## December 31, 2008

### The Toric Variety Associated to the Weyl Chambers

#### Posted by John Baez

Happy New Year! I’m making a resolution to avoid starting work on new papers. I want to spend more time learning math, playing music, and having fun.

For a long time, whenever people said the phrase toric variety, I’d cover my ears and refuse to listen to what they had to say. Since I knew nothing of algebraic geometry, I thought ‘toric varieties’ were just one more of those specialized concepts — like Fano varieties and del Pezzo surfaces — that were devised solely for the purpose of demonstrating one’s superior erudition in this esoteric subject.

That’s changed. Now I love toric varieties, and I have a question about them.

But I won’t explain what they are, because then everyone would know.

Just kidding. I do hope to explain toric varieties someday. As usual, they’re a lot nicer and a lot easier to understand than anybody admits. Jim Dolan and I have been busy learning about them over the last month. But I’m not quite ready to give them away yet.

Well… I can’t resist just giving away just one one clue: toric varieties are the varieties you can define using equations that don’t involve addition or subtraction — just multiplication! So, they fit in very nicely with Deitmar’s approach to the field with one element, where he generalizes algebraic geometry from commutative rings to commutative monoids, by seeing what we can do without addition. Addition turns out to be vastly overrated. Doing algebraic geometry without addition is an incredibly simple, beautiful idea — so now I love toric varieties!

(Hmm, maybe this why Deitmar he wrote a paper on F1-schemes and toric varieties! I hadn’t really noticed that before. The sneaky devil!)

Anyway: in the usual game, people like to construct a toric variety from something called a ‘fan’. (I’ll explain this someday, honest — but not today.) And, any simple Lie algebra gives an obvious fan, consisting of the Weyl chambers in its Cartan subalgebra. So, there’s a way to get a toric variety from a simple Lie algebra! This should be important and interesting.

But: what are these toric varieties like? Can you get them from simple Lie algebras some other way?

There seems to be a paper on this subject:

• Claudio Procesi, The toric variety associated to Weyl chambers, Mots, 153-161, Lang. Raison. Calc., Hermès, Paris, 1990.

But, I don’t know how to get ahold of it.

Math Reviews has this to say:

Given a root system $\Phi$ in a finite-dimensional real Euclidean space $V$, let $W$ be the Weyl group and $M$ the $\mathbb{Z}$-lattice in $V$ generated by $\Phi$. Then

$N = \{v\in V | (v,\alpha)\in \mathbb{Z} for all \alpha\in\Phi\}$

is the lattice in $V$ dual to $M$, where the parentheses denote the inner product in $V$. Each set $\Sigma$ of simple roots in $\Phi$ determines a Weyl chamber

$C_{\Sigma} = \{v\in V| (v,\alpha)\geq 0 for all \alpha\in\Sigma \}$

and the set $\Delta$ of the faces of the Weyl chambers is a fan for $N$. The set $\Psi$ of fundamental weights is exactly the set of primitive elements in $N$ spanning the one-dimensional cones in the fan $\Delta$.

The fan $\Delta$ for $N$ determines a (complex) toric variety $X$, which turns out to be smooth and projective. Since the Weyl group $W$ is an automorphism group of the fan $\Delta$, the graded cohomology ring $H^*(X,\mathbb{Q})$ of $X$ with rational coefficients has a natural $W$-module structure.

The author describes the character of the $W$-module in terms of other $W$-modules. The key ingredients of the proof are as follows: (1) description, due to Jurkiewicz and Danilov [cf. V. I. Danilov, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247; MR0495499 (80g:14001) (Theorem 10.8 and Remark 10.9)], of $H^*(X,\mathbb{Q})$ as a quotient of the Stanley-Reisner ring $\overline{A}$ of the simplicial sphere determined by the fan $\Delta$; (2) $\overline{A}$ is a free module over the symmetric algebra $B$ of the $\mathbb{Q}$-vector space $M\otimes_{\mathbb{Z}}\mathbb{Q}$; (3) the determination, due to A. M. Garsia and D. W. Stanton [Adv. Math. 51 (1984), no. 2, 107–201; MR0736732 (86f:20003)], of the character of the natural $W$-module structure on the Stanley-Reisner ring $\overline{A}$.

For the root systems $A_n$, the author also gives a recursive formula with respect to $n$.

Hmm. It seems Procesi’s fan consists of the set of faces of the Weyl chambers, instead of the Weyl chambers themselves. Both fans sound interesting. But either way, this stuff doesn’t make me go “Aha! Now I understand the meaning of this toric variety!”

Now I wish I hadn’t put my hands over my ears every time someone told me anything about toric varieties. I’m sorry… I apologize. Help!

Posted at December 31, 2008 10:12 PM UTC

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### Re: The Toric Variety Associated to the Weyl Chambers

Isn’t this toric variety the GIT quotient of the conjugation action of the adjoint form of G on the wonderful compactification of G?

