basics on torus embeddings;...

I

Basics on torus embeddings; examples

1 Torus embeddings over the complex numbers

We wish to review here quickly some results of TE I† and to give a moreexplicit description of the complex varieties obtained via certain real spaces ofhalf the dimension.

Let T be an algebraic torus, i.e., T ∼= Gnm for some n, and let

M = Hom(T,Gm) ,

the character group of T , and

N = Hom(Gm,T ) ,

the group of ‘one-parameter’ subgroups of T (in the algebraic sense).Then M ∼= Zn and N ∼= Zn, and there is a natural non-degenerate pairing

〈·, ·〉 : M ×N −→ Z of determinant 1. All this is valid over any field k. Whenk = C, however, T can be described analytically as T/π , where T is a complexvector space and π is a discrete subgroup, generating T over C and isomorphicto Zn. Here T is the universal covering space of T and π is π1(T ) acting on Tvia translations. Note, however, that for all a ∈ π the map

φa : C −→ T

λ �−→ λ ·ainduces a map

φa : C/Z −→ T/π = T ,

and that C/Z ∼= Gm canonically via λ �−→ e2πiλ . Thus every a ∈ π inducesφa ∈ N, and this is easily checked to be an isomorphism between π and N.Thus π is just N up to a canonical identification. Since T = π ⊗C, it followsthat we have canonical maps:

† Recall this reference from p. x.

1

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-0-521-73955-9 - Smooth Compactifications of Locally Symmetric Varieties, Second EditionAvner Ash, David Mumford, Michael Rapoport and Yung-Sheng TaiExcerptMore information

http://www.cambridge.org/9780521739559

http://www.cambridge.org


2 I Basics on torus embeddings; examples

(i) N ∼= the usual topological π1 of T ;

(ii) N ⊗C ∼= the universal covering space of T ;

(iii) (N ⊗C)/N ∼= T .

We abbreviate N ⊗C by NC and N ⊗R by NR.Next, in the isomorphism NC/N ∼= T , consider the subgroup corresponding

to NR/N: this a compact real torus, and is the maximal compact subgroup ofT . We denote it by Tc (short for “compact torus”). Moreover, NR ⊂ NC has anatural complement, viz. iNR, and, by quotienting, iNR injects into NC/N. Inother words, we get a canonical decomposition

NC/N ∼= (NR/N)× (iNR) ,

and hence (dividing by i in the second factor)

T ∼= Tc ×NR .

We denote the projection T −→ NR by “ord,” which is then defined by

ord(x+ iy modN) = y , for all x,y ∈ NR .

If α ∈ M, and Xα : T −→ C∗ is the corresponding function (as in TE I, itis useful to think of M as an additive group, and hence to adopt exponentialnotation for the characters regarded as functions on T ), we obtain the formula

Xα(x+ iy modN) = e2πi(〈α,x〉+i〈α,y〉) , for all x,y ∈ NR ;

hence

|Xα(z)| = e−2π〈α,ord z〉 , for all z ∈ T .

Next, in TE I, Ch. I, §1, we define embeddings of T in normal affine varietiesXσ , with the action of T on itself extending to an action of T on Xσ , wheneverσ ⊂ NR is a closed rational polyhedral cone not containing a line. Recall that

Xσ = SpecC[. . . ,Xα , . . .]α∈M∩σ ;

here σ ⊂ MR is the dual cone to σ , so M ∩ σ is a sub-semigroup of M. Inorder to study convergence in the classical topology and other details on Xσ ,it will be convenient to introduce here the topological space (in the classical,not Zariski, topology) obtained by dividing Xσ by Tc. This will look like NR

with points at infinity added. Let us first construct these embeddings, whichwe call Nσ , of NR and then show there is a map ord : Xσ −→ Nσ inducing ahomeomorphism Xσ /Tc

∼−→ Nσ .






1 Torus embeddings over the complex numbers 3

The simplest way to define Nσ is via a basis α1, . . . ,αm of the semigroupσ ∩M. Then define

i : NR −→ Rm>0,

x �−→ (e−2π〈α1,x〉, . . . ,e−2π〈αm,x〉),

and let

Nσ = closure of iNR in Rm≥0 .

