Cohomology rings of compactifications of toric arrangements
aa r X i v : . [ m a t h . A T ] J un COHOMOLOGY RINGS OF COMPACTIFICATIONS OFTORIC ARRANGEMENTS
CORRADO DE CONCINI, GIOVANNI GAIFFI
Abstract.
Some projective wonderful models for the complement of atoric arrangement in a n -dimensional algebraic torus T were constructedin [3]. In this paper we describe their integer cohomology rings bygenerators and relations. Introduction
Let T be a n -dimensional algebraic torus T over the complex numbers,and let X ∗ ( T ) be its character group, which is a lattice of rank n .A layer in T is the subvariety K Γ ,φ = { t ∈ T | χ ( t ) = φ ( χ ) , ∀ χ ∈ Γ } where Γ is a split direct summand of X ∗ ( T ) and φ : Γ → C ∗ is a homomor-phism.A toric arrangement A is given by finite set of layers A = {K , ..., K m } in T ; if for every i = 1 , ..., m the layer K i has codimension 1 the arrangement A is called divisorial .In [3] it is shown how to construct projective wonderful models for thecomplement M ( A ) = T − S i K i . A projective wonderful model is a smoothprojective variety containing M ( A ) as an open set and such that the com-plement of M ( A ) is a divisor with normal crossings and smooth irreduciblecomponents. We recall that the problem of finding a wonderful model for M ( A ) was first studied by Moci in [18], where a construction of non pro-jective models was described.In this paper we compute the integer cohomology ring of the projectivewonderful models by giving an explicit description of their generators andrelations. This allows for an extension to the setting of toric arrangementsof a rich theory that regards models of subspace arrangements and wasoriginated in [4], [5]. In these papers De Concini and Procesi constructedwonderful models for the complement of a subspace arrangement, providingboth a projective and a non projective version of their construction. In [5]they showed, using a description of the cohomology rings of the projectivewonderful models to give an explicit presentation of a Morgan algebra, thatthe mixed Hodge numbers and the rational homotopy type of the comple-ment of a complex subspace arrangement depend only on the intersection Date : June 8, 2018. lattice (viewed as a ranked poset). The cohomology rings of the models ofsubspace arrangements were then studied in [20], [12], were some integerbases were provided, and also, in the real case, in [7], [19]. Some combina-torial objects (nested sets, building sets) turned out to be relevant in thedescription of the boundary of the models and of their cohomology rings:their relation with discrete geometry were pointed out in [8], [13]; the caseof complex reflection groups was dealt with in [14] from the representationtheoretic point of view and in [2] from the homotopical point of view.The connections between the geometry of these models and the Chowrings of matroids were pointed out first in [9] and then in [1], where theyalso played a crucial role in the study of some log-concavity problems.As it happens for the case of subspace arrangements, in addition to theinterest in their own geometry, the projective wonderful models of a toricarrangement A may also spread a new light on the geometric propertiesof the complement M ( A ). For instance, in the divisorial case, using theproperties of a projective wonderful model, Denham and Suciu showed in[6] that M ( A ) is both a duality space and an abelian duality space.Let us now describe more in detail the content of the present paper.In Section 2 we briefly recall the construction of wonderful models ofvarieties equipped with an arrangement of subvarieties : this is a general-ization, studied by Li in [17], of the De Concini and Procesi’s constructionfor subspace arrangements. Its relevance in our setting is explained by thefollowing remark. In [3], as a first step, the torus T is embedded in a smoothprojective toric variety X . This toric variety, as we recall in Section 5, ischosen in such a way that the set made by the connected components of theintersections of the closures of the layers of A turns out to be an arrange-ment of subvarieties L ′ and one can apply Li’s construction in order to geta projective wonderful model.More precisely, there are many possible projective wonderful models as-sociated to L ′ , depending on the choice of a building set for L ′ .We devote Section 3 to a recall of the definition and the main proper-ties of building sets and nested sets of arrangements of subvarieties. Thesecombinatorial objects were introduced by De Concini and Procesi in [4] andtheir properties in the case of arrangements of subvarieties were investigatedin [17]. If G is a building set for L ′ we will denote by Y ( X, G ) the wonderfulmodel constructed starting from G .In Section 4, given any arrangement of subvarieties in a variety X , we fo-cus on its well connected building sets : these are building sets that satisfy anadditional property, that will be crucial for our cohomological computations.In Section 6 we recall a key lemma, due to Keel, that allows to compute thecohomology ring of the blowup of a variety M along a center Z provided thatthe restriction map H ∗ ( M ) → H ∗ ( Z ) is surjective. In this result the Chernpolynomial of the normal bundle of Z in M plays a crucial role. Then we goback to the case of toric arrangements and, given a smooth projective toricvariety X associated to the toric arrangement A , we describe the properties OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 3 of some polynomials in H ∗ ( X, Z ) that are related to the Chern polynomialsof the closures of the layers of A in X .In Section 7 we prove our main result (Theorem 7.1): we provide a presen-tation of the cohomology ring H ∗ ( Y ( X, G ) , Z ) by generators and relations,as a quotient of a polynomial ring over H ∗ ( X, Z ), whose presentation is wellknown. A concrete choice for the generators that appear in our theorem isprovided in Section 8. We recall that a description of the cohomology of awonderful model of subvarieties as a module was already found by Li in [16].Finally, in Section 9 we provide a presentation of the cohomology ringsof all the strata in the boundary of Y ( X, G ).2. Wonderful models of stratified varieties
In this section we are going to recall the definitions of arrangements ofsubvarieties, building sets and nested sets given in Li’s paper [17]. We willgive these definitions in two steps, first for simple arrangements of subva-rieties, then in a more general situation. We are going to work over thecomplex numbers, hence all the algebraic varieties we are going to considerare complex algebraic varieties.
Definition 2.1.
Let X be a non singular variety. A simple arrangement ofsubvarieties of X is a finite set Λ = { Λ i } of nonsingular closed connectedsubvarieties Λ i , properly contained in X , which satisfy the following condi-tions:(i) Λ i and Λ j intersect cleanly , i.e. their intersection is nonsingular and forevery y ∈ Λ i ∩ Λ j we have T Λ i ∩ Λ j ,y = T Λ i ,y ∩ T Λ j ,y (ii) Λ i ∩ Λ j either belongs to Λ or is empty. Definition 2.2.
Let Λ be a simple arrangement of subvarieties of X . Asubset G ⊆ Λ is called a building set for Λ if for every Λ i ∈ Λ − G theminimal elements in { G ∈ G : G ⊇ Λ i } intersect transversally and theirintersection is Λ i . These minimal elements are called the G -factors of Λ i . Definition 2.3.
