On Toric Varieties

Inauguraldissertationder Philosophisch-naturwissenschaftlichen Fakultat

der Universitat Bernzur Erlangung der Doktorwurde

vorgelegt von

Stephan Fischli

von Basel (BS) und Nafels (GL)

Leiter der Arbeit: Prof. Dr. P. ManiMathematisches Institut

Von der Philosophisch-naturwissenschaftlichen Fakultatauf Antrag des Herrn Prof. Dr. P. Mani angenommen

Bern, den 27. Februar 1992 Der Dekan:Prof. Dr. U. Wurgler

Selbstverlag, Bern, 1992

Preface

I first learned about toric varieties when I read Stanley’s proof of McMullen’s famousg-conjecture (see [10]). In this proof Stanley interprets the components of the h-vector of a rational simplicial convex polytope as the dimensions of the cohomologygroups of the associated toric variety. Then, by applying the hard Lefschetz theoremto the toric variety and using results from commutative algebra, he deduces theconjectured conditions for the h-vector. It is this connection between the topologyof toric varieties and the combinatorial geometry of convex polytopes (or fans ingeneral) which has fascinated me since then.

The idea of this thesis was to describe the cohomology (with integral coefficients)of a toric variety in terms of the underlying fan which already contains all of theinformation. To this aim we establish in section 2 a spectral sequence for toricvarieties whose E2-term is determined, up to multilinear algebra, by the cones of theunderlying fans. As an application we show how this spectral sequence can be usedto compute the cohomology of a 3-dimensional toric variety. Before, in section 1,we present a topological definition of toric varieties due to R. MacPherson andconstruct a finite CW-cell decomposition. Finally, in section 3, we give a topologicalclassification of the smooth 2-dimensional toric varieties. This section, which is jointwork with David Yavin, may also serve as a source of examples of toric varieties.

At this point I wish to thank all the people who have supported me during the timewhen I was writing this thesis. Especially I am indebted to Peter Mani, my advisor,for having aroused my interest in many mathematical problems (as well as in non-mathematical things like Charlie Parker). It is also due to him that I met some otherinteresting mathematicians like Prof. Ewald and Markus Eikelberg who introducedto us the algebraic geometry of toric varieties, and David Yavin who gave us aninsight into their topological structure. The resulting collaboration with David letme experience how lively mathematics can be. I also wish to thank Victor Batyrevand Urs Wurgler for the stimulating discussions I had with them, and the NationalScience Foundation for the financial support which made this thesis possible.

i

List of symbols

Bd unit ball in Rd

Sd−1 unit sphere in Rd

Σ complete fan in Rd

σ cone of the fan Σσ cell of Bd dual to the cone σσ⊥ subspace of Rd orthogonal to the cone σσ∨ subgroup of Zd orthogonal to the cone σT d d-dimensional (real) torusTσ subtorus of T d generated by the cone σ

Tσ quotient of T d modulo the subtorus Tσ

pτ,σ canonical projection of Tτ onto Tσ (for τ ⊂ σ)XΣ toric variety associated to the fan Σp projection of XΣ onto Bd

By the interior and the boundary of a subspace A ⊂ X, denoted by intA and ∂A,we always mean its relative interior, respectively boundary, with respect to its closureclA in X.

ii

Contents

1 Preliminaries 1

1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 CW-cell decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 A cohomology spectral sequence for toric varieties 5

2.1 Functors on fans and associated cohomology . . . . . . . . . . . . . . 5

2.2 The spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Classification of smooth 2-dimensional toric varieties 13

3.1 Cellular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 The intersection form . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Characterisation of the intersection form . . . . . . . . . . . . . . . . 15

3.4 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

A Spectral sequences 19

B Tensor products and exterior algebra 21

C Symmetric bilinear forms 23

iii

1 Preliminaries

1.1 Definitions

A set σ ⊂ Rd is called a rational polyhedral cone in Rd, if there exist finitely manyvectors v1, . . . , vn ∈ Zd such that σ = R≥0v1 + . . . + R≥0vn (where R≥0 denotes thenon-negative real numbers). The dimension of σ is defined to be the dimension of thesubspace spanσ, and σ is called simplicial if the generating vectors v1, . . . , vn canbe chosen linearly independent (and hence n = dim σ). If H ⊂ Rd is a hyperplanewhich contains the origin 0 ∈ Rd such that σ lies in one of the closed halfspaces ofRd bounded by H , then the intersection σ ∩H is again a rational polyhedral conewhich is called a face of σ. If {0} is a face of σ, we say that σ has a vertex at 0.

Definition 1.1 A fan Σ in Rd is a finite set of rational polyhedral cones in Rd

satisfying the following conditions:(i) Every cone σ ∈ Σ has a vertex at 0;(ii) If τ is a face of a cone σ ∈ Σ, then τ ∈ Σ;(iii) If σ, σ′ ∈ Σ, then σ ∩ σ′ is a face both of σ and σ′.Σ is called simplicial if it consists of simplicial cones, and Σ is called complete if⋃

σ∈Σ σ = Rd. Let Σ(i) denote the set of i-dimensional cones in Σ and ai theirnumber.

Example 1.1 Let P ⊂ Rd be a d-dimensional convex polytope with rational ver-tices and assume that 0 ∈ intP . Then the cones

⋃x∈F R≥0x spanned by the faces

F of P , together with the cone {0}, form a complete fan in Rd which is simplicialwhenever the polytope P is simplicial. Such a fan which is generated by a convexpolytope is called polytopal.

In the following we will always assume a fan to be complete. Thus the sets obtainedby intersecting each cone σ ∈ Σ, σ �= {0}, with the unit sphere Sd−1 ⊂ Rd forma spherical complex C. Let C′ be the barycentric subdivision of C, and for σ ∈ Σ,σ �= {0}, let σ be the union of all simplices in C′ whose vertices are barycentersof elements in C which contain σ ∩ Sd−1. Note that int σ is a homology cell ofdimension d − dim σ. For σ = {0} we set σ = Bd, the unit ball of Rd, and callΣ = {σ| σ ∈ Σ} the dual complex of Σ.

Let T d denote the d-dimensional torus Rd/Zd. Each k-dimensional cone σ ∈ Σ spansa k-dimensional subspace of Rd which, since it has a rational basis, maps under thecanonical projection Rd → T d to the k-dimensional subtorus

Tσ = (span σ + Zd)/Zd ∼= span σ/(span σ ∩ Zd).

1

Definition 1.2 The toric variety XΣ associated to the complete fan Σ in Rd isdefined to be the quotient space Bd×T d/∼ where ∼ is the equivalence relation givenby

(x, t) ∼ (x′, t′) ⇐⇒ x = x′ and t− t′ ∈ Tσ where x ∈ int σ.

We call d the dimension of XΣ and denote by p the projection of XΣ onto Bd.

Note that over each interior point of a face σ ⊂ Bd the torus T d is collapsed by therelation ∼ to the quotient

Tσ = T d/Tσ∼= Rd/(span σ + Zd).

Since span σ ∩ Zd is a direct summand of Zd, i.e. there is a subgroup G ⊂ Zd suchthat Zd = (span σ∩Zd)⊕G, it follows that Tσ

∼= (spanG)/G is a torus of dimensiond− dim σ = dim σ and T d ∼= Tσ ⊕ Tσ.

The proofs of the following properties can be found e.g. in [1].