It is very easy to see that this space is independent of the choice of basis (checkthat if you add to the αi one more α , then Nσ does not change). If we let NR

act on Rm by

x · (y1, . . . ,ym) = (e−2π〈α1,x〉y1, . . . ,e−2π〈αm,x〉ym) ,

then Nσ is the closure of the orbit of (1,1, . . . ,1). In particular, NR acts onNσ , extending its action on itself by translation. Exactly as in the theory oftorus embeddings (see TE I, Ch. I, §1, Theorem 2), we can decompose Nσ intoNR-orbits; these will correspond bijectively to the faces of σ , and each onewill contain a unique point (y1, . . . ,ym) with yi = 0 or 1 for all i. Explicitly, forevery face τ of σ , the corresponding orbit is:

O(τ) =

{(y1, . . . ,ym) ∈ Nσ | yi = 0 if αi > 0 on Intτ

yi �= 0 if αi ≡ 0 on Intτ

}= NR-orbit of ετ = (ε1, . . . ,εm) ,

where

εi ={

0 if αi > 0 on Intτ1 if αi ≡ 0 on Intτ .

This can be proven following TE I, substituting the following lemma for theuse of k[[t]].

Lemma 1.1 If {xk} is a sequence in NR and S ⊂ {1, . . . ,m} satisfies

limk−→∞

αi(xk) = λi , i ∈ S,

limk−→∞

αi(xk) = ∞ , i /∈ S ,

then

(a) there is some y ∈ NR with αi(y) = 0 for i ∈ S; αi(y) > 0 for i /∈ S ;

(b) there is some z ∈ NR with αi(z) = λi for i ∈ S.

Proof Left to reader.







Now if we map Xσ into Rm≥0 as follows:

T � �

��

ord �� NR� �

��Xσ

f ��

� Nσ

Rm≥0

⊃

f (x) = (|Xα1(x)|, . . . , |Xαm(x)|) ,

we get a commutative diagram. Since T is dense in Xσ , it follows that f definesa map

ord : Xσ −→ Nσ

and that ord(gx) = ord(x) for all g∈ Tc. Conversely, if ord(x1) = ord(x2), it fol-lows that |Xα(x1)|= |Xα(x2)| for all α ∈ σ ∩M, from which it follows readilythat x1 = gx2 for some g ∈ Tc. Note that if Oτ ⊂ Xσ is the orbit correspondingto τ , then ord−1(O(τ)) = Oτ .

For some purposes, it is convenient to have a coordinate-invariant way ofdescribing Nσ as NR plus a set of ideal points at infinity. To describe Nσ thisway, for every face τ of σ , let

L(τ) = smallest linear space containing τ .

Then L(τ) is the stabilizer of ετ , so we get:

NR/L(τ) ∼−→ O(τ)

x �−→ x · ετ .

Let x + ∞ · τ ∈ Nσ denote x · ετ (where x1 + ∞ · τ = x2 + ∞ · τ if and only ifx1 − x2 ∈ L(τ)). The reason for this notation is as follows: decompose NR =N′

R ⊕L(τ), choose any sequence xn = yn + zn ∈ NR = N′R ⊕L(τ), and choose

any y ∈ N′R. Then one sees easily that

[lim

n−→∞xn = y+∞ · τ in Nσ

]⇐⇒

[lim

n−→∞yn = y and, for every

w ∈ L(τ), zn ∈ τ +w if n � 0 .

]

Heuristically, we have added a lower-dimensional vector space isomorphic toNR/L(τ) of ideal points x + ∞ · τ obtained by starting at x and moving out to






1 Torus embeddings over the complex numbers 5

infinity in the direction determined by the cone τ .

Our convergence condition may be rephrased by saying that a fundamentalsystem of neighborhoods of y+∞ · τ in NR is given by

U0ε ,w(y+∞ · τ) = y+w+Bε + τ ,

for any w ∈ L(τ) and any ε > 0, where Bε denotes the ε-ball around 0 (takeany metric on NR). More generally, with this notation, a fundamental systemof neighborhoods of y+∞ · τ in Nσ is given by

Uε ,w(y+∞ · τ) = U0ε ,w(y+∞ · τ)∪

⋃τ ′ face of τ

(y+w+Bε + τ +∞ · τ ′) .