Let G be a building set for a simple arrangement Λ . A nonempty subset T ⊆ G is called G -nested if for any subset { A , ..., A k } ⊂ T (with k > ) of pairwise non comparable elements, A , . . . , A k are the G -factors of an element in Λ . We remark that in Section 5.4 of [17] the following more general definitionsare provided, to include the case when the intersection of two strata is adisjoint union of strata.
Definition 2.4. An arrangement of subvarieties of a nonsingular variety X is a finite set Λ = { Λ i } of nonsingular closed connected subvarieties Λ i ,properly contained in X , that satisfy the following conditions:(i) Λ i and Λ j intersect cleanly; CORRADO DE CONCINI, GIOVANNI GAIFFI (ii) Λ i ∩ Λ j is either equal to the disjoint union of some of the Λ k ’s or it isempty. Given an open set U ⊂ X , and a family Λ of subvarieties of X , by therestriction Λ | U of Λ to U we shall mean the family of non empty intersectionsof elements of Λ with U . Definition 2.5.
Let Λ be an arrangement of subvarieties of X . A subset G ⊆ Λ is called a building set for Λ if there is an open cover { U i } of X suchthat:a) the restriction of the arrangement Λ to U i is simple for every i ;b) G | U i is a building set for Λ | U i . We have first introduced the notion of arrangement of subvarieties andthen defined a building set for the arrangement. However it is often conve-nient to go in the opposite direction and first introduce the notion of buildingset and use it to define the corresponding arrangement.
Definition 2.6.
A finite set G of connected subvarieties of X is called abuilding set if the set of the connected components of all the possible intersec-tions of collections of subvarieties from G is an arrangement of subvarieties Λ (the arrangement induced by G ) and G is a building set for Λ . Let us now introduce the notion of G -nested set in the more general con-text of (not necessarily simple) arrangements of subvarieties. Definition 2.7.
Let G be a building set for an arrangement Λ . A subset T ⊆ G is called G -nested if there is an open cover { U i } of X such that, forevery i , G | U i is simple and T | U i is G | U i -nested. Remark 2.1.
We notice that, according to the definition above, if somevarieties G , G , .., G k ∈ G have empty intersection, then they cannot belongto the same G -nested set. Once we have an arrangement Λ of a nonsingular variety X and a buildingset G for Λ, we can construct a wonderful model Y ( X, G ) by considering (byanalogy with [5]) the closure of the image of the locally closed embedding X − [ Λ i ∈ Λ Λ i → Y G ∈G Bl G X where Bl G X is the blowup of X along G .In [17], Proposition 2.8, one shows: Proposition 2.1.
Let G be a building set in the variety X . Let F ∈ G bea minimal element in G under inclusion. Then the set G ′ consisting of theproper transforms of the elements in G is a building set in Bl F X .Proof. In fact Li shows this for a building set of a simple arrangement. Butsince the definition of building set is local, one can easily adapt his proof(see also Section 5.4 of [17]). (cid:3)
OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 5
Using this in [17], Theorem 1.3 and the discussion following it, one shows
Theorem 2.1.
Let G be a building set of subvarieties in a nonsingularvariety X . Let us arrange the elements G , G , ..., G m of G in such a waythat for every ≤ i ≤ N the set G h = { G , G , . . . , G h } is building. Thenif for each ≤ h ≤ m , we set X = X and X h := Y ( X, G h ) , for h > , wehave X h = Bl ˜ G h X h − , where ˜ G h denotes the dominant transform of G h in X h − . Remark 2.2.
1. We notice that any total ordering of the elements of abuilding set G = { G , . . . , G m } which refines the ordering by inclusion, thatis i < j if G i ⊂ G j , satisfies the condition of Theorem 2.1.2. In particular using the above ordering we deduce that Y ( X, G ) is ob-tained from X by a sequence of blow ups each with center a minimal elementin a suitable building set. For every element G ∈ G we denote by D G itsdominant transform, that is a divisor of Y ( X, G ) . To finish this section let us mention a further result of Li describing theboundary of Y ( X, G ) in terms of G -nested sets: Theorem 2.2 (see [17], Theorem 1.2) . The complement in Y ( X, G ) to X − S Λ i ∈ Λ Λ i is the union of the divisors D G , where G ranges among theelements of G . An intersection of these divisors is nonempty if and only if { T , ..., T k } is G -nested. If the intersection is nonempty it is transversal. Some further properties of building sets
In this section we collect a few facts of a technical nature which willbe used later. Let Λ be an arrangement of subvarieties in a connectednonsingular variety X . Let G be a building set for Λ and let F be a minimalelement in G . Let us denote by e X the blowup Bl F X and, for every subvariety D , let us call e D the transform of D .Let us first recall the following lemma from [17] (originally stated forΛ simple arrangement, but valid also for the general case due to its localnature). Lemma 3.1 (see [17] Lemma 2.9) . Let G be a building set for Λ , and let F be a minimal element in G . Let consider the blowup e X = Bl F X , and let A, B, A , A , B , B be nonsingular subvarieties of X .1. Suppose that A A and A A , and suppose that A ∩ A = F andthe intersection is clean. Then e A ∩ e A = ∅ . In the blowup of a variety M along a center F the dominant transform of a subvariety Z coincides with the strict transform if Z F (and therefore it is isomorphic to theblowup of Z along Z ∩ F ) and to π − ( Z ) if Z ⊂ F , where π : BL F M → M is theprojection. CORRADO DE CONCINI, GIOVANNI GAIFFI
2. Suppose that A and A intersect cleanly and that F ( A ∩ A . Then e A ∩ e A = ^ A ∩ A .3. Suppose that B and B intersect cleanly and that F is transversal to B , B and B ∩ B . Then e B ∩ e B = ^ B ∩ B .4. Suppose that A is transversal to B , F is transversal to B and F ⊂ A .Then e A ∩ e B = ^ A ∩ B . The following simple lemma will be useful later.
Lemma 3.2.
Let G be a building set for Λ , and let U be an open set as inthe Definition 2.5. Let us consider two subsets { H , ..., H k } and { G , ..., G s } of G . If H = U ∩ T i =1 ,...,k H i = ∅ and H = U ∩ \ i =1 ,...,k H i ⊂ G = U ∩ \ j =1 ,...,s G j then the connected component of T i =1 ,...,k H i that contains H is containedin the connected component of T j =1 ,...,s G j that contains G .Proof. First we notice that H and G are connected by the Definition2.5. The statement follows since H is a dense open set of the connectedcomponent of T i =1 ,...,k H i that contains it. (cid:3) Proposition 3.1.