Posted by: Jason Starr on December 31, 2008 11:34 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Cool! If that’s true, I sure wish I knew what it meant!

I guess I can pretend to understand all the phrases in that sentence except “the wonderful compactification of $G$”. In my feeble attempts to look up the answer to my question, I started bumping into this “wonderful” terminology — but I don’t know what it means yet.

Could you briefly explain this compactification and why it’s so bloody wonderful? Also: why you think that its geometric invariant theory quotient by the adjoint action is the toric variety coming from the fan built from the Weyl chambers?

but I really prefer it when people tell me stuff, since they leave out unnecessary information.

Posted by: John Baez on January 1, 2009 2:15 AM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

It seems from this Invitation to Toric Topology that your toric varieties occupy merely one vertex of a tetrahedron - ACST:

• A – algebraic geometry, toric varieties
• C – combinatorial geometry, polyhedra
• S – symplectic geometry
• T – toric topology

The fourth vertex seems to be the newest.

Posted by: David Corfield on January 1, 2009 1:35 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

That’s an interesting review article, David — it seems pretty readable so far. It’s heavily biased towards homotopy theory, and that’s an aspect of toric geometry I haven’t thought about at all.

Posted by: John Baez on January 3, 2009 5:43 AM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Which interesting review article?

Posted by: jim stasheff on January 3, 2009 2:11 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Actually I think what I said is wrong. The correct statement (I hope) is that the closure of a maximal torus in the wonderful compactification is your toric variety. And then the GIT quotient turns out to be the quotient of the toric variety by the induced action of the Weyl group. This “compactifies” the classical statement that every conjugacy class of semisimple elements in G intersects any fixed maximal torus T in an orbit for the Weyl group action on T. So the quotient of the set of semisimple elements by the conjugation action equals the quotient of T by the Weyl group action (which is easier to understand since the Weyl group is just a finite group).

Here is my understanding of the wonderful compactification very quickly. Left and right multiplication define an action of G x G on G. The wonderful compactification G’ of G (I’m not sure how to do LaTeX in HTML) is a projective variety containing G as a dense open subset such that: (1) G’ is smooth, (2) the complement of G in G’ is a simple normal crossings divisor in G’, and (3) the action of G x G on G extends to an action on G’. I believe G’ is minimal among compactifications satisfying (1), (2) and (3); every other compactification is obtained by blowing up G’.

Posted by: Jason Starr on January 1, 2009 1:48 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

A more concrete definition of the wonderful compactification is this:

We have a map $G \to \mathrm{Grass}(2n, n)$ where $n=\dim G$ given by sending $x\in G$ to the subspace of $\mathbf{g}\oplus\mathbf{g}$ (where $\mathbf{g}$ is the Lie algebra) of the form $(v,\mathrm{Ad}_x v)$. This map is an inclusion if $G$ is adjoint type.

The wonderful compactification $G'$ is the closure of the image of this map in the Grassmannian. Remember, being a Lie subalgebra is a closed condition, so all the points in this variety are Lie subalgebras of $\mathbf{g}\oplus\mathbf{g}$. These subalgebras aren’t terribly difficult to describe, (for example, in the paper of Evens and Lu).

Posted by: Ben Webster on January 2, 2009 10:44 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Thanks very much, Jason and Ben!

Putting your remarks together, it sounds like the toric variety I’m interested in looks like this:

Take a complex simple Lie group $G$ of adjoint type — say, $PSL(n,\mathbb{C})$ for starters. Take a maximal torus $T$.

(I think by this Jason must have meant a maximal abelian subgroup of $T$. In Lie theory, a ‘maximal torus’ $T$ often means a maximal compact abelian subgroup. But he’s gonna compactify his $T$ and get a complex variety, which fails to make sense on two counts if $T$ is a compact real subgroup of $G$. So, I’ll assume his $T$ is a maximal abelian subgroup, which Lie theorists like to call $A$ — and thus a torus in the toric geometry sense.)

(On second thought, toric geometry should make sense over any field, so we could try to get a real variety. But let’s stick with complex varieties.)

Then, map $T$ into $Grass(2n, n)$ by sending $x \in T$ to the space of all vectors of the form $(v, Ad_x v)$. And then, take the closure of this in $Grass(2n, n)$ to get our variety $\overline{T}$.

I presume $\overline{T}$ becomes a toric variety by virtue of $T$ itself acting on $\overline{T}$ in some obvious way, extending the action

$t : \{(v, Ad_x v)\} \mapsto \{(Ad_t v, Ad_{t x} v)\}$

I would like to understand what points we’re gluing on to $T$ to get $\overline{T}$. At least in some examples, this should be easy to visualize.

Posted by: John Baez on January 2, 2009 11:25 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Why not ask Procesi for a copy?