For instance, if NR = R2 and σ is the positive quadrant, we get the following







picture:

Next recall that in TE I, Ch. I, §2, we glue the affine varieties Xσ together:whenever {σα} is a rational partial polyhedral decomposition of NR, meaning

(i) if σ is a face of σα , then σ = σβ , for some β ;

(ii) for all α , β , the cone σα ∩σβ is a face of σα and σβ ,

then we can glue the Xσα together, obtaining a scheme X{σα}. In TE I, weasked that {σα} be a finite set, so that X{σα} was a variety. This is in facttotally irrelevant: for any set {σα} as above, we get an X{σα} as before, exceptthat it may require an infinite number of affines to cover it. Now X{σα} isalways a separated normal irreducible scheme, locally of finite type over C andcontaining T as an open dense subset. In exactly the same way, we glue the Nσα

together into a topological space N{σα}, which is NR plus a large number ofideal vector spaces situated at infinity in many different directions. Moreover,we glue the ord maps together into one map:

ord : X{σα} −→ N{σα} .

For instance, X{σα}, as a set, is the disjoint union of T -orbits Oσα , one for eachα; likewise N{σα} as a set is the disjoint union of NR-orbits O(σα), one foreach α , and ord−1(O(σα)) = Oσα .






2 The functor of a torus embedding 7

2 The functor of a torus embedding

In order to make some of our later constructions of compactifications D/Γpurely algebraic and valid for schemes over any ground fields, it will be usefulto learn what functor a torus embedding represents. This also gives us anotherview of what torus embeddings are. First some notations and definitions.

(1) If S is a scheme and X is a set, XS denotes the constant sheaf on S with stalkX .

(2) Every semigroup or sheaf of semigroups will have an identity element e oridentity section e.

(3) If A1, A2 are semigroups, a homomorphism φ : A1 −→ A2 is called strict ifφ(e1) = e2 and φ(x) invertible implies x invertible. If A1, A2 are sheavesof semigroups on S, we require that, for every s ∈ S, the map on stalksφs : A1,s −→ A2,s is strict.

(4) If S is a scheme, then O(×)S will be the semigroup sheaf (OS, mult.).

The result is:

Theorem 2.1 Let T be a torus over k and T ⊂ X{σα} a torus embedding, whereσα ⊂ N(T )R are polyhedral cones. For any k-scheme S, let F{σα}(S) be theset of pairs (Σ,π) consisting of a sub-semigroup sheaf Σ ⊂ M(T )S and a strict

homomorphism π : Σ−→O(×)S such that, for all s∈ S, we have Σs = σα ∩M(T )

for some α . Then there are canonical isomorphisms, functorial in k-schemesS:

Hom k(S,X{σα}) ∼= F{σα}(S) .

Proof We first show how to associate a pair (Σ,π) to a morphism f : S −→X{σα}. Define:

Uα = f−1(Xσα ) ,

Σ = the union of the subsheaves (σα ∩M(T ))Uα of M(T )S .

Note that, for all s ∈ S, if f (s) ∈ Oα , then

s ∈Uβ ⇐⇒ f (s) ∈ Xσβ

⇐⇒ Oα ⊂ Xσβ

⇐⇒ σα is a face of σβ

⇐⇒ σβ ∩M(T ) ⊆ σα ∩M(T ) ;

hence the stalk of Σ at s is the union of the subsets σβ ∩M(T ) of M(T ) forall σβ with face σα , i.e., just σα ∩M(T ). Hence if r ∈ Σs, then r ∈ σα , so







Xr is defined on Xσα and f ∗(Xr) is defined at s. Therefore we can define

π : Σ −→ O(×)S by

π(r) = f ∗(Xr) .

Note that

π(r) invertible in OS,s ⇐⇒ π(r)(s) �= 0

⇐⇒ Xr( f (s)) �= 0

⇐⇒ Xr �≡ 0 on Oα

⇐⇒ r ≡ 0 on σα

⇐⇒−r ∈ σα ∩M(T )

⇐⇒ r invertible in Σs ,

hence π is a strict homomorphism.Next, let us start with (Σ,π) and define a morphism f . Define open sets Uα

by

Uα = {s ∈ S | σα ∩M(T ) ⊂ Σs} .

These form an open covering of S such that if σα is a face of σβ , then Uβ ⊂Uα .Next define

fα : Uα −→ Xσα = Speck[. . . ,Xr, . . .]r∈σα∩M(T )

via f ∗α(Xr) = π(r) for all r ∈ σα ∩M(T ): this is correct since such an r isin Γ(Uα ,Σ) and since π(r1 + r2) = π(r1) · π(r2). Now, for any α and β , letσγ = σα ∩σβ , which is a face of σα and σβ . Then

Uα ∩Uβ = {s ∈ S | σα ∩M(T ) ⊂ Σs and σβ ∩M(T ) ⊂ Σs} .