Let G be a building set for Λ . Let us fix an open set U as in the Definition 2.5 (for brevity, in what follows every object will berestricted to U but we are going to omit the symbol of restriction, for instancewe will denote by G the set G ∩ U for every G ∈ G ). Let G , G ∈ G benot comparable. Then either G ∩ G = ∅ , or G ∩ G ∈ G or G ∩ G istransversal.Proof. Let us suppose G ∩ G = ∅ . We know by the definition of buildingset that(1) G ∩ G = H ∩ H ∩ ... ∩ H k where the H j ’s are the minimal elements in G that contain G ∩ G andthe intersection among the H j ’s is transversal. We can suppose, up toreordering, that H ⊂ G .If we also have H ⊂ G then H ⊂ G ∩ G , while from the equality (1)we have G ∩ G ⊂ H . This means that G ∩ G = H and therefore itbelongs to G .If, on the other hand, H is not contained in G , we can suppose, up toreordering, that H ⊂ G . Then H ∩ H ⊂ G ∩ G while from the equality(1) we have G ∩ G ⊂ H ∩ H . This means that G ∩ G = H ∩ H sothat in particular k = 2.Since the intersection H ∩ H is transversal, then also G ∩ G is transver-sal. Indeed once one fixes a point y ∈ H ∩ H , the set of linear equations OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 7 that describe the tangent space T H i ,y includes the set of equations that de-scribe T G i ,y . Since the intersections are clean and all the involved varietiesare smooth this implies in particular that G = H and G = H . (cid:3) Corollary 3.1 (see Lemma 2.6 in [17]) . Let G be a building set. Let F be aelement in G .1. If F is minimal, for any G ∈ G , either G contains F , or F ∩ G = ∅ , or F ∩ G is transversal.2. Let K be an element of the arrangement induced by G such that none ofits G factors contains F . Assume that H = K ∩ F also has F as one ofits G factors. Then the intersection of K and F is transversal.Proof. First we notice that, by Lemma 3.2, for every open set U as in theDefinition 2.5, F ∩ U is empty or it is minimal also for the restriction of G to U . Therefore it is sufficient to prove our statement locally (and from nowon we will think of every object as intersected with U ).So (1) is an immediate consequence of Proposition 3.1 since if F G and F ∩ G = ∅ , then F ∩ G / ∈ G by minimality of F .As for (2), since G is building, we can write H = B ∩ .. ∩ B j ∩ F where B , ..., B j , F (with j ≥
1) are the G factors of H and their intersectionis transversal.Let G be a G factor of K . Since G contains H but does not contain F ,it must contain one of the B i ’s. It follow that S = B ∩ · · · ∩ B j ⊂ K . Wededuce that, since H = K ∩ F = S ∩ F,K and F intersect cleanly and S and F intersect transversally, also K and F intersect transversally. (cid:3) Well connected building sets
In the computation of the cohomology of compact wonderful models wewill need some building sets that have an extra property.
Definition 4.1.
A building set G is called well connected if for any subset { G , ..., G k } in G , the intersection G ∩ G ∩ ... ∩ G k is either empty, orconnected or it is the union of connected components each belonging to G . Remark 4.1.
In particular, if G is well connected and F ∈ G is minimal,we have that for every G ∈ G the intersection G ∩ F is either empty orconnected. Notice that, for example, if Λ is an arrangement of subvarieties then Λitself is a, rather obvious, example of a well connected building set.
CORRADO DE CONCINI, GIOVANNI GAIFFI
As another example, if Λ is simple then clearly every building set for Λ iswell connected.The following two propositions are going to be crucial in our inductiveprocedure. Let X be a smooth variety and G = { G , ..., G m } a well connectedbuilding set of subvarieties of X whose elements are ordered in a way thatrefines inclusion. Proposition 4.1.
For every k = 1 , ..., m , the set G k = { G , ..., G k } is a wellconnected building set.Proof. Let us prove that G k is building.First we check what happens ‘locally’. We fix an open set U as in theDefinition 2.5 and in what follows we will consider the restriction of everyobject to U .Since G is building, we know that every intersection G j ∩ · · · ∩ G j s ofelements of G k is equal to the transversal intersection of the minimal elements B , ..., B h of G that contain G j ∩ · · · ∩ G j s . Up to reordering we can assumethat the set { B , . . . , B r } for some r ≤ s consists of those among the B ′ i swhich are contained in at least one among the G j t ’s . Notice that necessarily r \ i =1 B j = h \ i =1 B j = G j ∩ · · · ∩ G j s . Since the intersection of B , . . . , B h is transversal, we clearly have that r = h and so we deduce that for each j ≤ h , there is an a ≤ k with B j = G a .Going back from the local to the global setting, we observe that with theargument above we have proven that B j ∩ U = G a ∩ U . Since intersectingwith U preserves inclusion relations by Lemma 3.2, we immediately deducethat B i ∈ G k for each i = 1 , . . . , h .A similar reasoning also shows that G k is well connected. (cid:3) Let us consider the variety Z := G m . Let us take the family H = { H , . . . , H u } of non empty subvarieties in Z which are obtained as con-nected components of intersections G i ∩ Z with i < m .Let us remark that, since G is well connected, if G i ∩ Z is non connected(and of course non empty) its connected components belong to G so thateach of them equals some G j ( Z . We deduce that we do not need toadd the connected components of the disconnected intersections G i ∩ Z . Inparticular u ≤ m − H in such a way that if for each 1 ≤ i ≤ u , we set s i ≤ m − H i = G s i ∩ Z , s i < s j as soon as i < j . Proposition 4.2.