Theorem 1.1 Let XΣ be the toric variety associated to the complete fan Σ in Rd.(i) XΣ is compact and simply connected.(ii) XΣ is a smooth manifold if and only if Σ is simplicial and every d-dimensionalcone in Σ can be generated by a basis of Zd (in which case the fan Σ is called regular).(iii) If f is a unimodular transformation of Rd, then the toric variety XΣ′ associatedto the fan Σ′ = {f(σ)| σ ∈ Σ} is homeomorphic to XΣ.

1.2 CW-cell decomposition

Next we construct a finite CW-cell decomposition of a toric variety XΣ. For eachpair of incident cones τ ⊂ σ in the underlying fan Σ let pτ,σ denote the canonical

projection Tτ → Tσ. Then XΣ is homeomorphic to the topological sum∑

σ∈Σ σ×Tσ

in which two points are identified whenever they correspond under a map idB2×pτ,σ.Therefore, if we define for each cone σ ∈ Σ a CW-cell decomposition of the torusTσ such that the projection maps pτ,σ are all cellular, the products of the cells int σ

with the cells in the decomposition of the corresponding torus Tσ form a CW-celldecomposition of the toric variety

XΣ =⋃

σ∈Σ

int σ×Tσ.

Lemma 1.1 Let XΣ be the toric variety associated to the complete fan Σ in Rd.There exists a finite CW-cell decomposition of XΣ such that p−1(σ) is a CW-sub-complex for all cones σ ∈ Σ.

2

Proof For each cone σ ∈ Σ let πσ : T d → σ⊥/πσ(Zd) be the map induced by theorthogonal projection πσ of Rd onto the subspace

σ⊥ = {x ∈ Rd| x · y = 0 ∀ y ∈ σ}.

Since ker πσ = Tσ there is an isomorphism πσ : Tσ → σ⊥/πσ(Zd). Thus a finiteCW-cell decomposition of the torus Tσ is equivalent to a CW-cell decomposition ofthe subspace σ⊥ which is periodic with respect to the group πσ(Zd) such that thereare only finitely many classes of periodic cells. Such a decomposition of σ⊥ can bedefined as follows.

For a non-zero vector v ∈ σ⊥ let H(v) = {x ∈ σ⊥| x · v ∈ Z} be the band of parallelhyperplanes generated by v. If V ⊂ σ⊥ is a finite set of non-zero vectors whichspan σ⊥, the closures of the components of σ⊥ \ ⋃

v∈V H(v) are convex polytopeswhich, together with their faces, form a polyhedral complex P(V ) with support σ⊥.If in addition the vectors v ∈ V are integral, the corresponding hyperplane bandsH(v) are πσ(Zd)-periodic and hence the polyhedral complex P(V ) too. Note that,if V ′ ⊂ σ⊥ is another finite set of non-zero vectors which contains V , the complexP(V ′) is a refinement of P(V ).

For each pair of incident cones τ ⊂ σ we now have the following commutativediagram

Tτπτ−−−→∼= τ⊥/πτ (Zd)

pτ,σ

�� πτ,σ

Tσπσ−−−→∼= σ⊥/πσ(Zd)

where πτ,σ is the map induced by the orthogonal projection πτ,σ : τ⊥ → σ⊥. There-

fore, in order to obtain finite CW-cell decompositions of the tori Tσ such that theprojection maps pτ,σ are cellular, we have to define for each cone σ ∈ Σ a finite setVσ ⊂ σ⊥ of non-zero integral vectors which span σ⊥, such that the projection mapsπτ,σ are cellular with respect to the corresponding polyhedral complexes P(Vσ).Since the composition of two cellular maps is cellular and πθ,σ = πτ,σ ◦ πθ,τ forθ ⊂ τ ⊂ σ, it is sufficient to consider pairs of incident cones τ ⊂ σ whose dimensionsdiffer by 1.

We proceed inductively and assume that we have already defined the sets Vτ for allcones τ ∈ Σ of dimension k. (For the cone {0} we may start with the set of co-ordinate vectors which generates the standard CW-cell decomposition of the torusT d.) Let σ be a (k + 1)-dimensional cone and τ ⊂ σ a k-dimensional face. Forevery pair of linearly independent vectors v1, v2 ∈ Vτ which not both lie in σ⊥, theorthogonal projection of the intersection H(v1) ∩ H(v2) onto σ⊥ yields a band ofparallel hyperplanes in σ⊥ generated by an integral vector v′ ∈ σ⊥. (In fact v′ is a

3

linear combination of the vectors v1 and v2 with coprime integral coefficients.) LetVτ,σ be the set of all vectors v′ obtained in this way. Then the projection map πτ,σ

is cellular with respect to the complexes P(Vτ ) and P(Vτ,σ). Thus, if we define Vσ

to be the union of the sets Vτ,σ where τ ranges over the faces of σ of codimension 1,P(Vσ) is the common refinement of the complexes P(Vτ,σ) and hence all the projec-tion maps πτ,σ are cellular, as required.

That p−1(σ) is a CW-subcomplex of XΣ follows from the fact that p−1(σ) is a closedsubspace which consists of cells of XΣ. 2

Remark Though the CW-cell decomposition of XΣ is finite, it seems hard toexplicitly compute the corresponding cellular homology of XΣ, except in the lowdimensional case (see section 3).

4

2 A cohomology spectral sequence for toric vari-

eties

Let p : E → B be a fibration over a pathwise and simply connected base space B.Then, by the Leray-Serre theorem, there exists a spectral sequence which convergesto the cohomology of the total space E and whose E2-term is isomorphic to thecohomology of B with coefficients in the cohomology of a fibre p−1(b), b ∈ B.In the case of a toric variety XΣ, the projection p : XΣ → Bd is not a fibrationsince the tori Tσ

∼= p−1(x), x ∈ int σ ⊂ Bd, do not all have the same cohomology.Nevertheless there exists a spectral sequence similar to that of a fibration. But inorder to describe its E2-term, we have to introduce the notion of a functor on the fanΣ and its associated cohomology which represents the different cohomology groupsof the tori Tσ and the transition maps between them. Since the cohomology of thetorus Tσ is isomorphic to the exterior algebra on the subgroup of Zd orthogonal tothe cone σ, we obtain an alternative description of the E2-term.

2.1 Functors on fans and associated cohomology

A fan Σ can be regarded as a category whose objects are the cones of Σ and whosemorphisms are the inclusions between them. Thus, a contravariant functor F froma fan Σ to a category C is a map which associates to each cone σ ∈ Σ an objectF (σ) ∈ C and to each inclusion τ ⊂ σ a morphism F (τ, σ) : F (σ) → F (τ), suchthat F (σ, σ) = idF (σ) and F (θ, σ) = F (θ, τ) ◦ F (τ, σ) for θ ⊂ τ ⊂ σ.

Example 2.1 Let H∗Σ be the functor which associates to each cone σ ∈ Σ the coho-

mology ring H∗(Tσ) of the torus Tσ, and to each inclusion τ ⊂ σ the homomorphismp∗τ,σ : H∗(Tσ)→ H∗(Tτ ) induced by the canonical projection map pτ,σ : Tτ → Tσ.

Example 2.2 Let∧∗

Σ be the functor which associates to each cone σ ∈ Σ theexterior algebra

∧∗(σ∨) on the subgroup σ∨ = σ⊥ ∩ Zd of Zd, and to each inclusionτ ⊂ σ the induced inclusion map i∧τ,σ :

∧∗(σ∨)→ ∧∗(τ∨).

