Objects In Toric Geometry From Different Perspectives

Start with an algebraic torus \(T\), not toric variety for now. The character lattice \(M\) of \(T\) can be thought of in the following equivalent ways:

1. \(M = \Hom(T, \mathbb G_m)\) in the category of algebraic groups
2. \(M\) is the set of monomial functions on \(T\) which constant coefficient (“M” for “monomial”)
3. \(M\) is the set of invertible functions on \(T\) modulo \(\mathbb C^\times\)
4. \(M\) is the degree one part of \(A^*(BT) = A^*_T(\pt) = \Sym(M)\)

To show the equivalence of \((1)\) and \((2)\), show that every regular function \(f:T\to \mathbb C\) can be written as a finite sum of characters \(\chi:T\to \mathbb G_m = \mathbb C^\times\).

We could instead start with a lattice (finite rank \(\mathbb Z\)-module) \(M\), and then we could recover \(T\) by setting \(T = \Spec \mathbb C[M] \cong \Spec \mathbb C[t^{\pm}_1,...,t^{\pm}_r]\) where \(r = \rank M\). Note that

\begin{align*} \Spec \mathbb C[M] = \Spec \left\{\sum t^\chi ~ \mid ~ \chi \in M\right\}, \end{align*}

and if we choose a basis \(e_1,...,e_r\) for \(M\) and \(\chi = a_1e_1 + ... + a_re_r\), then by choosing coordinates \((t_1,...,t_r) \in T\) we get \(\chi(t_1,...,t_r) = t_1^{a_1}\cdot \dots\cdot t_r^{a_r}\). This is how to interpret \(t^\chi\) as a function on \(T\).

If we had started with \(T\) instead, then a character \(\chi\) would by definition be a function \(\chi:T\to \mathbb G_m\subset \mathbb C\), in which case the formal symbol \(t^\chi\in \mathbb C[M]\) is a regular function on \(T\) simply by setting \(t^\chi(p) = \chi(p)\).

For \((3)\), we may need the algebraic closedness of \(\mathbb C\).

Now start with an affine toric variety \(X = \Spec R\) with torus \(T\), and consider: how does \(R\) relate to \(\mathbb C[m] = \mathcal O_T\)? The quick answer: \(R\) is the subring of \(\mathbb C[M]\) obtained by deleting all functions which don’t extend to \(X\), because they don’t converge. Said another way, \(\mathbb C[M]\) is obtained from \(R\) by localizing at certain functions.

More precisely, let \(Z = X\setminus T\). Then \(Z\hookrightarrow X\) is a closed immersion, so \(Z\) corresponds to some ideal \(I\subset R\). Since a toric variety is finite type and hence Noetherian, \(I = (f_1,...,f_n)\) for some functions \(f_i \in R\).

Let \(j:T\to X\) be the inclusion. Then \(j^{-1}\mathcal O_T^\times \cap R\) consists of all functions on \(X\) which are both invertible on \(T\) and are defined on \(R\), that is, its the set of functions \(f:X\to \mathbb C\) which, if they have a zero anywhere, it occurs outside of the open set \(T\) (obviously this is the divisorial log structure on \(X\) with respect to \(D = X\setminus T\)). Let’s restrict our attention to just those functions which have at least some zero. We do this by quotienting by the units in \(R\), and we get a set

\begin{align*} P = j^{-1}\mathcal O^\times_T \cap R/R^\times. \end{align*}

It turns out \(P\) is a monoid under multiplication; if I take two functions \(f_1,f_2\in P\) then \(f_1 + f_2\) may have a new zero, but \(f_1\cdot f_2\) vanishes only on \(X\setminus T\). Notice that since \(\mathbb C^\times \subset R^\times\),

\begin{align*} P \subset j^{-1}\mathcal O^\times_T/\mathbb C^\times = M, \end{align*}

so as a matter of fact, \(P\) is a submonoid of \(M\).

Punchline:

\begin{align*} \left(\text{toric variety} ~ T\hookrightarrow X \right) ~ \rightsquigarrow \left(\text{monoid} ~ P \subseteq M\right) \end{align*}

Notice a property of this monoid: if \(\chi\in P\), then it is a non-vanishing function \(f:T\to \mathbb G_m\) since \(P\subset M\). However it is also a regular function on \(X\) well defined up to scaling by \(R^\times\); we can fix a representative by demanding that \(\chi(1) = 1\), where \(1\) is the unit in \(T\) (this then ensures it is a group homomorphism too). Once we’ve done this, we see that by definition, \(\chi\) extends to \(X\) – of course it does, it started life as a function on \(X\). The claim is then that

\begin{align*} X = \Spec \mathbb C[P]. \end{align*}

This is the case. It is easier to show that \(R = \mathbb C[P]\). The inclusion \(\mathbb C[P] \subset R\) is easy, it follows from what we have just described – for the other direction, start with a regular function \(f\in R\), restrict it to \(T\), then write it in terms of characters in \(M \) (possible since \(\mathcal O_T = \mathbb C[M]\)). These characters must all extend to \(X\), hence are contained in \(P\), giving \(R\subseteq \mathbb C[P]\). So a more straightforward definition of \(P\) might be that “\(P\) is the submonoid of \(M\) containing all characters which extend from a map \(T\to \mathbb G_m\) to a map \(X\to \mathbb C\).”

Reverse direction If we instead start with a torus \(T\) with character lattice \(M\), we can build \(X\) by setting \(X = \Spec \mathbb C[P]\).