The family of subvarieties H = { H , ..., H u } in Z isbuilding and well connected.Proof. Let us prove that H is building. By definition of building set, itsuffices to prove this locally, i.e. in U ∩ Z for any of the open sets U that OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 9 appears in the definition of the building set G . So we fix such an U andassume that X = U .As before, we write for each i = 1 , . . . , u , H i = G s i ∩ Z with G s i ∈ G m − .Let H = H i ∩ · · · ∩ H i ℓ be a nonempty intersection of elements of H .Since H is also an intersection of elements of G we can write H = H i ∩ · · · ∩ H i ℓ = G s i ∩ · · · ∩ G s iℓ ∩ Z = G j ∩ · · · ∩ G j k , where G j , ..., G j k are the minimal elements of G that contain H and theirintersection is transversal in X .Consider the set I = { s i , . . . , s i ℓ , m } and J = { j , . . . , j k } . In I × J wetake the subset S consisting of those pairs ( a, b ) such that G a ⊃ G b . Byeventually reordering the indices, we can assume that the projection of S onthe second factor equals { j , . . . , j k ′ } , for some k ′ ≤ k . On the other hand,by minimality, the projection of S on the first factor is surjective and wecan further assume that Z ⊃ G j k ′ .We claim that k = k ′ . Indeed if k ′ where less than k , H = H i ∩ · · · ∩ H i ℓ = G j ∩ · · · ∩ G j ′ k and the intersection G j ∩ · · · ∩ G j k would not be transversal.Let β s , for every 1 ≤ s ≤ k , be such that H β s ∈ H is the connectedcomponent of G j s ∩ Z that contains H .Then we have H β ∩ · · · ∩ H β k = H. We set d = k − Z = G j k , d = k otherwise. In both cases we then easilysee that H is the transversal intersection H β ∩ · · · ∩ H β d . Finally let us observe that H β , H β , ..., H β d are the minimal elements in H containing H . This is obvious if d = 1. If d >
1, assume by contradictionthat there is an element H ′ ∈ H and an index s ∈ { , ..., d } such that H ⊆ H ′ ( H β s . The last inclusion implies that H ′ = G ′ ∩ Z ( G j s ∩ Z for some G ′ ∈ G . From this in particular it follows that Z is not contained in G ′ and that G j s * G ′ . Now, since the elements G j , ..., G j k are the minimalelements of G that contain H , G j ν ⊆ G ′ for some 1 ≤ ν ≤ k . Since Z is notcontained in G ′ , j ν = m .Then we have H β ν ⊆ G j ν ∩ Z ⊆ G ′ ∩ Z = H ′ . But H ′ ( H β s , so we deduce H β ν ( H β s which is a contradiction, since we know that their intersectionis transversal.This completes the proof that H is building.Let us now prove that H is well connected (this proof is not local). Firstwe observe that by definition the elements of H are connected. Then let H = H i ∩ · · · ∩ H i t be a nonempty intersection of elements of H . Since H is also an intersection of elements of G , by the well connectedness of G , if H is not connected then it is the disjoint union of connected components thatbelong to G . Let G s be such a component: since it is contained in Z then s < m and G s = G s ∩ Z belongs to H . This proves that all these connectedcomponents belong to H . (cid:3) Remark 4.2.
In the proof of the proposition above we have shown thatif H = H i ∩ · · · ∩ H i ℓ then H is equal to the transversal intersection of H β , ..., H β d . In particular we have shown that, for every γ = 1 , ..., d , H β γ is a connected component of G j γ ∩ Z and G j γ is included in some of the G s i , ..., G s iℓ . With the chosen ordering of H = { H , . . . , H u } , this impliesthat each one of the β j ’s is ≤ max { i , ..., i l } . Therefore we have proven thatfor each ≤ i ≤ u , the arrangement of subvarieties H i = { H , ..., H i } in Z is building and well connected. Proposition 4.3.
Let ≤ s ≤ m − and let ≤ i ≤ u be such that s i ≤ s < s i +1 . Then the proper transform of Z in X s equals Z i . Proof.
We first treat the case in which s i < s < s i +1 . In this case there aretwo possibilities(1) G s ∩ Z = ∅ .(2) G s ∩ Z = ∅ and each of its connected components lies in G .In the first case there is nothing to prove. In the second case, by assumptionwhen we reach X s − we have already blown up each of the connected com-ponents of G s ∩ Z . Since we know that the intersection G s ∩ Z is clean, byLemma 3.1.(1) the transforms of Z and G s in X h − have empty intersectionand clearly also in this case there is nothing to prove.If s = s i again we have two cases(1) G s i ⊂ Z .(2) The intersection G s i ∩ Z is transversal and does not lie in G .Let us denote by ˜ H i and ˜ G s i the proper transforms of H i and G s i in X s i − By Remark 2.2.2, X s i is obtained from X s i − by blowing ˜ G s i which is aminimal element in a suitable building set.Thus our statement in case (1) follows, using induction from the fact that Z i = Bl ˜ G si Z i − = Bl ˜ H i Z i − .In case (2), since H i = G s i ∩ Z , by induction and Lemma 3.1 we deducethat ˜ H i = ˜ G s i ∩ Z i − . So by Corollary 3.1.(1), and the minimality of ˜ G s i ina suitable building set, the intersection ˜ H i = ˜ G s i ∩ Z i − is transversal andthe proper transform of Z in X s i equals Bl ˜ H i Z i − = Z i as desired. (cid:3) Recollections on the construction of projective wonderfulmodels of a toric arrangement
We are now going to consider a special situation. We consider a n -dimensional algebraic torus T over the complex numbers and we denoteby X ∗ ( T ) its character group.Let us take V = hom Z ( X ∗ ( T ) , R ) = X ∗ ( T ) ⊗ Z R , X ∗ ( T ) being the lattice hom Z ( X ∗ ( T ) , Z ) of one parameter subgroups in T . OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 11