For each cone σ ∈ Σ we define its orientation group Oσ to be the free abelian groupgenerated by the pairs (σ, o), (σ, o) modulo the relation (σ, o) + (σ, o) = 0 where oand o are the two orientations of σ. (Note that Oσ

∼= Z.) For a face τ ⊂ σ of codi-mension 1 let ωτ,σ : Oσ → Oτ denote the homomorphism given by [(σ, o)] �→ [(τ, o′)]where o′ is the orientation of τ induced by o.

5

Definition 2.1 Let Σ be a complete fan in Rd and F a contravariant functor fromΣ to the category of abelian groups. For 0 ≤ s ≤ d we set

Cs(Σ, F ) =⊕

σ∈Σ(d−s)

Oσ ⊗ F (σ).

Then a coboundary map δs : Cs(Σ, F )→ Cs+1(Σ, F ) is defined by

δs|Oσ⊗F (σ) =∑

τ∈Σ(d−s−1)

τ⊂σ

ωτ,σ ⊗ F (τ, σ).

We call C∗(Σ, F ) = (Cs(Σ, F ), δs)s∈Z the cochain complex of the fan Σ associatedto the functor F . Let H∗(Σ, F ) denote the corresponding homology.

Example 2.3 Let GΣ be the constant functor which associates to each cone σ ∈ Σthe fixed abelian group G and to each inclusion τ ⊂ σ the identity map on G. Thenthe associated cochain complex C∗(Σ, GΣ) is isomorphic to the augmented cellularchain complex of the spherical complex C = {σ ∩ Sd−1| σ ∈ Σ, σ �= {0}} withcoefficients in G. Therefore H0(Σ, GΣ) ∼= G and Hs(Σ, GΣ) = 0 for s �= 0.

The cochain complex C∗(Σ, F ) is natural with respect to the functor F in the fol-lowing sense.

Lemma 2.1 Let F1, F2 be two contravariant functors from the fan Σ to the categoryof abelian groups. If there exists a natural transformation Φ from F1 to F2, thenΦ induces a chain map f : C∗(Σ, F1) → C∗(Σ, F2). Furthermore, if Φ is a naturalequivalence, then f is an isomorphism and hence H∗(Σ, F1) ∼= H∗(Σ, F2).

Proof Let Φ be a natural transformation from F1 to F2, i.e. a map which associatesto each cone σ ∈ Σ a homomorphism Φσ : F1(σ)→ F2(σ) such that Φτ ◦ F1(τ, σ) =F2(τ, σ) ◦ Φσ for every inclusion τ ⊂ σ. For 0 ≤ s ≤ d we define a homomorphismf s : Cs(Σ, F1)→ Cs(Σ, F2) by

f s|Oσ⊗F1(σ) =∑

σ∈Σ(d−s)

idOσ ⊗ Φσ.

It follows that f s+1 ◦ δs1 = δs

2 ◦ f s for all s, hence f = (f s)s∈Z is a chain map. If Φ isa natural equivalence, i.e. the homomorphisms Φσ are isomorphisms, the maps f s

are isomorphisms, too. 2

Let σ ∈ Σ be a k-dimensional cone. Then Tσ is a torus of dimension d− k and σ∨ isa free abelian group of rank d− k, hence the cohomology ring H∗(Tσ) is isomorphicto the exterior algebra

∧∗(σ∨) (see example B.2). Moreover, the corresponding iso-morphisms can be chosen simultaneously for all cones σ ∈ Σ such that the followinglemma holds.

6

Lemma 2.2 Let Σ be a complete fan in Rd. There is a natural equivalence betweenthe functors H∗

Σ and∧∗

Σ.

Proof For each cone σ ∈ Σ we have to find an isomorphism Φσ : H∗(Tσ)→ ∧∗(σ∨)such that for every inclusion τ ⊂ σ the following diagram is commutative:

H∗(Tσ)p∗τ,σ−−−→ H∗(Tτ )

Φσ

�∼= Φτ

�∼=

∧∗(σ∨)i∧τ,σ−−−→ ∧∗(τ∨)

We first define the isomorphism Φ = Φ{0} : H∗(T d) → ∧∗(Zd). For a vector v ∈ Zd

let cv ∈ H1(T d) be the homology class represented by the cycle which is obtained byprojecting the segment {tv| 0 ≤ t ≤ 1} ⊂ Rd onto the torus T d. The map v �→ cv

defines an isomorphism from Zd to H1(Td) and since H1(T d) ∼= Hom(H1(T

d),Z),there is a vector Φ(α) ∈ Zd for each cohomology class α ∈ H1(T d) such that

α(cv) = Φ(α) · v ∀ v ∈ Zd.

Let Φ : H∗(T d) → ∧∗(Zd) be the induced ring isomorphism. For a cone σ ∈ Σ,σ �= {0}, we now define Φσ : H∗(Tσ) → ∧∗(Zd) to be the composition of the ringhomomorphism p∗σ : H∗(Tσ) → H∗(T d) induced by the canonical projection pσ :T d → Tσ, followed by Φ. We show that Φσ is injective and that im Φσ =

∧∗(σ∨).The commutativity of the diagram will then follow from the fact that p∗σ = p∗τ ◦ p∗τ,σ

for τ ⊂ σ.

(i) Since the homology of the torus Tσ is free and of finite rank, it follows fromT d ∼= Tσ ⊕ Tσ that H∗(T d) ∼= H∗(Tσ) ⊗ H∗(Tσ) by the Kunneth theorem. Underthis isomorphism the homomorphism p∗σ : H∗(Tσ)→ H∗(T d) is given by α �→ 1⊗ αwhere 1 ∈ H∗(Tσ) is the unit element. Therefore p∗σ is injective and Φσ = Φ ◦ p∗σas well.

(ii) Since Φσ is a ring homomorphism and H∗(Tσ) is generated by H1(Tσ) just as∧∗(σ∨) is generated by∧1(σ∨) ∼= σ∨, it is sufficient to show that im Φσ|H1(T d) = σ∨.

From ker pσ = Tσ it follows that ker pσ∗|H1(T d) = {cv ∈ H1(Td)| v ∈ span σ ∩ Zd}.

Thus for α ∈ H1(T d) we have

Φ(p∗σα) · v = (p∗σα)(cv) = α(pσ∗(cv)) = 0 ∀ v ∈ span σ ∩ Zd

and hence Φσ(α) = Φ(p∗σα) ∈ σ∨. Reversely, if Φ(β) ∈ σ∨ for some β ∈ H1(T d),then β(cv) = 0 for all v ∈ span σ ∩ Zd and hence ker pσ∗|H1(T d) ⊂ ker β. Thus there

exists a homomorphism α : H1(Tσ)→ Z such that β = α ◦ pσ∗|H1(T d) = p∗σ(α). 2

7

Remark Since H∗(Tσ) and∧∗(σ∨) both are graded rings and the isomorphism Φσ

respects their gradations, there also exists a natural equivalence for each 0 ≤ t ≤ dbetween the functors H t

Σ and∧t

Σ which associate to each cone σ ∈ Σ the groupsH t(Tσ) and

∧t(σ∨), respectively.

2.2 The spectral sequence

Let XΣ be the toric variety associated to the complete fan Σ in Rd. By taking thepreimages Xs = p−1(Bs) where Bs is the union of the cells int σ ⊂ Bd of dimension≤ s, we obtain a natural filtration

∅ = X−1 ⊂ X0 ⊂ X1 ⊂ . . . ⊂ Xd = XΣ.