But if Σs = σδ ∩M(T ), then[Σs ⊃ σα ∩M(T ) and

Σs ⊃ σβ ∩M(T )

]⇐⇒ σδ ⊃ σα and σδ ⊃ σβ

⇐⇒ σδ ⊂ σα and σδ ⊂ σβ

⇐⇒ σδ ⊂ σγ

⇐⇒ Σs ⊂ σγ ∩M(T ) ,

so Uα ∩Uβ = Uγ . Finally, it is clear from the definition that fα = res fβwhenever Uα ⊂ Uβ . Therefore the fα patch together to form a morphismf : S −→ X{σα}.

It is now straightforward to check that these two procedures – associating a(Σ,π) to an f and associating an f to a (Σ,π) – are inverse to each other: weleave this to the reader.






3 Toroidal embeddings over the complex numbers 9

For instance, we find:

X{σα}(k) ∼= {(α,π) | π : σα ∩M(T ) −→ k(×) strict homomorphism} .

If k = C, one can easily prove also that

N{σα} ∼= {(α,ρ) | ρ : σα ∩M(T ) −→ R(×)≥0 strict homomorphism}

∼= {(α,σ) | σ : σα ∩M(T ) −→ R∪{∞} strict homomorphism} ,

where R∪{∞} is a semigroup via +. Here

ord : X{σα} −→ N{σα}

is given by

ρ(x) = |π(x)| ,σ(x) = − logρ(x) .

3 Toroidal embeddings over the complex numbers

We wish to review here quickly some results of TE I, Ch. II, indicating waysto interpret them over C, and generalizing them slightly. A pair

U ⊂ X ,

where U is a Zariski-open subset of a normal variety X , was called a toroidalembedding if, for all x ∈ X , we have that (X ,U) is formally isomorphic at x to(Xσ ,T ) at some t ∈ Xσ (for some torus embedding T ⊂ Xσ ). Equivalently, thismeans that there is an etale correspondence between X and Xσ , relating x andt, with U and T corresponding open sets. Over C, a pair

U ⊂ X ,

where X is an analytic space and U is open in the complex topology, will becalled a toroidal embedding if, for all x ∈ X , there exists a small neighborhoodWx ⊂X of x such that (Wx,Wx∩U) is isomorphic to (Vt ,Vt ∩T ) for some neigh-borhood Vt ⊂ Xσ of some t ∈ Xσ (for some torus embedding T ⊂ Xσ ). WhenX , U are varieties, this coincides with the previous definition. Now, this im-plies immediately that Wx has a canonical stratification {Yα,x} into non-singularlocally closed analytic strata with Y α,x normal: let Ei be the irreducible com-ponents of Wx \Wx ∩U , and let the Yα,x be the sets⋂

i∈I

Ei \⋃i/∈I

Ei .

We shrink Wx if necessary, so that these Yα,x are connected. As x varies, thesestrata patch up on overlaps, so we can uniquely stratify the whole of X into







{Yα}, where the Yα are connected, locally closed, non-singular analytic strata,and where Yα ∩Wx is a union of the Yβ ,x. However, it may happen that

Yα ∩Wx ⊃ more than one Yβ ,x .

This means that there is a path in X starting and ending in Wx and lying all inone stratum, but linking two distinct local strata:

Since this will mean that Y α has more than one branch through x, it is equiva-lent to Y α being non-normal. As in TE I, p. 57, we say that (X ,U) has or hasnot self-intersection according to whether Yα ∩Wx can be more than one localstratum, or Yα ∩Wx is always one local stratum. In TE I, we stuck with (X ,U)’swithout self-intersection. However, there is a class of toroidal embeddings withself-intersection that are almost as nice and that arise in the examples we willtreat. Suppose Yβ1,x and Yβ2,x are part of the same global stratum Yα . Locallyat x there is a unique stratum Yβ3,x such that

Y β3,x = Y β1,x ∩Y β2,x .

Let Yβ3,x define a global stratum Yγ . We say that (X ,U) is without monodromyif Yγ has a neighborhood W such that Yβ1,x and Yβ2,x lie in different componentsof Yα ∩W . To visualize this, note that, for every path in Yγ beginning andending at x, we can uniquely propagate the germ of analytic space Y β1,x alongthis path. If this germ can be taken to Y β2,x by such a path, then, for every






basics on torus embeddings;...

Documents