Posted by: jim stasheff on January 1, 2009 11:08 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

I could, but I seem to be doing better asking for help on this blog. It’s not his paper I need, so much as a general understanding of this toric variety. And this, in turn, is just an excuse to ponder toric varieties and see how they’re related to other stuff.

Posted by: John Baez on January 2, 2009 11:28 PM | Permalink | Reply to this

### Re: Book

Looks like we have that book here in Göttingen. I’m sure we can get hold of it when you’re coming over.

Posted by: ssp on January 2, 2009 10:35 AM | Permalink | Reply to this

### Re: Book

Good idea!

Posted by: John Baez on January 2, 2009 11:26 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

“It seems Procesi’s fan consists of the set of faces of the Weyl chambers, instead of the Weyl chambers themselves. Both fans sound interesting.”

Here I suspect that you are slightly confused. Procesi is being precise at the expense of being clear.

If we are dealing with the Weyl group A_2, the fan of “faces of the Weyl chambers” has 6 two-dimensional cones, 6 one-dimensional cones and 1 zero-dimensional cone. It corresponds to a complete toric variety of complex dimension 2. If I were dealing with A_3, these numbers would be (24, 36, 14, 1). I suspect that this is the fan you are thinking of when you talk about the fan of the “Weyl chambers themselves”.

Let’s recall some definitions: A polyhedral cone is the intersection of finitely many closed halfspaces whose boundary contains the origin. If C is a cone, a face of C is a subset of C which can be described as { x \in C: lambda(x) =0 } where lambda is a linear function which is nonnegative on C. In particular, C is a face of itself. So, when Procesi says that he is interested in the “fan of faces of the Weyl chambers”, the Weyl chambers themselves are faces of this fan.

Why does he use this language? Recall that the definition of a fan is a nonempty set X of cones such that:

(1) If C is in X, every face of C is also in X.

(2) If Y is any subset of X, and C an element of Y, then bigcap_{D in Y} D is a face of C.

So, properly, the set of Weyl Chambers is not a fan, while the set of faces of Weyl chambers is. Thus, Procesi’s language is correct.

Define a maximal face of a fan to be a face which is not contained in any other faces. A fan is determined by its set of maximal faces. I think that most mathematicians who work with fans think of a fan as the set of its maximal faces. (I do.) I suspect that you have started doing the same, and that this is why you thought of the set of Weyl chambers as a reasonable fan.

Posted by: DavidS on January 2, 2009 5:23 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Thanks! I was being silly. So, Procesi is indeed talking about the fan I’m interested in, and the toric variety it determines.

If you’re the DavidS that I think you are, I’d be glad to take advantage of your half-hearted offer to do expository blog posts on 1) toric varieties and 2) aspects of tropical geometry. Following your advice, I won’t hold my breath… but I’m especially curious about the relation between toric varieties and tropical geometry. Tropical geometry seems to be about schemes defined over the tropical rig. Toric geometry seems to be about schemes defined over ‘the field with one element’. In both cases we’re taking algebraic geometry and stretching it to work over what Durov would call a generalized ring. But I’m missing a sense of how toric and tropical geometry are supposed to interact within this big framework.

Posted by: John Baez on January 2, 2009 6:19 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

I have seen that you were interested in categorifying symplectic geometry. Did you take a look at the notion of moment map (or I guess maybe “momentum” map since you are on the west coast ;-))?

I ask this question because an alternative definition of (symplectic) toric manifold, is (taken from A. Cannas da Silva Lecture Notes in Mathematics (springer) 1764):

“compact connected symplectic manifold, with an effective hamiltonian action of a torus T of dimension one half of the dimension of the manifold, and with a choice of a corresponding moment map”.

In short: toric manifold=moment map.

Hence if you can categorify moment maps, I guess you have chances to also categorify toric manifolds.

By the way, if you are free next week, and
want to visit Oberwolfach ;-):

Title: Toric Geometry
Organisers: Klaus Altmann, Berlin
Victor Batyrev, Tübingen
Yael Karshon, Toronto
Date: January 4th - January 10th, 2009
ID: 0902

Posted by: yael on January 2, 2009 8:20 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Rather than a geographic distinction,
I think it would be good if there were a technical distinction between moment map
and
momentum map

also sometimes moment map means equivariant moment map

in the context of constrained Hamiltonians (cf.
BFV construction)
there is a map M –> R^n which plays
the role of a moment map but R^n does
not carry the structure of (the dual of) a Lie algebra

it does satisfy a weakened condition
so I have dubbed it a momap

Posted by: jim stasheff on January 3, 2009 2:09 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

“I think it would be good if there were a technical distinction between moment map and
momentum map”

unfortunately they mean exactly the same, and they are supposed to be equivariant.

