some finite-dimensional projective space, as in “Projective morphisms before Proj”. It is a restriction on the selected fan data, not an assertion that every smooth complete fan is projective.

Let $N\cong {\mathbb{Z}}^n$ be a lattice, $M=\mathrm{Hom}(N,\mathbb{Z})$ its dual, and $\mathrm{\Sigma }$ a smooth complete fan in $N_{\mathbb{R}}$ whose associated toric variety $X_{\mathrm{\Sigma }}$ is projective. For each one-dimensional cone $\rho \in \mathrm{\Sigma }(1)$, let $v_{\rho }\in N$ be its primitive lattice generator. Set $A_{\mathrm{\Sigma }}=\{0\}\cup \{v_{\rho }:\rho \in \mathrm{\Sigma }(1)\}$. For coefficients $c_a\in {\mathbb{C}}^{\ast }$, the associated Laurent polynomial on the mirror algebraic torus $T^{\vee }=\mathrm{Hom}(N,{\mathbb{C}}^{\ast })$, whose character lattice is $N$, is

$W_{\mathrm{\Sigma }}(z)={\sum }_{a\in A_{\mathrm{\Sigma }}}{c}_a{z}^a,$

where $z^a(t)=t(a)$ is the character determined by $a\in N$ and $t\in T^{\vee }$. Thus the fan lattice of $X_{\mathrm{\Sigma }}$ is the character lattice of its mirror torus; no identification of $N$ with its dual $M$ is assumed. An admissible toric Landau–Ginzburg datum additionally names the coefficient choices, the zero fibre $M_0=W_{\mathrm{\Sigma }}^{-1}(0)$, and any regularity, symplectic, or Lefschetz conditions actually required for the selected Fukaya construction. This definition does not assert that generic coefficients satisfy those conditions, that $M_0$ is smooth, or that a homological mirror equivalence holds.

The lattice, affine schemes, principal-localization open immersions, compatible gluing and finite-dimensional projectivity use the following exact suppliers. The smooth-coordinate calculations above check the localization and cocycle hypotheses. No genericity, Lefschetz or mirror-equivalence existence result is implicit.

Free abelian group on a set

Affine schemes and their coordinate rings

A principal localization identifies its spectrum with a distinguished open

Gluing affine schemes along compatible open isomorphisms

Projective morphisms before Proj