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An algebraic torus over a field $k$ is an algebraic group $T$ whose base change to a separable closure $k_s$ is isomorphic to ${\mathbb{G}}_m^n$. Its dimension is $n$. An algebraic torus already isomorphic to ${\mathbb{G}}_m^n$ over $k$ is a split algebraic torus. Toric geometry described by an ordinary toric fan uses a split algebraic torus.

The real algebraic group $x^2+y^2=1$, with multiplication inherited from complex multiplication, is a one-dimensional nonsplit example. Over the complex numbers, $z=x+iy$ identifies its coordinate ring with $\mathbb{C}[z,z^{-1}]$. Complex conjugation acts on its geometric character lattice of an algebraic torus by $m\mapsto -m$, so it has no nontrivial real-defined algebraic torus character. Algebraic tori of dimension one therefore need not be the real multiplicative group.^[ref]

For an algebraic torus $T$ over $k$ and a chosen separable closure $k_s$, its geometric character lattice of an algebraic torus isIt is a free abelian group of rank $\dim T$, with its natural action of $\mathrm{Gal}(k_s/k)$. Characters of an algebraic torus defined over $k$ form the fixed subgroup of this action; they need not constitute the full geometric toric lattice.

For a split algebraic torus $T={\mathbb{G}}_m^n$, each algebraic torus character is a Laurent monomial $t_1^{m_1}\cdots t_n^{m_n}$, so $X^{\ast }(T)\cong {\mathbb{Z}}^n$. The real norm-one algebraic torus has geometric algebraic torus character toric lattice $\mathbb{Z}$ with complex conjugation acting by minus one, so its subgroup of real-defined characters of an algebraic torus is zero.^[ref]

The geometric cocharacter lattice of an algebraic torus $T$ over $k$ isusing a separable closure $k_s$. Composition with an algebraic torus character gives a homomorphism ${\mathbb{G}}_m\to {\mathbb{G}}_m$, necessarily $z\mapsto z^a$ for an integer $a$. This defines the perfect pairingFor a split algebraic torus, the cocharacter with exponent vector $(n_1,\dots ,n_r)$ is $z\mapsto (z^{n_1},\dots ,z^{n_r})$ and the pairing is ${\sum }_i{m}_i{n}_i$. The two geometric toric lattices carry compatible Galois group actions; $k$-defined cocharacters are the fixed elements. This is the duality used for the toric lattice $N$ and its dual $M$.^[ref]

Toric geometry studies algebraic varieties with an action of an algebraic torus having a dense open orbit isomorphic to that algebraic torus. For normal algebraic varieties with a split algebraic torus, toric fans encode the geometry using integral toric cones and their intersections.

The simplest chart is the affine line: the multiplicative algebraic torus acts by scaling, with one open orbit and the origin as a fixed point. The projective line adds another fixed point at infinity. Its toric fan has the zero toric cone and the positive and negative half-lines in the real extension of the rank-one toric lattice.

The dictionary turns geometric questions into finite arithmetic data: smoothness criterion for a toric variety is checked by primitive toric ray generators extending to a lattice basis, properness by a complete fan, and line bundles by compatible integral linear functions on toric cones. The normal variety and split algebraic torus conventions matter; an arbitrary algebraic variety with a dense algebraic torus need not be described by the same ordinary toric fan construction.^[ref]

A normal split toric variety is an algebraic variety containing a split algebraic torus $T$ as a dense open subset, with the multiplication action of $T$ extended to the whole algebraic variety. The toric fan construction gluesfor toric cones of a toric fan, along the open charts corresponding to common faces. Gordan lemma makes each chart finitely generated, and saturation of the toric lattice monoid makes the chart a normal variety.

The first-quadrant toric cone gives affine space ${\mathbb{A}}^n$; the zero toric cone gives the algebraic torus ${\mathbb{G}}_m^n$. The two half-lines in rank one give the projective line. These examples show that a toric algebraic variety need not be complete or have any fixed points.

Some authors allow nonnormal toric algebraic varieties as well. Such algebraic varieties need not arise from ordinary toric fans; the restriction to normal varieties is essential in the stated toric fan classification.^[ref]