Then, setting V C = hom Z ( X ∗ ( T ) , C ) = X ∗ ( T ) ⊗ Z C , we have a naturalidentification of T with V C /X ∗ ( T ) and we may consider a χ ∈ X ∗ ( T ) as alinear function on V C . From now on the corresponding character e πiχ willbe usually denoted by x χ .Now, let A be the toric arrangement A = {K Γ ,φ , ..., K Γ r ,φ r } in T asdefined in the Introduction, where the Γ i are split direct summands of X ∗ ( T )and the φ i ’s are homomorphisms φ i : Γ i → C ∗ .Remark that K Γ ,φ is a coset with respect to the torus H = ∩ χ ∈ Γ Ker ( x χ ).Now we consider the subspace V Γ = { v ∈ V | h χ, v i = 0 , ∀ χ ∈ Γ } . Notice thatsince X ∗ ( H ) = X ∗ ( T ) / Γ, V Γ is naturally isomorphic to hom Z ( X ∗ ( H ) , R ) = X ∗ ( H ) ⊗ Z R .In [3] (see Proposition 6.1) it was shown how to construct a projectivesmooth T - embedding X = X ∆ whose fan ∆ in V has the following property.For every Γ i there is an integral basis of Γ i , χ , . . . , χ s , such that, for everycone C of ∆ with generators r , . . . , r h , up to replace χ i with − χ i for some i , the pairings h χ i , r j i are all ≥ ≤
0. The basis χ , . . . χ s is called an equal sign basis for Γ i .Moreover we remark that ∆ can be chosen in such a way that for everylayer K Γ ,φ , obtained as a connected component of the intersection of someof the layers in A , the lattice Γ has an equal sign basis. Given such a ∆, wewill say that X = X ∆ is a good toric variety for A .In such a toric variety X consider the closure K Γ ,φ of a layer. This closureturns out to be a toric variety, whose explicit description is provided by thefollowing result from [3]. Theorem 5.1 (Proposition 3.1 and Theorem 3.1 in [3]) . For every layer K Γ ,φ , let H be the corresponding subtorus and let V Γ = { v ∈ V | h χ, v i =0 , ∀ χ ∈ Γ } . Then,1. For every cone C ∈ ∆ , its relative interior is either entirely contained in V Γ or disjoint from V Γ .2. The collection of cones C ∈ ∆ which are contained in V Γ is a smooth fan ∆ H .3. K Γ ,φ is a smooth H -variety whose fan is ∆ H .4. Let O be a T orbit in X = X ∆ and let C O ∈ ∆ be the corresponding cone.Then(a) If C O is not contained in V Γ , O ∩ K Γ ,φ = ∅ .(b) If C O ⊂ V Γ , O ∩ K Γ ,φ is the H orbit in K Γ ,φ corresponding to C O ∈ ∆ H . Let us denote by Q ′ (resp. Q ) the set whose elements are the subvarieties K Γ i ,φ i of X (resp. the subvarieties K Γ i ,φ i and the irreducible components ofthe complement X − T ). We then denote by L ′ (resp. L ) the poset made byall the connected components of all the intersections of some of the elementsof Q ′ (resp. Q ). In [3] (Theorem 7.1) we have shown that the family L is anarrangement of subvarieties in X . As a consequence also L ′ , being containedin L and closed under intersection, is an arrangement of subvarieties. We notice that the complement in X of the union of the elements in L isequal to M ( A ), and it is strictly contained in the complement of the unionof the elements in L ′ .In the sequel of this paper we will focus on the wonderful model Y ( X, G )obtained by choosing a (well connected) building set G for L ′ . Let us nowexplain our choice.As a consequence of Theorem 5.1 we deduce that the elements of L areexactly the non empty intersections K Γ ,φ ∩ O 6 = ∅ . This means that they areindexed by a family of triples (Γ , φ, C O ) with φ ∈ hom(Γ , C ∗ ), and C O ⊂ V Γ .The triples ( { } , , C O ) index the closures of T orbits in X .The intersection K Γ φ ∩ O is transversal. Furthermore, since X is smooth, if the cone C O = C ( r i , . . . r i h ),where the r i j are the rays of C O , we have that O is the transversal intersec-tion of the divisors D r ij . We deduce: Proposition 5.1.
Let G be a building set for the arrangement of subvarieties L ′ in X . Then G + = G ∪ { D r } r ∈R is a building set for L .Proof. We have seen that an element of S ∈ L is the transversal intersection S = K Γ ,φ ∩ \ r ∈ J D r , with J a, possibly empty, subset of R We know that, in a suitable open set U , K Γ ,φ is the transversal intersectionof the minimal elements in G containing it. Since G + = G ∪ { D r } r ∈R , thesame holds for S with respect to G + .On the other hand we observe that the connected components of anyintersection of elements of G + belong to L , by the definition of L .This clearly means that L is the arrangement induced by G + and that G + is a building set for L . (cid:3) As a consequence of the proposition above, we can construct Y ( X, G + ),which is a projective wonderful model for the complement M ( A ) = X − [ G ∈G + G = X − [ A ∈L A. Now we observe that the varieties Y ( X, G ) and Y ( X, G + ) are isomorphic.To prove this for instance one could order G + in the following way: oneputs first the elements of G ordered in a way that refines inclusion, then theelements D r in any order. As we know from Theorem 2.1, Y ( X, G + ) canbe obtained as the result of a series of blowups starting from X . After thefirst |G| steps we get Y ( X, G ), then the centers of the last |R| blowups aredivisors so Y ( X, G + ) is isomorphic to Y ( X, G ). OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 13
To finish our recollection on projective models and toric varieties, we needto describe explicitly the restriction map in cohomology j ∗ : H ∗ ( X, Z ) → H ∗ ( K Γ ,φ , Z ) , induced by the inclusion, for a layer K Γ ,φ .Let us first recall the following well known presentation of the cohomologyring of a smooth projective toric variety by generators and relations. LetΣ be a smooth complete fan and let X Σ its associated toric variety. Takea one dimensional face in Σ. This face contains a unique primitive ray r ∈ hom Z ( X ∗ ( T ) , Z ). We denote by R the collection of rays. We have: Proposition 5.2. (see for example [10] , Section 5.2.) H ∗ ( X Σ ) , Z ) = Z [ c r ] r ∈R /L Σ where L Σ is the ideal generated bya) c r c r · · · c r k if the rays r , ..., r k do not belong to a cone of Σ .b) P r ∈R h β, r i c r for any β ∈ X ∗ ( T ) .Furthermore the residue class of c r in H ( X Σ , Z ) is the cohomology class ofthe divisor D r associated to the ray r for each r ∈ R . By abuse of notationwe are going to denote by c r its residue class in H ( X Σ , Z ) . Let us consider as before a toric arrangement A in a torus T , and a goodtoric variety X = X ∆ for A . We can apply the proposition above to both X and the closure of a layer K Γ ,φ . Let us remark that by Theorem 5.1.4,if r / ∈ V Γ then the divisor D r does not intersect K Γ ,φ , while if r ∈ V Γ thedivisor D r intersects K Γ ,φ in the divisor corresponding to r . We deduce: Proposition 5.3.