But your generalization sounds interesting. Is there an analogue of the classification of toric manifolds, if one replace moment maps by momaps?

In particular, what is known about the geometry of the image of the momap?

I ask this question because in the case of toric manifolds, there is a one to one correspondence between toric manifolds and Delzant polytopes: the image
of a moment map is always a convex polytope, but it is a Delzant polytope, only when the manifold is toric. On the other hand, given a Delzant polytope, there is an algorithm to construct a toric
manifold whose image by the moment map is
the given Delzant polytope.

Posted by: yael on January 3, 2009 2:50 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

I don’t know of any work on generalized Delzant for momaps. It’s certainly natural for this blog as momaps involve our beloved `higher structure’. Wiki doesn’t seem to have an article on BFV formalism, though BV is covered. Briefly, the original setting for BFV is that of a symplectic manifold W with a set of first class constraints, which is to say, the set generaates an ideal of the algebra $C^\infty$(W) which is closed under the poison bracket. The momap is given by the set of generators.

Posted by: jim stasheff on January 4, 2009 1:58 AM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

John Baez wrote: “… one more of those specialized concepts — like Fano varieties and del Pezzo surfaces — that were devised solely for the purpose of demonstrating one’s superior erudition in this esoteric subject.”

Maybe this was just a light-hearted comment, but actually Fano varieties are very simple and fundamental, at least from the point of view of complex differential geometry. Fano manifolds are precisely those complex projective manifolds having a family of hyperplane sections which are Calabi-Yau manifolds. In fact most of the Calabi-Yau manifolds we know about (though certainly not all) are constructed as hyperplane sections of toric Fano manifolds (or more generally, toric Fano orbifolds). Del Pezzo surfaces are just the Fano manifolds which happen to be surfaces (i.e., complex dimension 2).

Posted by: Jason Starr on January 4, 2009 2:56 AM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

It might be worth adding that Fano varieties (and their 2d version, del Pezzo surfaces) are simply projective varieties (complex submanifolds of projective space) that have positive Ricci curvature - equivalently the top exterior power of the tangent bundle is positive (ample). Nothing very esoteric about them.

Of course there’s nothing very esoteric about a toric variety - a complex projective manifold with a torus action with a dense orbit. Here “torus” has its usual meaning from Lie theory - a group which over the algebraic closure becomes a product of multiplicative groups. Since we’re already over C this is just a product of $C^\times$’s.

Posted by: David Ben-Zvi on January 4, 2009 4:27 AM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

I’m sure John’s comment was light-hearted, but at the same time maybe it’s worth noting how subjective the notion of ‘esoteric’ is — especially given the very varied backgrounds of the Café’s clientele. In other words, it’s perfectly possible that a concept might be (as Jason puts it)

very simple and fundamental, at least from the point of view of complex differential geometry

and at the same time utterly intimidating.

I guess a concept seems esoteric if, in order to understand it, you first have to understand a lot of other things. Thus, you have no immediate point of contact with the concept itself. It’s not a judgement on the value of the concept; it’s really a very subjective thing, mostly depending on one’s personal state of knowledge. I’m sure there are lots of concepts that I think are wonderful and natural and many other people regard as esoteric.

In this case, for instance, in order to understand Jason’s definition —

Fano manifolds are precisely those complex projective manifolds having a family of hyperplane sections which are Calabi-Yau manifolds

— I’d first have to understand the following:

1. ‘Complex projective manifold’. (I think I know what a complex projective variety is, and I guess I know what a complex manifold is, but I don’t know what a complex projective manifold is.)
2. ‘Hyperplane section’.
3. ‘Family of hyperplane sections’. (I suspect that ‘family’ can’t mean what I expect it to mean, because if it did then you could take the empty family and every complex projective manifold would be a Fano manifold, which presumably isn’t the case.)
4. ‘Calabi-Yau manifold’.

I’m not asking for explanations of these terms. This is just by way of demonstrating how the concept might seem esoteric or difficult or inaccessible even if it’s perfectly natural from the right point of view.

Posted by: Tom Leinster on January 4, 2009 2:54 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

I completely agree with what Tom is saying: what is esoteric or fundamental is very relative. But within algebraic geometry or complex differential geometry, Fano varieties and Del Pezzo surfaces are not esoteric.

Here’s a categorical question. There is a purely categorical notion related to Calabi-Yau manifolds, Calabi-Yau categories. Is there a “Fano category”?

Posted by: Jason Starr on January 4, 2009 9:01 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Jason,

Absolutely: in fact you can recover the canonical
(or in this case anticanonical) ring of a variety (and in particular its Kodaira dimension) from its derived category. (Bondal, Orlov, Kawamata I think are the relevant names - this is explained nicely in Ch.6 of Huybrechts’ book.) So that gives a purely categorical version of being Fano… however it is not as interesting as the CY condition: any Fano category is equivalent to the derived category of a Fano variety, and this equivalence is unique up to shift, tensor by line bundle, and automorphism of the variety. This is very different from the CY case, where there are many “nongeometric” CY categories and many nontrivial (Fourier–Mukai type) equivalences between them.