By lemma 1.1 the sets Xs are CW-subcomplexes of XΣ, hence there is an associatedspectral sequence (Er, dr)r≥1 with initial term Es,t

1∼= Hs+t(Xs, Xs−1) and differential

d1 : Es,t1 → Es+1,t

1 corresponding to the connecting homomorphism in the long exactcohomology sequence of the triple (Xs+1, Xs, Xs−1) (see example A.1). Since thefiltration of XΣ is finite the spectral sequence degenerates, i.e. eventually becomesconstant, and the limit term E∞ is isomorphic to the bigraded module associatedto the filtration F of H∗(XΣ) defined by

F sH∗(XΣ) = ker[H∗(XΣ)→ H∗(Xs−1)].

By computing the E1-term and its differential d1 we will prove the following

Theorem 2.1 Let XΣ be the toric variety associated to the complete fan Σ in Rd.There exists a spectral sequence (Er, dr)r≥1 with Es,t

2∼= Hs(Σ,

∧tΣ) which converges

to the cohomology of XΣ.

Proof In order to compute the E1-term, we may write the pair (Xs, Xs−1) as theunion of the pairs (p−1(σ), p−1(∂σ)) where σ ranges over the (d − s)-dimensionalcones of Σ. Since the pairs (p−1(σ), p−1(∂σ)) are CW-subcomplexes of (Xs, Xs−1)and the intersection of every two of them has trivial cohomology (as a pair consistingof equal components), it follows by induction from the Mayer-Vietoris sequence that

Hs+t(Xs, Xs−1) ∼=⊕

σ∈Σ(d−s)

Hs+t(p−1(σ), p−1(∂σ)).

Under this isomorphism the differential d1 : Hs+t(Xs, Xs−1) → Hs+t+1(Xs+1, Xs)induces for each (d−s)-dimensional cone σ ∈ Σ and each face τ ⊂ σ of codimension1 a map Hs+t(p−1(σ), p−1(∂σ)) → Hs+t+1(p−1(τ), p−1(∂τ )) which is equal to thecomposition (i∗)−1 ◦ δ∗ in the first row of the following diagram:

8

Hs+

t(p −1(σ

),p −1(∂

σ))

i ∗←−−−−−−−−−−−−−−−−−−−−−−−−

∼=H

s+t(p −

1(∂τ),p −

1(∂τ\

intσ))

δ ∗−−−→

Hs+

t+1(p −

1(τ),p −

1(∂τ))

f ∗σ

�

∼=(1

)

�

f ∗τ |p −1(∂

τ)

(2)

∼=

�

f ∗τ

Hs+

t(σ×T

σ ,∂σ×

Tσ )

(id×p

τ,σ

) ∗−−−−−→

Hs+

t(σ×T

τ ,∂σ×

Tτ )

i ∗←−−−∼=

Hs+

t(∂τ×

Tτ ,(∂

τ\in

tσ)×

Tτ )

δ ∗−−−→

Hs+

t+1(τ×

Tτ ,∂

τ×T

τ )

λσ

�

∼=(3

)λ ′ �

∼=(4

)∼=

�

λ ′′(5

)∼=

�

λτ

Hs(σ

,∂σ)⊗

Ht(T

σ )id⊗

p ∗τ,σ

−−−−−→H

s(σ,∂

σ)⊗

Ht(T

τ )i ∗⊗

id←−−−∼=

Hs(∂

τ,∂

τ\in

tσ)⊗

Ht(T

τ )δ ∗⊗

id−−−→

Hs+

1(τ,∂

τ)⊗

Ht(T

τ )

µσ ⊗

id

�

∼=(6

)µ

σ ⊗id

�

∼=(7

)∼=

�

µτ ⊗

id

Oσ ⊗

Ht(T

σ )id⊗

p ∗τ,σ

−−−−−→O

σ ⊗H

t(Tτ )

ωτ,σ ⊗

id−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

Oτ ⊗

Ht(T

τ )

In the diagram on the previous page δ∗ always denotes the connecting homomor-phism in the long exact cohomology sequence of the corresponding triple of spacesand i∗ the homomorphism induced by the inclusion of the corresponding pair ofspaces. Since these inclusions are all of the form (A,A ∩ B) ⊂ (A ∪ B,B) where Aand B are CW-subcomplexes of A∪B, they induce isomorphisms in cohomology byexcision. We now describe the vertical isomorphisms and show the commutativityof the diagram, level by level.

(i) For each cone σ ∈ Σ we define a map fσ : (σ×Tσ, ∂σ×Tσ) → (p−1(σ), p−1(∂σ))by fσ(x, t) = (x, pσ,σ′(t)) where σ′ ∈ Σ is the unique cone for which x ∈ int σ′. Since

fσ is the identity map on int σ×Tσ, it is a relative homeomorphism between finiteCW-complexes and hence induces an isomorphism in cohomology. The commuta-tivity of square (1) follows from the commutativity of the following diagram

p−1(σ) ↪−−−−−−−−−−−→ p−1(∂τ )

fσ

fτ |∂τ×Tτ

σ×Tσid×pτ,σ←−−−− σ×Tτ ↪→ ∂τ×Tτ

and square (2) is commutative because fτ induces a chain map between the corre-sponding chain complexes.

(ii) Since the homology of the torus Tσ is free and of finite rank, there exists by theKunneth theorem an isomorphism λσ : H∗(σ, ∂σ) ⊗ H∗(Tσ) → H∗(σ×Tσ, ∂σ×Tσ)given by α⊗β �→ α×β. The isomorphisms λ′ and λ′′ are analogously defined. Sincethe cohomology cross product is natural (see [9], 5.6.2), the squares (3) and (4) arecommutative and the commutativity of square (5) follows from [9], 5.6.6.

(iii) If we fix an orientation of the sphere Sd−1 ⊂ Rd, then every orientation o of thecone σ ∈ Σ induces a unique orientation of the cell σ ⊂ Sd−1, which is representedby a generating element α ∈ Hs(σ, ∂σ) where s = dim σ. The map o �→ α inducesan isomorphism µσ : Oσ → Hs(σ, ∂σ) such that square (7) is commutative. Thecommutativity of square (6) is trivial.

By taking the composition of the isomorphisms f ∗σ , (λσ)−1 and (µσ ⊗ id)−1 we see

that

Es,t1∼= Hs+t(Xs, Xs−1) ∼=

⊕σ∈Σ(d−s)

Oσ ⊗H t(Tσ) = Cs(Σ, H tΣ)

and that the differential d1 : Es,t1 → Es+1,t

1 corresponds under this isomorphismto the coboundary map δs of the cochain complex C∗(Σ, H t

Σ). Therefore Es,t2∼=

Hs(Σ, H tΣ) and since there is a natural equivalence between the functors H t

Σ and∧t

Σ

(see lemma 2.2 and the remark after), the proof is completed. 2

10

Corollary 2.1 Es,t1 is a free abelian group of rank ad−s

(st

)for 0 ≤ s ≤ d, 0 ≤ t ≤ s,

and Es,t1∼= 0 otherwise. In particular (Er, dr) is a first quadrant spectral sequence.