The restriction map j ∗ : H ∗ ( X, Z ) → H ∗ ( K Γ ,φ , Z ) . is surjective and its kernel I is generated by the classes c r with r ∈ R suchthat r / ∈ V Γ . A result of Keel and Chern polynomials of closures oflayers
Let us as before consider a toric arrangement A in the torus T . As werecalled in Section 5, we can and will choose X = X ∆ to be a good toricvariety associated to A and take the arrangement L ′ of subvarieties in X .Let us fix a well connected building set G = { G , ..., G m } for L ′ , orderedin such a way that if G i ( G j then i < j .Our goal is to describe the cohomology ring H ∗ ( Y ( X, G ) , Z ) by generatorsand relations. For this we are going to use the following result due to Keel.Let Y be a smooth variety, and suppose that Z is a regularly embeddedsubvariety of codimension d (we denote by i : Z → Y the inclusion). Let Bl Z ( Y ) be the blowup of Y along Z , so we have a map π : Bl Z ( Y ) → Y ,and let E = E Z be the exceptional divisor. Theorem 6.1 (Theorem 1 in the Appendix of [15]) . Suppose that the map i ∗ : H ∗ ( Y ) → H ∗ ( Z ) is surjective with kernel J , then H ∗ ( Bl Z Y ) is isomor-phic to H ∗ ( Y )[ t ]( P ( t ) , t · J ) where P ( t ) ∈ H ∗ ( Y )[ t ] is any polynomial whose constant term is [ Z ] andwhose restriction to H ∗ ( Z ) is the Chern polynomial of the normal bundle N = N Z Y , that is to say i ∗ ( P ( t )) = t d + t d − c ( N ) + · · · + c d ( N ) This isomorphism is induced by π ∗ : H ∗ ( Y ) → H ∗ ( Bl Z Y ) and by sending − t to [ E ] . In order to use Theorem 6.1, we need to introduce certain polynomialswith coefficient in H ∗ ( X, Z ).For every G := K Γ ,φ ∈ L ′ , we set Λ G := Γ. Setting B = H ∗ ( X, Z ) wechoose a polynomial P G ( t ) = P XG ( t ) ∈ B [ t ] that satisfies the following twoproperties:(1) P G (0) is the class dual to the class of G in homology.(2) the restriction map to H ∗ ( G, Z )[ t ] sends P G ( t ) to the Chern polyno-mial of N G X .We will say that such a polynomial is a good lifting of the Chern polynomialof N G X . Let I be the kernel of the restriction map j ∗ : H ∗ ( X, Z ) → H ∗ ( G, Z ) . Lemma 6.1.
The ideal ( tI, P G ( − t )) ⊂ B [ t ] does not depend on the choiceof P G ( t ) .Proof. Let Q G ( t ) be another polynomial satisfying (1) e (2). From (1) weknow that P G ( t ) − Q G ( t ) has constant term equal to 0. Moreover from (2) wededuce that every coefficient of P G ( t ) − Q G ( t ) belongs to I so P G ( t ) − Q G ( t ) ∈ ( tI ). (cid:3) Let us now consider two elements
G, M ∈ L ′ , with G ⊂ M . Let uschoose a polynomial P MG ( t ) ∈ H ∗ ( M, Z )[ t ] that is a good lifting of the Chernpolynomial of N G M (i.e. it satisfies the properties (1) and (2) in H ∗ ( M, Z ))and let us denote by P MG ( t ) a lifting of P MG ( t ) to H ∗ ( X, Z )[ t ]. The existenceof such polynomial follows immediately from Proposition 5.3.Let us now consider a well connected building set G = { G , . . . , G m } forthe arrangement of subvarieties L ′ in X (see Section 5), ordered in a waythat refines inclusion.Now, for every pair ( i, A ) with i ∈ { , ..., m } , and A ⊂ { , ..., m } suchthat if j ∈ A then G i ( G j , we can define the following polynomial in H ∗ ( X, Z )[ t , . . . , t m ] = B [ t , . . . , t m ].Let us consider the set B i = { h | G h ⊆ G i } , and let us denote by M theunique connected component of T j ∈ A G j that contains G i (if A = ∅ we put OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 15 M = X ). Then, after choosing all the polynomials P MG i as explained before,we put: F ( i, A ) = P MG i ( X h ∈ B i − t h ) Y j ∈ A t j . We also include as special cases the pairs (0 , A ) where A is such that T j ∈ A G j = ∅ , and we define the polynomials: F (0 , A ) = Y j ∈ A t j . Proposition 6.1.
Let I m be the ideal in B [ t , . . . , t m ] generated by1. the products t i c r for every ray r ∈ ∆ that does not belong to V Λ Gi (i.e. h r, ·i does not vanish on Λ G i );2. the polynomials F ( i, A ) defined above.Then I m does not depend on the choice of the polynomials P MG i .Proof. We will prove the statement by induction on m . We notice that if m = 1 the statement is true by the Lemma 6.1 (the ideal I coincides withthe ideal I in the lemma).Let then m ≥ I m − ⊂ B [ t , . . . , t m − ]which by the inductive hypothesis does not depend on the choice of thepolynomials P MG i ’s (where i < m ). We will denote by I ′ m − its extension to B [ t , . . . , t m ].The polynomials F ( i, A ) belong to I ′ m − unless m ∈ A or i = m . In thelatter case A = ∅ and the same proof as in Lemma 6.1 implies that the idealdoes not depend on the choice of the polynomial P G m .In the first case ( m ∈ A ), if we consider two liftings P MG i and Q MG i wenotice that the restriction of their difference P MG i − Q MG i to H ∗ ( M )[ t ] hasconstant term equal to 0, while the restriction to H ∗ ( G i )[ t ] is 0.Let z be equal to P MG i (0) − Q MG i (0). By construction z belongs to the idealgenerated by the c r ’s such that r does not belong to V Λ M , that is to say, h r, ·i does not vanish on Λ M . Now we observe that the lattice Γ = P j ∈ A Λ G j hasfinite index in Λ M . If h r, ·i vanished on Λ G j for every j ∈ A then it wouldvanish on Γ and therefore on Λ M .It follows that if r does not belong to V Λ M then it exists j ∈ A such that r does not belong to V Λ Gj . This implies that Q j ∈ A t j z belongs to the idealgenerated by the monomials in (1) . To conclude it is sufficient to noticethat the coefficients of P MG i ( P h ∈ B i − t h ) − Q MG i ( P h ∈ B i − t h ) − z belong to theideal generated by the c r ’s such that r / ∈ V Λ Gi , and therefore, for the samereasoning as above, P MG i ( P h ∈ B i − t h ) − Q MG i ( P h ∈ B i − t h ) − z belongs to I ′ m − . (cid:3) Presentation of the cohomology ring
Let us consider a toric arrangement A in the torus T . As recalled inSection 5, let X = X ∆ be a good toric variety associated to the chosen toricarrangement, and let us consider the arrangement L ′ of subvarieties in X .Fix now a well connected building set G = { G , ..., G m } for L ′ , ordered insuch a way that if G i ( G j then i < j .Our goal is to describe the cohomology ring H ∗ ( Y ( X, G ) , Z ) by genera-tors and relations. For any pair ( G, M ) ∈ L ′ × L ′ with G ⊂ M , we fix apolynomial P MG ∈ H ∗ ( X, Z )[ t ] = B [ t ] as explained in the preceding section.We also fix the polynomials P XG ∈ B [ t ]. This means in particular that wehave fixed a choice for the polynomials F ( i, A ) ∈ B [ t , ..., t m ]. Then we canstate our main theorem: Theorem 7.1.