Posted by: David Ben-Zvi on January 4, 2009 9:20 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Could there be a better definition of Fano category, not all of which arise from Fano varieties? Not every Calabi-Yau manifold is a hyperplane section of a Fano variety. But maybe there is another definition of Fano category such that every Calabi-Yau category is a “hyperplane section” of a Fano category.

Posted by: Jason Starr on January 4, 2009 10:49 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Which characteristic property of Fano do you wish to encode? For example one can discuss categories with exceptional collections (such as representations of a quiver) as Fano-type categories (I believe Kawamata proved that at least all toric Fanos have exceptional collections?). There’s a nice way (exploited to great effect by Seidel and others) to cut out a family of Calabi-Yau categories as hyperplane sections (you construct a new category by trivializing the Serre functor — the role of an anticanonical section is taken up by morphisms between the trivial and the Serre functor), and the exceptional objects become spherical objects I believe. So one can ask your question, can any Calabi-Yau category be constructed using this procedure from a category with an exceptional collection?
That sounds cool..

My point was just that if we take the usual definition (ample anticanonical bundle), it has a natural categorical formulation – we know what takes the place of the canonical bundle (Serre functor), and we can ask for its (anti)ampleness categorically (unless I’m misremembering my Huybrechts, which is likely). This is precisely the categorical route to defining Calabi-Yau category (one with trivial[ized] Serre functor). It’s just that in the Fano case this gives nothing interesting.

Posted by: David Ben-Zvi on January 4, 2009 11:20 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

I have a somewhat more reasonable question than that last one. Some canonically embedded curves are hyperplane sections of K3 surfaces, but most are not (at least when the genus is sufficiently large). Could it be that for every positive genus curve the derived category of coherent sheaves on the curve is a “hyperplane section” of a Calabi-Yau category?

Posted by: Jason Starr on January 4, 2009 11:07 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Jason wrote:

Maybe this was just a light-hearted comment…

Of course! As was my comment that these concepts “were devised solely for the purpose of demonstrating one’s superior erudition”. I don’t actually believe that. Honest.

But my point was: before one takes some time to learn a subject and get an intuition for it, standard math terminology can seem like an armored wall designed to prevent understanding. After putting in the necessary work, everything changes as if by magic… and one tends to forget ones former ignorant self.

This is why most mathematicians are so lousy at explaining things. We’re great at giving precise definitions and theorems, but when it comes to the really hard part — the business of developing intuition — we all too often let our poor listeners fend for themselves. We forget all the work we put in to develop these intuitions ourselves.

So, I was trying to remind mathematicians that the terms they now understand and love remain baffling and often intimidating to people outside their speciality.

For example: before I learned a bit about toric varieties, they seemed like abstruse gizmos. A “normal variety that contains a torus as a dense open subset, together with an action of that torus that extends the action of the torus on itself” - that’s nothing I ever woke up wanting to learn about! But now I see that toric varieties are like a kid’s sandbox, a kind of ‘baby’ version of algebraic geometry where everything is easier. Algebraic geometry where you don’t need to add!

And here’s another example. In my former, ignorant state, if I asked someone what a Fano varieties were, and they said

“precisely those complex projective manifolds having a family of hyperplane sections which are Calabi-Yau manifolds”

I would say

“Oh no! I asked the definition of one concept I don’t understand, and it turns out to depend on three more! So, it’s like cutting off the heads of the hydra: for each head I cut off, three more grow back. I’d better give up now, before it gets even worse!”

In my current, slightly less ignorant state, I’d say

“Hmm, I wonder how that relates to the definition of Fano variety I already know!”

And this counts as progress.

Posted by: John Baez on January 6, 2009 8:05 PM | Permalink | Reply to this

### Devolving terminology

Another problem as terminology (d)evolves:
Henri Cartan formalized as a g-algebra where g is a Lie algebra in terms of 3 derivations: d, contraction and Lie derivative. Decades later it was reivented as a Leibniz pair and now I see it being called a calculus! though this may be a contraction of Cartan contraction.

Posted by: jim stasheff on January 7, 2009 1:57 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

A simplest description of the toric variety associated with Weyl’s chambers of a complex semisimple group G is that it is the closure of a generic orbit of its maximal complex torus T in the flag variety G/B of G (here B is a Borel subgroup).

Toric Fano varieties were used by Victor Batyrev to construct Calabi-Yau as their hyperplane sections. The fan of Weyl chambers is centrally symmetric and usually doesn’t produce a toric Fano. Actually all centrally symmetic Fano can be classified in a way similar to root systems, see:

V E Voskresenskii, A A Klyachko, MATH USSR IZV, 1985, 24 (2), 221-244.