2.3 Applications

We show how the spectral sequence of theorem 2.1 can be used to compute theEuler characteristic of a toric variety XΣ and the low and high dimensional coho-mology groups. Recall that the limit term of the spectral sequence is related to thecohomology of XΣ by

Es,t∞ ∼= F sHs+t(XΣ)/F s+1Hs+t(XΣ)∼= ker[Hs+t(XΣ)→ Hs+t(Xs−1)]/ ker[Hs+t(XΣ)→ Hs+t(Xs)]

Theorem 2.2 The Euler characteristic χ(XΣ) of a d-dimensional toric variety XΣ

is equal to the number ad of d-dimensional cones in the fan Σ.

Proof For every term Er of the spectral sequence we define its Euler characteristicby χ(Er) =

∑s,t(−1)s+t rkEs,t

r . By corollary 2.1 we have

χ(E1) =∑d

s=0

∑st=0(−1)s+tad−s

(st

)= ad.

From [9], 4.3.14, it follows that χ(Er+1) = χ(H(Er)) = χ(Er) and hence we obtainby induction χ(Er) = ad for all r. Since the spectral sequence degenerates, i.e.E∞ ∼= Er for some r, we also have χ(E∞) = ad. On the other hand, it follows from

∑s+t=q rkEs,t

∞ =∑d

s=0 rk(F sHq(XΣ)/F s+1Hq(XΣ)) = rkHq(XΣ)

that χ(E∞) = χ(XΣ). 2

Theorem 2.3 Let XΣ be a d-dimensional toric variety. Then the low dimensionalcohomology groups of XΣ are given by

Hq(XΣ) ∼=

Z q = 00 q = 1ker[δ1 : C1(Σ,

∧1Σ)→ C2(Σ,

∧1Σ)] q = 2

H2(Σ,∧1

Σ) q = 3

and the high dimensional groups by

Hq(XΣ) ∼=

Za1−d ⊕ ∧d−2(Zd)/∑

τ∈Σ(1)

∧d−2(τ∨) q = 2d− 2∧d−1(Zd)/∑

τ∈Σ(1)

∧d−1(τ∨) q = 2d− 1Z q = 2d

11

Proof Since∧0(σ∨) ∼= Z for all cones σ ∈ Σ, it follows from example 2.3 that

Es,02∼= Hs(Σ,

∧0Σ) ∼=

{Z s = 00 s �= 0

By corollary 2.1 we know in addition that Es,t2∼= 0 for t < 0, s > d and t > s.

This implies that for s + t ≤ 3 and s + t ≥ 2d − 2 the groups Es,tr , r ≥ 2, are

constant, for the differentials from and to these groups are all trivial, and henceEs,t

∞ ∼= Es,t2 . On the other hand, the groups Es,t

∞ with s + t = q determine the qthcohomology group of the toric variety XΣ up to group extensions. In particular, ifthere is only one non-trivial group Es,t

∞ with s + t = q, this group must coincidewith Hq(XΣ). This is the case for q = 0, 2, 3, 2d − 1, 2d where the respective non-trivial groups are E0,0

2 , E1,12 , E2,1

2 , Ed,d−12 and Ed,d

2 . For q = 1 all the groups Es,t2

with s + t = q are trivial and hence H1(XΣ) is trivial, too. For q = 2d − 2 thereare two non-trivial groups Es,t

2 with s + t = q and we first obtain the isomorphismEd−1,d−1

2∼= H2d−2(XΣ)/Ed,d−2

2 . But since Ed−1,d−12

∼= ker[d1 : Ed−1,d−11 → Ed,d−1

1 ] isfree, H2d−2(XΣ) splits into the direct sum Ed−1,d−1

2 ⊕ Ed,d−22 . Finally the proof can

be completed by interpreting the corresponding groups Es,t2∼= Hs(Σ,

∧tΣ). 2

Remarks (i) The formula for H2(XΣ) is a generalization of a result by M. Eikel-berg (see [2], 7.3 Folgerung, 7.7 Satz), since it also holds for toric varieties whoseunderlying fans are not polytopal.(ii) The high dimensional cohomology groups do not depend on the combinatorialstructure of the fan Σ but on its generating vectors.(iii) All the groups which occur in the theorem can easily be calculated withinmultilinear algebra. Thus the theorem enables us in particular to determine thecohomology of a 3-dimensional toric variety.

Example 2.4 Let Σ be the fan generated by the cube [−1, 1]3 ⊂ R3 (see example1.1) and Σ′ the fan obtained from Σ by replacing the generating vector v = (1, 1, 1)with the vector v′ = (2, 1, 1). Then the associated toric variety XΣ′ has the followingcohomology groups:

Hq(XΣ′) ∼=

0 q = 2Z q = 3Z5 q = 4Z/2Z q = 5

This example shows that the second cohomology group of a toric variety can vanish,if the underlying fan is not polytopal.

12

3 Classification of smooth 2-dimensional toric va-

rieties

In this section we use the CW-cell decomposition of section 1 to explicitly calculatethe intersection form of a smooth 2-dimensional toric variety XΣ. Applying Freed-man’s characterisation of 4-manifolds by their intersection form, we conclude thatXΣ is either homeomorphic to the complex projective plane CP 2, the product ofspheres S2×S2 or the connected sum of CP 2 with a finite number of −CP 2. Thisresult could also be deduced from a theorem of Oda ([8], theorem 8.2) which usestechniques from algebraic geometry and combinatorial properties of regular fansin R2.

3.1 Cellular homology

Let XΣ be a 2-dimensional (not necessarily smooth) toric variety associated to thecomplete fan Σ in R2. We assume that the 1-dimensional cones τ1, . . . , τn of Σ arenumbered counterclockwise and that the cone τi is generated by the primitive vectorvi = (vi1, vi2) ∈ Z2 (i.e. vi1 and vi2 are coprime integers). Furthermore let σi be the2-dimensional cone which is generated by the vectors vi and vi+1. (The indexingfrom 1 to n is always meant to be cyclic.) For convenience we may represent thedual complex Σ as the face complex of a polygon P , which we hence designate asthe dual of the cone {0}.

Σ HHHH

H

��

��

AAAA

τi

τi+1

σi

Σ

���@@@��

�

τi

τi+1 σi

P

Figure 1: A 2-dimensional fan Σ and its dual complex Σ

We first describe for each cone 3 ∈ Σ the CW-cell decomposition of the torus T�

which we have used in the proof of lemma 1.1. Recall that T�∼= 3⊥/π�(Z2) where

3⊥ is the subspace of R2 orthogonal to the cone 3 and π� the orthogonal projectionof R2 onto 3⊥. Let I denote the interval {t ∈ R| 0 < t < 1}. Then the standardCW-cell decomposition of the torus T{0} ∼= T 2 consists of the cells c0, c1

1, c1

2and

c2 which are represented by the subsets {(0, 0)}, I×{0}, {0}×I and I×I of R2,respectively. For 1 ≤ i ≤ n the 1-dimensional torus Tτi

is decomposed into a 0-cell

13

which we identify with c0 and the 1-cell c1i which is represented by the segment

Iwi ⊂ τ⊥i , where wi ∈ R2 is the vector orthogonal to vi such that det(vi, wi) = 1.

Finally, the 0-dimensional tori Tσi, 1 ≤ i ≤ n, consist of a single 0-cell which we

identify again with c0.