The cohomology ring H ∗ ( Y ( X, G ) , Z ) is isomorphic to thepolynomial ring B [ t , . . . , t m ] modulo the ideal J m generated by the followingelements:1. The products t i c r , with i ∈ { , ..., m } , for every ray r ∈ R that does notbelong to V Λ Gi .2. The polynomials F ( i, A ) , for every pair ( i, A ) with i ∈ { , ..., m } and A ⊂ { , ..., m } such that if j ∈ A then G i ( G j , and for the pairs (0 , A ) where A is such that T j ∈ A G j = ∅ .The isomorphism is given by sending, for every i = 1 , ..., m , t i to the pullback under the projection π i : Y ( X, G ) → X i = Bl ˜ G i X i − of the class of theexceptional divisor in X i .Proof. As a preliminary remark, let us observe that the ideal generated bythe relations in the statement of the theorem, according to Proposition 6.1,does not depend on the choice of the polynomials F ( i, A ). In this proofwe will use the following notation: if a polynomial g is another choice for F ( i, A ) we will write g ∼ F ( i, A ).The proof of the theorem is by induction on the cardinality m of G . Thecase when m = 0 is obvious.Let us now suppose that the statement of the theorem is true for anyprojective model constructed starting from a toric arrangement A ′ in a torus T ′ , and then choosing a good toric variety for A ′ and a well connectedbuilding set with cardinality ≤ m − Y ( X, G m − ). Let us use thenotation of Section 4 and in particular set Y ( X, G m − ) = X m − and Z = G m . Now, in order to get Y ( X, G ) we have to blowup X m − along the propertransform of Z which by Proposition 4.3 is equal to Z u .Since G is a building set for L ′ , we know that Z is the closure of a layer K Γ ,φ ⊂ T , which is a coset with respect to the subtorus H = ∩ χ ∈ Γ Ker ( x χ )of T . Up to translation, we identify K Γ ,φ ⊂ T with H . OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 17
Under this identification we get the arrangement A H in H , given by theconnected components of the intersections A ∩ K Γ ,φ for every A ∈ A . Noticethat X ∗ ( H ) = X ∗ ( T ) / Γ.We know that Z is the H -variety associated to the fan ∆ H , consistingof those cones in ∆ which lie in V Λ Z . From this it is immediate to checkthat Z is a good toric variety for A H . If we denote by L ′ H its correspondingarrangement of subvarieties, we also have, by Proposition 4.2, that H is awell connected building set for L ′ H . Thus since u ≤ m −
1, we can alsoassume that our result holds for H ∗ ( Z u , Z ).To be more precise we can assume that the cohomology ring H ∗ ( X m − , Z )is isomorphic to the polynomial ring B [ t , . . . , t m − ] modulo the ideal J m − generated by(1) The products t i c r , with i ∈ { , ..., m − } , for every ray r ∈ R thatdoes not belong to V Λ Gi .(2) The polynomials F ( i, A ), for every pair ( i, A ) with i ∈ { , ..., m − } and A ⊂ { , ..., m − } such that if j ∈ A then G i ( G j , and for thepairs (0 , A ) where A is such that T j ∈ A G j = ∅ .The isomorphism is given by sending, for every i = 1 , ..., m − t i to thepull back under the projection π i : X m − → X i = Bl ˜ G i X i − of the class ofthe exceptional divisor in X i .As far as Z u is concerned we need to fix some notation.Following what we have done for X and G , for every pair ( i, A ) with i ∈ { , ..., u } , and A ⊂ { , ..., u } such that if j ∈ A then H i ( H j , we definethe polynomial F Z ( i, A ) in H ∗ ( Z, Z )[ z , . . . , z u ], as follows.We consider the set C i = { h | H h ⊆ H i } , and we denote by M the uniqueconnected component of T j ∈ A H j that contains H i (if A = ∅ we put M = Z ).Then we restrict the polynomials P MH i to H ∗ ( Z, Z )[ t ] and we denote theserestrictions by P MH i ,Z . We put: F Z ( i, A ) = P MH i ,Z ( X h ∈ C i − z h ) Y j ∈ A z j . As before we include the pairs (0 , A ) with T j ∈ A H j = ∅ , and we set: F Z (0 , A ) = Y j ∈ A z j . Then, setting B ′ = H ∗ ( Z, Z ), we can assume that cohomology ring H ∗ ( Z u , Z )is isomorphic to the polynomial ring B ′ [ z , . . . , z u ] modulo the ideal S gen-erated by(1) The products z i c r , with i ∈ { , ..., u } , for every ray r ∈ ∆ that doesnot belong to V Λ Hi .(2) The polynomials F Z ( i, A ), for every pair ( i, A ) with i ∈ { , ..., u } and A ⊂ { , ..., u } such that if j ∈ A then H i ( H j , and for thepairs (0 , A ) where A is such that T j ∈ A H j = ∅ . The isomorphism is given by sending, for every i = 1 , ..., u , z i to the pullback under the projection π i : Z u → Z i = Bl ˜ H i Z i − of the class of theexceptional divisor in Z i .Let us now consider the homomorphisms j ∗ : H ∗ ( X m − , Z ) → H ∗ ( Z u , Z )and ι ∗ : H ∗ ( X, Z ) → H ∗ ( Z, Z )induced by the respective inclusions. We now remark that by the discussionin the proof of Proposition 4.3, we get that, denoting by [ t j ] (resp. [ z i ]) theimage of t j (resp. z i ) in H ∗ ( X m − , Z ) (resp. H ∗ ( Z u , Z )), j ∗ ([ t i ]) = ( j = s i [ z i ] if j = s i From this we deduce that j ∗ is surjective and, if we define f : B [ t , . . . , t m − ] → B ′ [ z , . . . , z u ] f ( a ) = ι ∗ ( a ) if a ∈ Bf ( t i ) = ( j = s i z i if j = s i , we get a commutative diagram: B [ t , . . . , t m − ] B ′ [ z , . . . , z u ] H ∗ ( X m − , Z ) H ∗ ( Z u , Z ) fp qj ∗ where p and q are the quotient maps. At this point we can apply Theorem6.1.We deduce that H ∗ ( Y ( X, G , Z ) is isomorphic to B [ t , . . . , t m ] /L where theideal L = ( J m − , t m ker ( q ◦ f ) , P Z u ( − t m )).In order to proceed, we need an explicit description of the generators forthe ideal ker ( q ◦ f ). From the definition of f and our description of therelations for H ∗ ( Z u , Z ) we deduce that ker q ◦ f is generated by(1) The elements c r , for every ray r ∈ ∆ which does not belong to V Λ Z .