One can get some idea of using toric varieties in physics from:

A. Knutson and E. Sharpe, Sheaves on Toric Varieties for Physics.

A. Knutson and E. Sharpe, Equivariant Sheaves.

Posted by: Alex on January 8, 2009 2:33 PM | Permalink | Reply to this

### Re: The Toric Variety Associated to the Weyl Chambers

Alex wrote approximately:

The simplest description of the toric variety associated to the Weyl chambers of a complex semisimple group $G$ is that it is the closure of a generic orbit of its maximal complex torus $T$ in flag variety $G/B$ of $G$ (here $B$ is a Borel subgroup).

Thanks! It’s interesting that you cite some papers by Allen Knutson and Eric Sharpe, because I met Allen at the AMS meeting here in DC a couple days ago, and he told me a bunch of stuff about toric varieties — including the fact you just mentioned.

This is indeed a very nice (I suspect maximally nice) answer to my original question.

Posted by: John Baez on January 8, 2009 3:27 PM | Permalink | Reply to this

### Re: toric varieties and F_1

Deitmar’s paper is a little gem. However, anyone with a categorical bias
will certainly also enjoy the general framework of Toen and Vaquie’s
seminal paper “Au-dessous de Spec Z”. While Deitmar works with spectra of
prime ideals just like in the books of Atiyah-MacDonald or Hartshorne,
Toen-Vaquie go for the functorial approach.

(In fact it appears that Grothendieck himself saw the functorial approach
as superior to the prime-ideal approach (apparently he said so in his 1970
talk in Buffalo), but that Dieudonne’ persuaded him to emphasise the
prime-ideal approach in order not to scare people away. (The introduction
to EGA1 still has a lot of representable-functor stuff but in the main text
this is less prominent.) Then came Hartshorne’s book which to this day is
considered a standard reference for scheme theory, and it hardly contains a
functor (I mean a representable functor). The emphasis on prime spectra
made the generalisation from schemes to stacks very cumbersome for
algebraic geometers, whereas the representable-functor viewpoint leads
directly to stacks and beyond.)

So (following Toen and Vaquie’, but glossing over many details), start with
a nice symmetric monoidal category C. Let Aff_C denote the opposite of the
category of commutative monoids in C, and define a notion of Zariski
topology by taking as open immersions the flat monos of finite
presentation. Now a scheme relative to C is a sheaf on Aff_C that admits a
Zariski open cover by representables. (This looks a lot like the
definition of stack, except that the notion of Zariski open has to be
introduced abstractly, and the sheaves are just set valued.) When C is the
category of abelian groups, the usual notion of scheme results (since a
commutative monoid in Ab is just a commutative ring, and since flat ring
epis of finite presentation generate the usual Zariski topology).

The interesting new cases lie below Spec Z, escaping the realm of
commutative rings. When C is the category of sets we get a version of
‘schemes over F_1’. Toen and Vaquie’ show that toric varieties are defined
as schemes relative to Set, i.e. as schemes over F_1, and that you get
usual toric varieties after base change to a commutative ring.

Florian Marty, a student of Toen, has carried out some comparison between
the Deitmar approach and the functorial approach.

(As you can imagine, Toen and Vaquie go on to a homotopical version of the
theory, relative to a C with a Quillen model structure. This leads to
different forms of homotopical algebraic geometry, including ‘brave new
schemes’.)

However, other predictions about F_1, like geometry of flag manifolds over
finite fields F_q reducing to combinatorics when q goes to 1, do not seem
to come out of the settings of schemes relative to Set: although GL(n) can
be defined over F_1, it does not yield the usual GL(n) after base change
(you get only semidirect-product of the symmetric group with n copies of
the multiplicative group). Toen and Vaquie’ argue convincingly that what
is mythically called F_1 is really more than one thing. Another model is
given by taking C to be the category of commutative monoids (N-modules).
In this case the GL(n) over N does base change to the GL(n) over Z, and
hence this model seems more appropriate for other dreams about F_1.

Posted by: Joachim Kock on January 10, 2009 10:26 PM | Permalink | Reply to this

### Re: toric varieties and F_1

So (following Toen and Vaquie’, but glossing over many details), start with a nice symmetric monoidal category $C$. Let $Aff_C$ denote the opposite of the category of commutative monoids in C, and define a notion of Zariski topology by taking as open immersions the flat monos of finite presentation. Now a scheme relative to $C$ is a sheaf on $Aff_C$ that admits a Zariski open cover by representables.

Wow! I’d been trying to develop an idea like this myself, motivated by the usual notion of scheme ($C = AbGp$) and Deitmar’s schemes over $\mathbb{F}_1$ ($C = Set$). But I was working heuristically — I hadn’t gotten as far as finding the right Zariski topology.