The product cells int 3×c now form a CW-cell decomposition of the toric varietyXΣ and if we provide them with appropriate orientations, the boundaries of thecorresponding cellular chains (for which we use the same notation) are given by

∂0 (int σi×c0) = 0

∂1 (int τi×c0) = int σi×c0 − int σi−1×c0

∂2 (int τi×c1i) = 0

∂2 (intP×c0) =∑

(int τi×c0)∂3 (intP×c1

1) =∑

(−vi2)(int τi×c1i)

∂3 (intP×c12) =

∑vi1(int τi×c1

i)∂4 (intP×c2) = 0

(1)

Note that the multiplicities of the chain int τi×c1i in the boundaries of the two

3-chains come from the images πτi(1, 0) = −vi2wi and πτi

(0, 1) = vi1wi. By evaluat-ing the above boundary maps we finally obtain the following

Lemma 3.1 Let XΣ be a 2-dimensional toric variety associated to the completefan Σ in R2 whose 1-dimensional cones are generated by the primitive vectorsv1, . . . , vn ∈ Z2. Then the homology of XΣ is given by

Hq(XΣ) ∼=

Z q = 0, 4Zn/(Zv∗1 + Zv∗2) ∼= Zn−2 ⊕ Z/mZ q = 20 otherwise

where v∗1 = (v11, . . . , vn1), v∗2 = (v12, . . . , vn2), and m is the greatest common divisor

of the determinants det(vi, vj), 0 ≤ i < j ≤ n. In particular H2(XΣ) ∼= Zn−2 if XΣ

is smooth.

3.2 The intersection form

In the following we assume that the 2-dimensional toric variety XΣ is smooth. Thusby property (ii) of theorem 1.1 we have

det(vi, vi+1) = 1 (1 ≤ i ≤ n) (2)

and in view of property (iii) we may assume in addition that vn−1 = (1, 0) andvn = (0, 1). By (1) we see that the group of 2-cycles of XΣ is generated by the

14

chains int τi × c1i , 1 ≤ i ≤ n, and the corresponding homology classes zi ∈ H2(XΣ)

satisfy the following relations:

v11z1 + . . . + vn−2,1zn−2 + zn−1 = 0v12z1 + . . . + vn−2,2zn−2 + zn = 0.

(3)

In order to determine the intersection numbers zi · zj , we first observe that thespheres p−1(τi) and p−1(τj) which represent the classes zi and zj, respectively, donot intersect in XΣ if their generating cones τi and τj are not adjacent. Thereforewe have

zi ·zj = 0 (1 < |i− j| < n− 1).

Second, the spheres p−1(τi) and p−1(τi+1) intersect in the unique point which liesover the vertex σi of P . Since det(vi, vi+1) = 1, it can be seen that this intersectionis transversal and hence

zi ·zi+1 = ±1 (1 ≤ i ≤ n)

where the signs are all equal and only depend on the orientation of XΣ. In thefollowing we fix them to be +1.

Third, by multiplying both relations (3) with zi and taking a suitable linear combi-nation of the resulting equations, we obtain the self intersection numbers

zi ·zi = − det(vi−1, vi+1) (1 ≤ i ≤ n)

where we have also used the smoothness condition (2).

3.3 Characterisation of the intersection form

Having calculated the intersection form of XΣ, we now characterise it up to isomor-phism, i.e. up to a change of basis of H2(XΣ). By (3) the classes z1, . . . , zn−2 forma basis of H2(XΣ), hence the rank of the intersection form equals n− 2.

In order to determine the signature, we first have to calculate the principal mi-nors Dk = det((zi ·zj)1≤i,j≤k) of the intersection matrix. (Henceforth we will notdistinguish between the intersection form and its matrix.) From the results of theprevious subsection it follows that D1 = det(v2, vn) and

Dk = − det(vk−1, vk+1)Dk−1 −Dk−2 (2 ≤ k ≤ n− 2)

where we have set D0 = 1. By induction one can easily prove, e.g. by using theGrassmann-Plucker relation in R2, that

Dk = (−1)k+1 det(vk+1, vn) (1 ≤ k ≤ n− 2).

15

From this equation we see that if none of the vectors vk is equal to −vn = (0,−1),then all the principal minors are non-zero and have alternating signs, except for theunique pair (Dk−1, Dk) for which the vectors vk and vk+1 lie on opposite sides of they-axis. Hence by Jacobi’s theorem the signature of the intersection form equals 4−n.By a rule of Gundenfinger this still holds even if there exists a vector vk = (0,−1)in which case Dk−1 = 0 (see [4], note 1 on page 304).

If n = 3 there is only one possible vector v1 = (−1,−1) and the intersection formof XΣ given by the matrix (1) is positive definite. If n > 3 the absolute value ofthe signature of the intersection form of XΣ is less than its rank, hence the form isindefinite. Thus in order to characterise it, we finally have to determine its type.

The even type is only possible if n = 4. Indeed, if the intersection form of XΣ

is even, then all the determinants det(vi−1, vi+1), 1 ≤ i ≤ n − 2, must be even.Since the vectors vi are primitive, it follows that they all have one even and one oddcoordinate, thus they are contained in the lattice Γ = (1, 0) + Z(1, 1) + Z(−1, 1).Hence by Pick’s formula (see e.g. [5]) the area A(S) of the star-shaped polygonS =

⋃ni=1 conv{0, vi, vi+1} is given by

A(S)

det Γ= card(Γ ∩ intS) +

1

2card(Γ ∩ ∂S)− 1

where det Γ denotes the determinant of a basis of Γ. But by condition (2) thearea A(S) equals n

2and S does not contain any points of Γ other than its vertices.

Therefore the equality can hold only if n = 4.

In fact, if n = 4 and v1 = (−1, 0), v2 = (0,−1) the resulting form is even. On theother hand, every odd indefinite intersection form of rank n ≥ 4 can also be realised,e.g. by setting vi = (i− 2,−1), 1 ≤ i ≤ n− 2.

���

���

���

@@@ . . .

PPPPPPPPP

Figure 2: Representative fans with 3, 4 and n generators

Thus we have completely characterised the possible intersection forms of XΣ and wesummarize the results in the following

Theorem 3.1 A non-singular integral symmetric bilinear form B can be realised asthe intersection form of an oriented smooth 2-dimensional toric variety XΣ if andonly if either(i) rank(B) = 1 and B = (±1), or(ii) rank(B) = 2 and B is even, or(iii) rank(B) ≥ 2, |signature(B)| = rank(B)− 2 and B is of odd type.

16

3.4 Classification

In 1982 Freedman characterised topological 4-manifolds by showing that every non-singular integral symmetric bilinear form can be realised as the intersection form ofan oriented closed simply-connected 4-manifold, and that any two such manifoldsrealising the same form are homeomorphic if the form is even, whereas if the formis odd there are two homeomorphism classes, one with trivial and the other withnon-trivial Kirby-Siebenmann obstruction (see [3], theorem 1.5).

In our case it is easy to give representatives of 4-manifolds which realise the in-tersection forms described in the theorem of the previous subsection. Namely, letus consider the oriented complex projective plane CP 2 which has intersection form(+1). Then CP 2 with the opposite orientation, which we denote by −CP 2, hasintersection form (−1). Furthermore, if we take the connected sum of CP 2 with afinite number of copies of −CP 2, we obtain a 4-manifold whose intersection form isthe orthogonal sum of (+1) with a finite number of (−1) and hence satisfies condi-tion (iii) of the theorem. Finally, the even indefinite form of rank 2 is the intersectionform of the product of spheres S2×S2. All these manifolds are smooth and hencehave trivial Kirby-Siebenmann obstruction, and since the same is true for the toricvarieties in question, we can state the following

Corollary 3.1 The homeomorphism classes of a smooth 2-dimensional toric varietyare represented by the complex projective plane CP 2, the product of spheres S2×S2

and the connected sum of CP 2 with a finite number of copies of −CP 2.

