The n-Category Café

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

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.

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’.

Why not ask Procesi for a copy?

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

“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.

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).

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.

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.

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.

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.

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…)

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
MIMS TTT68 Link
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.