The interesting new cases lie below $Spec \mathbb{Z}$, escaping the realm of commutative rings.

Yeah! I always enjoy ‘subterranean’ mathematics, where you find interesting structure in situations that seemed too primitive to support it.

Toen and Vaquie’ show that toric varieties are defined as schemes relative to Set, i.e. as schemes over $\mathbb{F}_1$, and that you get usual toric varieties after base change to a commutative ring.

Okay, that seems to match Deitmar’s idea.

However, other predictions about $\mathbb{F}_1$, like geometry of flag manifolds over finite fields $\mathbb{F}_q$ reducing to combinatorics when $q$ goes to 1, do not seem to come out of the settings of schemes relative to Set.

Right. That was the secret inspiration for my question in this blog entry. I wanted flag varieties to be schemes over $\mathbb{F}_1$, but if schemes over $\mathbb{F}_1$ are toric varieties that means flag varieties should be toric varieties. They’re not. But there are some obvious ways to get toric varieties from the same data that give flag varieties! So, it’s worth pondering those… and in this post, I was asking about the special case of complete flag varieties.

This is why I loved what Alex and Allen told me: there’s a way to turn any (normal?) variety with a torus action into a toric variety — and if you do that to a complete flag variety, you get the toric variety I was asking about!

Toen and Vaquié argue convincingly that what is mythically called $\mathbb{F}_1$ is really more than one thing.

That seems to follow from the above.

Another model is given by taking $C$ to be the category of commutative monoids ($\mathbb{N}$-modules). In this case the $GL(n)$ over $\mathbb{N}$ does base change to the $GL(n)$ over $\mathbb{Z}$, and hence this model seems more appropriate for other dreams about $\mathbb{F}_1$.

Cool. But do dreams about flag varieties at $q = 1$ work well in this context? I think it’s very important to get them to work.
Posted by: John Baez on January 11, 2009 5:29 PM | Permalink | Reply to this

### Re: toric varieties and tropical geometry

Warning: this post probably contains some misunderstandings and
inaccuracies. I look forward to being corrected.

Toric varieties are in some sense the natural ambient space for tropical
varieties, or more precisely the ambient space of the complex varieties of
which tropical varieties are tropicalisations. At least this slogan seems
to be valid for curves, and probably more generally for hypersurfaces. One
definition of tropical curve is something like 1-dimensional polyhedral
complex embedded in R^2 with rational slopes and with a balancing condition
at every vertex. (I am sorry if this is not a complete definition; in
reality I assume you have already seen pictures of tropical curves.) The
embedding in R^2 determines a toric surface as follows: the slopes of the
‘tentacles’ reaching out to infinity define a polygon (with faces
orthogonal to the tentacles) , and this polytope defines a toric surface.
Moreover, the number of tentacles in each direction determines the scaling
of the polytope, which in turn is the degree data determining a line bundle
on the surface, and the tropical curve is the tropicalisation of a curve on
this toric surface given as the zero locus of a section of this line
bundle. If you draw the tropical curve randomly, changes are the surface
will be something complicated and uninteresting. To make the curve end up
in P^2, for example, it must have tentacles only in directions N, E, SW,
corresponding to the fact that the polytope defining P^2 is a right-angled
solid triangle. If the tropical curve has four tentacles in each direction
then the corresponding line bundle on P^2 is O(4), so the curve is a plane
quartic.

(This ‘independence of ambient space’ is one of the miracles of tropical
geometry: whereas in complex geometry people have learnt how to count
curves in P^2 and P^1 x P^1 or such simple examples, but quickly run their
head into a wall on any slightly more complicated surface, tropical
geometry (notably Mikhalkin) can count curves in any smooth toric surface,
because in any case the technique is to reduce the count to some
combinatorics, like lattice paths in a polygon, and the combinatorics can
be carried out independently of which toric surface the polytope happens to
define, in fact independently of toric geometry altogether, if you don’t
care why the objects you count are interesting.)

The above is an attempt at quickly explaining the relevance of toric
varieties to tropical geometry from the naive viewpoint of the drawings we
see on the internet. I think there are more direct explanations in terms
of Laurent polynomials and Newton polytopes, but I could not get them right
just now.

Posted by: Joachim Kock on January 10, 2009 10:42 PM | Permalink | Reply to this

### Re: toric varieties

Here is a viewpoint on toric varieties that it is worth perhaps to make
explicit, although it may be a bit exaggerated:

The purpose of toric varieities (like that of projective space) is to serve
as a convenient ambient space for hypersurfaces, complete intersections,
and subvarieties in general.