17

18

A Spectral sequences

A spectral sequence is a sequence of chain complexes each of which is the homologymodule of the preceding one. Associated to a spectral sequence there is a limit mod-ule, and the terms of the spectral sequence are regarded as approximations to thislimit module. A spectral sequence can be used e.g. to approximate the cohomologyof a topological space X by the relative cohomology modules of subspaces of X.Since this is the case in which we are interested, we describe cohomology spectralsequences rather than ordinary spectral sequences, though they are the same apartfrom the notation. Our exposition follows [6] and [9]. We first present the algebraicconcept of spectral sequences.

In the following we consider modules over a fixed principal ideal domain. Let E bea bigraded module, i.e. a family of modules (Es,t)s,t∈Z. A differential d : E → Eof bidegree (r, 1− r) is a family of homomorphisms d : Es,t → Es+r,t+1−r such thatd2 = 0. The homology module H(E, d) is the bigraded module defined by

Hs,t(E, d) = ker[d : Es,t → Es+r,t+1−r]/dEs−r,t−1+r.

Definition A.1 A (cohomology) spectral sequence (Er, dr)r≥1 is a sequence of bi-graded modules Er and differentials dr of bidegree (r, 1− r) such that

H(Er, dr) ∼= Er+1 ∀r.

The module E1 is called the initial term of the spectral sequence. In order to definethe limit term, we identify the module Er+1 with H(Er, dr) by the isomorphismabove. Then E2 = H(E1, d1) is a subquotient C1/B1 of E1, where C1 = ker d1

and B1 = im d1. In turn E3 = H(E2, d2) is a subquotient of C1/B1 and hence it isisomorphic to C2/B2, where B2 ⊂ C2 are submodules of C1 such that C2/B1 = ker d2

and B2/B1 = im d2. By induction we obtain a tower of submodules

B1 ⊂ B2 ⊂ B3 ⊂ . . . ⊂ C3 ⊂ C2 ⊂ C1

of E1 such that Er+1∼= Cr/Br. We set B∞ =

⋃r Br and C∞ =

⋂r Cr. Then

B∞ ⊂ C∞ and the quotient module E∞ = C∞/B∞ is called the limit term of thespectral sequence.

The spectral sequence (Er, dr) is said to be convergent, if for every pair (s, t) thereexists an index r(s, t) such that the differentials dr : Es,t

r → Es+r,t+1−rr are trivial

for r ≥ r(s, t). In this case Es,tr+1 is isomorphic to a quotient of Es,t

r , and Es,t∞ is the

direct limit of the sequence Es,tr(s,t) → Es,t

r(s,t)+1 → . . .

A particular example of a convergent spectral sequence is a first quadrant spectralsequence, i.e. a spectral sequence for which Es,t

r = 0 if s < 0 or t < 0 (for some r).

19

For such a spectral sequence there even exist indices r(s, t) such that the homomor-phisms in the sequence above are isomorphisms and hence Es,t

∞ ∼= Es,tr(s,t).

We now study the spectral sequence associated to a filtration of a cochain complex.A (decreasing) filtration F of a module A is a sequence of submodules (F sA)s∈Z

such that F sA ⊃ F s+1A. A filtration F determines an associated graded moduleG(A) defined by G(A)s = F sA/F s+1A. If A = (At)t∈Z is a graded module, thesubmodules F sA are graded by (F sA)t = F sA∩At and the associated module G(A)is bigraded by G(A)s,t = F sAs+t/F

s+1As+t.

A (decreasing) filtration of a cochain complex C∗ = (Cq, δq) is a sequence of cochainsubcomplexes (F sC∗)s∈Z such that F sC∗ ⊃ F s+1C∗. The filtration F is said to bebounded, if for every q there exist indices s1(q) < s2(q) such that F s1(q)Cq = Cq

and F s2(q)Cq = 0.

Theorem A.1 Let F be a filtration of a cochain complex C∗ = (Cq, δq). There isa spectral sequence (Er, dr) with initial term

Es,t1∼= Hs+t(F sC∗/F s+1C∗)

and differential d1 corresponding to the coboundary operator of the triple (F sC∗,F s+1C∗, F s+2C∗). If F is bounded the spectral sequence is convergent, and the limitterm E∞ is isomorphic to the bigraded module associated to the filtration of thecohomology module H(C∗) defined by

F sH(C∗) = im[H(F s−1C∗)→ H(C∗)].

(In this case we say that the spectral sequence converges to the cohomology of C∗.)

Example A.1 Let X be a CW-complex and (Xs)s∈Z an increasing filtration of Xconsisting of CW-subcomplexes. There is an induced (decreasing) filtration F ofthe cellular cochain complex C∗(X) given by the subcomplexes

F sC∗(X) = {c ∈ C∗(X)| c|C∗(Xs−1) = 0}

where C∗(X) is the cellular chain complex of X. If for every q-skeleton Xq of Xthere exist indices s1(q) < s2(q) such that Xs1(q) ⊂ Xq ⊂ Xs2(q), the filtrationF is bounded. In this case there is a convergent spectral sequence (Er, dr) withEs,t

1∼= Hs+t(Xs, Xs−1), d1 corresponding to the connecting homomorphism in the

long exact cohomology sequence of the triple (Xs+1, Xs, Xs−1) and

Es,t∞ ∼= ker[Hs+t(X)→ Hs+t(Xs−1)]/ ker[Hs+t(X)→ Hs+t(Xs)].

20

B Tensor products and exterior algebra

All modules in this appendix are modules over a fixed commutative ring R with aunit. Furthermore we assume that all R-algebras have a unit and that a homomor-phism between two R-algebras is unit preserving.

A graded algebra consists of a graded module A =⊕

q∈Z Aq and a homomorphismµ : A⊗ A→ A of degree 0, i.e. µ maps As ⊗ At to As+t. µ is called the product ofthe algebra and for a, a′ ∈ A we write aa′ = µ(a ⊗ a′). The algebra A is commu-tative if aa′ = (−1)deg a deg a′

a′a for all a, a′ ∈ A, where deg denotes the degree of anelement a ∈ A.

If A =⊕

q∈Z Aq and B =⊕

q∈Z Bq are graded algebras, their tensor product A⊗ Bis graded too by (A⊗ B)q =

⊕s+t=q As ⊗ Bt, and with the product

(a⊗ b)(a′ ⊗ b′) = (−1)deg b deg a′aa′ ⊗ bb′

A ⊗ B becomes a graded algebra which is associative or commutative whenever Aand B are.

Example B.1 Let R be a principal ideal domain. If X is a topological space, thenH∗(X;R) =

⊕q≥0 H

q(X;R) is a graded algebra whose product is the cup product.H∗(X;R) is associative and commutative, it is called the cohomology algebra of X.If Y is a second topological space whose homology module H∗(Y ;R) is free and offinite type, there is an isomorphism of graded algebras

H∗(X;R)⊗H∗(Y ;R) ∼= H∗(X×Y ;R)

by the Kunneth theorem (see [9], 5.6.1).

