Toric Geometry II – Toric Varieties

Last time we did a bunch of stuff with fans and polytopes and made a lot of definitions. This time, we get to use that stuff to do some algebraic geometry. First up, we’ll need to define some lattice in some vector space, otherwise nothing we did last time is applicable. Now, Matt’s recently started talking about rational varieties, and I’m going to do him one better. I not only just care about the case where our field is \mathbb{C}, but I’m only going to care about rational varieties with special nice group actions.

So now, let T=(\mathbb{C}^*)^n. That is, n-tuples of nonzero complex numbers. We’ll call objects of this type complex algebraic tori (Correction thanks to P. Clarke), or just tori, and a specific one is a torus. So, T is actually an affine variety, and we can describe it as the zero set of x_1x_2\ldots x_{n+1}-1 in \mathbb{C}^{n+1}.

So we want to get some lattices out of this. We manage this by defining a character of T to be a group homomorphism \chi:T\to \mathbb{C}^* and a 1-parameter subgroup to be \lambda:\mathbb{C}^*\to T. We’re actually going to require that these not just be group homomorphisms, but also be morphisms of varieties. Call the group of characters M and the group of 1-parameter subgroups N. Now, both of these happen to be isomorphic to \mathbb{Z}^n, because we can take (m_1,\ldots,m_n) to the character \chi^m(t_1,\ldots,t_n)=t_1^{m_1}\ldots t_n^{m_n}, and (u_1,\ldots,u_n) to the 1-parameter subgroup given by \lambda^u(t)=(t^{u_1},\ldots,t^{u_n}).

Even better, these two groups are naturally dual, because we get a map taking \chi\in M,\lambda\in N to \chi\circ \lambda:\mathbb{C}^*to \mathbb{C}^*, which is t\mapsto t^k. So we define this k to be \langle \chi,\lambda\rangle, and in coordinates, it’s just dot product. Additionally, both of these are lattices in the vector spaces M_{\mathbb{R}}=M\otimes_{\mathbb{Z}} \mathbb{R} and N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}. We’re going to make use of these dual lattices and dual vector spaces later.

Now, finally, our main definition. We’re going to actually restrict and not give the fully general definition, because the correspondences work better. A toric variety is a normal variety X of dimension n such that there is a Zariski open set isomorphic to T=(\mathbb{C}^*)^n such that the natural action of T on itself (by left multiplication) extends to an action on all of X.

First up, some examples: \mathbb{C}^n,\mathbb{P}^n are both toric, as is T, and products of toric varieties are toric. But that only begins to explain them.

We’ll first construct some affine toric varieties. Recall that, given a cone \sigma in N_{\mathbb{R}}, then the dual is \sigma^\vee=\{m\in M_{\mathbb{R}}|\langle m,u\rangle\geq 0\quad\forall u\in \sigma\}. Now we look at \sigma^{\vee}\cap M. Note that this will satisfy \forall m,m'\in \sigma^\vee\cap M we’ll have m+m'\in \sigma^\vee\cap M. So we look at the characters \{\chi^m|m\in \sigma^\vee\cap M\}, and define multiplication on them by \chi^m\cdot \chi^{m'}=\chi^{m+m'}, and take complex linear combinations of them, and give them this multiplication. This has the structure of a commutative \mathbb{C}-algebra, and in fact will be finitely generated and reduced, and we’ll denote the ring by \mathbb{C}[\sigma^\vee\cap M]. Thus, we get a variety X_\sigma=\mathrm{Spec}(\mathbb{C}[\sigma^\vee\cap M]).

Now, \mathbb{C}[\sigma^\vee\cap M] is a subring of \mathbb{C}[M], and so the variety corresponding to \mathbb{C}[M] is contained in X_\sigma. Now, \mathbb{C}[M] consists of all characters, and as M\cong \mathbb{Z}^n, this is isomorphic to \mathbb{C}[t_1^{\pm 1},\ldots,t_n^{\pm 1}], the coordinate ring of (\mathbb{C}^*)^n, so there’s an open subset of X_\sigma which is a torus, T. I’ll leave it as an exercise to check that the action extends.

So, this will actually be every single affine toric variety, but there are certainly non-affine ones. In general, every toric variety (by our definition) comes from not a cone, but rather a fan (possibly incomplete). It requires the following observation: if \tau\subset \sigma are cones, then \sigma^\vee\subset \tau^\vee, so X_\tau\subset X_\sigma and is in fact an open subvariety. So now, in general, given a fan \Sigma, we look at all the affine varieties X_\sigma for \sigma\in \Sigma, and then we glue them all together by gluing X_\sigma and X_\tau along X_{\sigma\cap \tau}. Now, THESE will be all the toric varieties.

Now we should do an example. Look at the fan pictured below:

We claim that it will give the toric variety \mathbb{P}^2. To see this, we first need to work out the affine opens. Now, X_{\sigma_0}=\mathrm{Spec}(\mathbb{C}[\sigma^\vee\cap \mathbb{Z}^2]). Now, \sigma_0^\vee consists of all ordered pairs of integers whose inner product with everything in the first quadrant is nonnegative. Thus, \sigma_0^\vee=\sigma, and so we have \mathbb{C}[\sigma^\vee\cap M]=\mathbb{C}[\mathbb{N}^2]=\mathbb{C}[x,y]. For \sigma_1, we get that the dual consists of integer linear combinations of (0,-1),(1,-1), and so we get \mathbb{C}[\frac{1}{y},\frac{x}{y}]. Similary, for \sigma_2 we get \mathbb{C}[\frac{1}{x},\frac{y}{x}]. This is the usual affine open cover of \mathbb{P}^2, and by working out the duals of the one-dimensional cones, we can compute the gluing maps, which are precisely the usual ones, so this fan gives \mathbb{P}^2.

We’ll stop here this time, and next time, we’ll talk about some properties of fans and cones and how we can use them to deduce properties of the toric varieties.

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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One Response to Toric Geometry II – Toric Varieties

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