For example, as has been mentioned, most known Calabi-Yau varieties are
‘known’ because they can be realised as complete intersections in toric
varieties. One of the first models for mirror symmetry was Batyrev’s, and
it was precisely about CY hypersurfaces in toric varieties, the mirror
being a family of hypersurfaces defined from the dual polytope (some
conditions on the polytope are needed in order for the dual to define CY as
well). Shortly after, Borisov made some generalisations to complete
intersection CYs. The point is that a lot of numerical data of complete
interesections in toric varieties is accessible just from the combinatorics.

There is another, less wellknown feature that (suitably nice) toric
varieties have in common with projective spaces, namely that they have a
homogeneous coordinate ring. (It seems parts of this discovery were first
made in symplective geometry by Audin and others, but the first really
clean algebraic account of it is due to David Cox.) The homogenenous
coordinate ring of projective space is N-graded. This N comes from the
fact that the effective-divisor class semigroup is N, since divisors are
classified by their degree. The homogenenous coordinate ring of a more
general toric variety (technically I think it should be simplicial, meaning
that every cone in the fan is spanned by the minimal number of rays) has a
variable for each ray of the fan, and it is graded in the effective-divisor
class semigorup, which in turn has some simple combinatorial discription in
terms of the fan. Furthermore, just as P^n can be described as C^{n+1}/C*
(where n+1 is the number of rays and C* is the character group of the
divisor class group), a (simplicial) toric variety can be presented as a
geometric quotient C^D/G where D is the set of rays in the fan, and G is
the algebraic group of characters of the divisor class group. With the
homogenenous coordinate ring of a toric variety you can mimic Serre’s
characterisation of coherent sheaves on projective space: every coherent
sheaf comes from a finitely generated graded module over the homogenenous
coordinate ring.

Although this beautiful theory is nearly 20 years old, it seems to me its
potential has not nearly been exploited enough. (I have not really tried
myself. It could be, of course, that the commutative algebra gets too
tough when it actually comes to computation…)

Posted by: Joachim Kock on January 10, 2009 11:07 PM | Permalink | Reply to this

### Tanspennine Toric Meeting

Sixty Eighth Meeting of the Transpennine Topology Triangle
School of Mathematics
Alan Turing Building
University of Manchester
Monday Monday 19 January 2009
Supported by the London Mathematical Society and MIMS
Programme

The talks will take place in the Frank Adams Seminar Rooms 1 and 2, on the first floor of the Alan Turing Building. This is on the east side of the campus, off Upper Brook Street and about 100m south of the junction with Booth Street East. It is about 15 minutes walk from Piccadilly station, through the old UMIST campus and under the Mancunian Way. The building is number 46 on the University Campus map.

Participants will meet for coffee from 1100AM onwards in the Atrium Bridge Common Room; the Frank Adams Rooms open into the common room.

Lunch may be taken in any of several local venues (such as the cafe in the atrium, or the vegetarian “On the Eighth Day”, for example), and we expect to visit a nearby restaurant for early-evening dinner.

* 11.00-11.30: COFFEE (Atrium Bridge)

* 11.30-12.30: Tony Bahri (Rider University, NJ USA)
Piecewise Polynomials and the Equivariant Cohomology of Toric Varieties
Toric varieties have a natural torus action. For smooth varieties, the integral equivariant cohomology with respect to this action is the Stanley-Reisner ring of the underlying fan. A description of this ring in terms of piecewise polynomials on the fan allows a generalization to a class of singular varieties which include weighted projective spaces. Unlike ordinary cohomology, the integral equivariant cohomology distinguishes among weighted projective spaces. [A report on joint work with Matthias Franz and Nigel Ray]

* 12.30-2.30: LUNCH

* 2.30-3.30: Al Kasprzyk (Kent)
Simplices in toric geometry: Fake weighted projective space
When considering toric Fano varieties, it is natural to think about simplices. In this talk I’ll concentrate on one-point lattice simplicies and their associated varieties: fake weighted projective space. I hope to illustrate the difference between fake and genuine weighted projective space, and give several example calculations.

* 4.00-5.00: Jon Woolf (Liverpool)
What are the homotopy groups of a stratified space?
The usual definition of homotopy groups makes perfect sense when the space is stratified, but are there subtler invariants which capture some of the extra information in the stratification? This talk will introduce two proposals (due respectively to MacPherson and Baez) for modifying the definition of the homotopy groups of a stratified space, illustrated by some simple examples. For MacPherson’s proposal I will discuss the analogue(s) of the fact that the fundamental group classifies covering spaces and for Baez’s I will discuss a conjectural relation to bordism theory.

Everyone who wishes to participate is welcome, particularly postgraduate students. We shall operate the usual criterea for assistance with travel expenses, but beneficiaries will need to complete the standard forms, and should come armed with NI numbers and details of UK bank accounts. Please email nigel.ray(at)manchester.ac.uk if you expect to attend, so that we can cater for appropriate numbers.

Posted by: jim stasheff on January 12, 2009 2:59 PM | Permalink | Reply to this

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