Definition B.1 Let⊗qA denote the q-fold tensor product of the module A with

itself. (In particular we set⊗0A = R and

⊗1A = A.) Then a product is defined onthe direct sum

⊗∗A =⊕

q≥0

⊗qA by

ab = (∑

q aq)(∑

q bq) =∑

q(∑

s+t=q as ⊗ bt)

where aq, bq ∈ ⊗qA are the homogeneous components of a and b, respectively. Thismakes

⊗∗A an associative graded algebra, called the tensor algebra on A.

Definition B.2 Let⊗∗A be the tensor algebra on the module A and I ⊂ ⊗∗A

the ideal generated by the elements a2, a ∈ A. Since I is graded, the quotient∧∗A =⊗∗A/I is graded too, and its component

∧qA of degree q is isomorphic to⊗qA modulo the submodule generated by the elements a1 ⊗ . . . ⊗ aq with ai = aj

21

for some i �= j. (In particular∧0A ∼= R and

∧1A ∼= A.) The product on⊗∗A

induces a product on∧∗A which is denoted by ∧. The relation a ∧ a = 0 implies

that a∧ a′ = −a′ ∧ a for all a, a′ ∈ A, and by induction on the degree it follows thatthe product ∧ is commutative. Thus

∧∗A is an associative and commutative gradedalgebra, called the exterior algebra on A.

The exterior algebra∧∗A can be characterised by the following universal property:

If f : A → X is a homomorphism into an algebra X such that (f(a))2 = 0 for alla ∈ A, then f can be extended uniquely to an algebra homomorphism from

∧∗A to X(where A has been identified with

∧1A ⊂ ∧∗A). This implies in particular that everyhomomorphism f between two modules A and B induces a unique homomorphismf∧ :

∧∗A→ ∧∗B such that f∧|A = f .

The module structure of∧∗A can be described as follows. If the module A is finitely

generated by the elements e1, . . . , en, then∧∗A is generated by the products

ei1 ∧ . . . ∧ eir (1 ≤ i1 < . . . < ir ≤ n)

Moreover, if A is free with basis {e1, . . . , en}, the products above form a basis of∧∗A and hence rk(∧∗A) = 2n.

The following lemma we need to compute the cohomology algebra of a torus.

Lemma B.1 If A,B are two modules, there is an isomorphism of graded algebras

∧∗A⊗ ∧∗B ∼= ∧∗(A⊕ B)

Example B.2 Let R be a principal ideal domain. By T d we denote the d-dimen-sional torus S1×. . .×S1 (d factors). Since the homology module of the circle S1 isfree and of finite type, there is an isomorphism H∗(T d;R) ∼= ⊗dH∗(S1;R) by theKunneth theorem. The cohomology algebra of the circle is generated by a singleelement α ∈ H1(S1;R) for which α ∪ α = 0 and therefore H∗(S1;R) ∼= ∧∗(R). Bythe previous lemma it follows that H∗(T d;R) is isomorphic to the exterior algebraon the free module of rank d over R.

22

C Symmetric bilinear forms

The main purpose of this appendix is to present the classification theorem of in-definite integral symmetric bilinear forms. For a thorough treatment of symmetricbilinear forms see [7].

Let R be a commutative ring with a unit and X a finitely generated free R-module.We always assume a symmetric bilinear form β : X×X → R to be non-singular, i.e.for each linear map ϕ : X → R there exists a uniquely determined element y0 ∈ Xsuch that ϕ is equal to the map x �→ β(x, y0). (For such a symmetric bilinear formthe notation x · y = β(x, y) is also used.) By the rank of a symmetric bilinearform we mean the rank of the space on which it is defined. Two symmetric bilinearforms β1 : X1×X1 → R and β2 : X2×X2 → R are isomorphic, if there exists anisomorphism f : X1 → X2 such that

β2(f(x), f(y)) = β1(x, y) ∀x, y ∈ X1.

In order to classify symmetric bilinear forms up to isomorphism, the following con-struction is useful.

Definition C.1 If β1 : X1×X1 → R and β2 : X2×X2 → R are two symmetric bili-near forms, their orthogonal sum is the symmetric bilinear form β which is definedon the direct sum X1 ⊕X2 by

β((x1, x2), (y1, y2)) = β1(x1, y1) + β2(x2, y2).

Example C.1 Let R be a principal ideal domain. If X is an oriented manifoldof dimension n = 2k, then the intersection numbers of homology classes define abilinear form on the free submodule of Hk(X;R), called the intersection form ofX. By Poincare duality the intersection form is non-singular and if k is even, it issymmetric. If Y is another oriented manifold of even dimension, then the intersectionform of the connected sum X#Y is the orthogonal sum of the intersection forms ofX and Y .

We now restrict to integral symmetric bilinear forms, i.e. R = Z. Before we canstate the main theorem we need some more definitions.

Definition C.2 Let β : X×X → Z be a symmetric bilinear form.(i) β is called positive (negative) definite, if x · x > 0 (x · x < 0) for all x ∈ X, andβ is called indefinite if x · x can take on both positive and negative values.(ii) Let {b1, . . . , bn} be an orthogonal basis of Q ⊗ X with respect to the inducedsymmetric bilinear form. Then the signature of β is defined to be the difference

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between the number of elements bi with bi · bi > 0 and the number of elements bj

with bj · bj < 0.(iii) β is said to be of odd type if there is an element x ∈ X with x · x odd, and β isof even type if there is no such element.

It can be shown that for every indefinite symmetric bilinear form there exists anelement x �= 0 with x · x = 0. This implies by induction that every indefinitesymmetric bilinear form of odd type is isomorphic to the orthogonal sum of bilinearforms of rank 1. Finally, by reducing the case of indefinite symmetric bilinear formsof even type to the case of odd forms, one obtains the following

Theorem C.1 Two indefinite integral symmetric bilinear forms are isomorphic ifand only if they have the same rank, signature and type.

Remark The classification of positive definite integral symmetric bilinear forms ismuch a harder problem. For further information see [7].

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References

[1] V.I. Danilov, The Geometry of Toric Varieties, Russian Math. Surveys 33(1978), 97–154

[2] M. Eikelberg, Zur Homologie torischer Varietaten, Dissertation, Ruhr-Univer-sitat Bochum, 1989

[3] M.H. Freedman, The Topology of 4-dimensional Manifolds, J. Diff. Geom. 17(1982), 357–453

[4] F.R. Gantmacher, The Theory of Matrices, Volume One, Chelsea PublishingCompany, 1959

[5] I.G. Macdonald, The Volume of a Lattice Polyhedron, Proc. Camb. Phil. Soc. 59(1963), 719–726

[6] S. MacLane, Homology, Springer 1963

[7] J. Milnor, D. Husemoller, Symmetric Bilinear Forms, Springer 1973

[8] T. Oda, Lectures on Torus Embeddings and Applications, Tata Inst. of Fund.Research, Springer 1978

[9] E.H. Spanier, Algebraic Topology, McGraw-Hill, 1966

[10] R.P. Stanley, The Number of Faces of a Simplicial Convex Polytope, Adv.Math. 35 (1980), 236–238

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Curriculum Vitae

I was born in Basel on June 6th, 1961 to Heidi and Joseph Fischli-Gass. When I wassix years old we moved to Bern. In autumn 1980 I began my studies in mathematicsat the University of Bern. As subsidiary subjects I enroled for computer science andastronomy. In spring 1988 I received my Master’s degree. Afterwards I worked forhalf a year for Swiss Reinsurance in London. Since 1989 I have been employed asan assistant by the Mathematics Institute at the University of Bern. My doctoralwork is part of a project supported by the National Science Foundation.

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