(2) The elements t j , with 1 ≤ j ≤ m − j / ∈ { s , ..., s u } .(3) The products t s i c r , with i ∈ { , ..., u } , for every ray r ∈ ∆ that doesnot belong to V Λ Hi .(4) For every ( s i , A ) with i ∈ { , ..., u } and A ⊂ { s , ..., s u } such that if s j ∈ A then H i ( H j , the elementsˇ F ( s i , A ) := P MH i ( X h ∈ B si − t h ) Y s j ∈ A t s j , OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 19 where M is the connected component of ∩ s j ∈ A H j that contains H i ,if A = ∅ , G m otherwise.Indeed f ( ˇ F ( s i , A )) = F Z ( i, A ) where A = { j | s j ∈ A } and there-fore it belongs to ker q .(5) The polynomials F (0 , A ) for the pairs (0 , A ) where A ⊂ { s , ..., s u } is such that T s j ∈ A H j = ∅ .Notice that the elements in (1) and (2) generate kerf .We want to show that L is equal to the ideal J m generated by the elementsdescribed in the statement of the theorem. Let us first show that J m ⊂ L .The generators of J m that do not contain t m belong to J m − and thereforeto L .A generator of the form t m c r , for a ray r ∈ R that does not belong to V Λ Z clearly lies in t m ker ( q ◦ f ).Take a generator of the form F ( j, A ) with m ∈ A and j >
0. Set A ′ = A \ { m } . Then F ( j, A ) = t m ( P MG j ( X h ∈ B j − t h ) Y ν ∈ A ′ t ν ) . If there is a ν ∈ A ′ such that ν is not one of the s i ’s, then P MG j ( X h ∈ B j − t h ) Y ν ∈ A ′ t ν ∈ kerf and we are done.Otherwise, set A = { i | s i ∈ A ′ } . Notice that since G j ⊂ Z necessarily G j = G s i for some 1 ≤ i ≤ t , and B j = { h | h = s k , H k ⊆ H i } . We deducethat f ( P MG j ( X h ∈ B j − t h ) Y ν ∈ A ′ t ν ) = F Z ( i, A )and therefore it belongs to ker q . Finally consider F (0 , A ) = Q ν ∈ A t ν , with m ∈ A . If there is a ν ∈ A ′ = A \ { m } such that ν is not one of the s i ’s,then Q ν ∈ A t ν ∈ kerf . Otherwise, set A = { i | s i ∈ A ′ } . We deduce that f ( F (0 , A )) = F Z (0 , A ) ∈ ker q, since \ i ∈ A H i = \ ν ∈ A ′ ( G ν ∩ Z ) = \ ν ∈ A G ν = ∅ . Finally in order to show that also F ( m, ∅ ) ∈ L we need the following wellknown Lemma 7.1.
Let W ⊂ W and W ⊂ W be regular imbeddings with normalbundles N W W and N W W . Set f W = BL W W and let f W denote thedominant transform of W .Then the canonical imbedding f W ⊂ f W is regular and denoting by π theprojection from f W to W ,1. If W ⊂ W N f W f W ∼ = π ∗ N W W ⊗ O ( − E ) where E is the exceptional divisor on f W .
2. If the intersection of W and W is transversal, N f W f W ∼ = π ∗ N W W . Proof.
For 1. see [11], B.6.10. The second part is easy. (cid:3)
By repeated use of this lemma, we easily get that F ( m, ∅ ) = P Z ( − X h ∈ B m t h ) = P Z u ( − t m ) ∈ L so that indeed L ⊇ J m . To finish, we need to see that L ⊆ J m . We first observe that J m − ⊂ J m . Furthermore, we have already seenthat P Z u ( − t m ) = F ( m, ∅ ) ∈ J m . It follows that we need to concentrate onthe generators of ker ( q ◦ f ) multiplied by t m . Following the list given abovewe consider:(1) The elements t m c r , for every ray r ∈ ∆ which does not belong to V Λ Z . These are also generators of J m and there is nothing to prove.(2) The products t m t j , with 1 ≤ j ≤ m −
1, and j is not one of the s i ’s. We notice that G j ∩ G m is either empty, and therefore t m t j = F (0 , { j, m } ) ∈ J m , or each connected component of G j ∩ G m belongsto G . Let G h be one of these components. Then the generator F ( h, { j, m } ) of J m is equal to t m t j since F ( h, { j, m } ) = t m t j P G h G h and P G h G h = 1. This finishes the proof that t m kerf ⊂ L .(3) The products t m t s i c r , with i ∈ { , ..., u } , for every ray r ∈ ∆ thatdoes not belong to V Λ Hi . There are two possibilities. If H i = G s i then V Λ Hi = V Λ Gsi and t s i c r is a generator of J m − .If H i is the transversal intersection of Z and G s i then V Λ Hi = V Λ Gsi ∩ V Λ Z . Therefore if r does not belong to V Λ Hi either it doesnot belong to V Λ Z , and then t m c r is a generator of J m that hasalready been considered in (1), or it does not belong to V Λ Gsi and t s i c r is a generator of J m − .(4) The elements t m ˇ F ( s i , A ), for every pair ( s i , A ) with i ∈ { , ..., u } and A ⊂ { s , ..., s u } such that if s j ∈ A then H i ( H j .If G s i ⊂ G m , that is G s i = H i , then, since M is the connectedcomponent of G m ∩ ( ∩ s j ∈ A G s j ) containing H i , it is clear that t m ˇ F ( s i , A ) = F ( s i , A ∪ { m } ) ∈ J m . Otherwise H i does not belong to G and it is the transversal in-tersection of G s i and G m (see Proposition 3.1), that are its G fac-tors. If A = ∅ , we observe that P XG si is a valid choice for P ZH i soˇ F ( s i , ∅ ) ∼ F ( s i , ∅ ).Therefore ˇ F ( s i , ∅ ) ∈ J m − and t m ˇ F ( s i , ∅ ) ∈ J m .Assume now A = ∅ . We claim that, denoting by M ′ the connectedcomponent of the intersection ∩ s j ∈ A G s j containing G s i , M is thetransversal intersection of M ′ and G m . OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 21
Take any t such that H i ⊆ H t . Then if G s t = H t ⊂ G m , since G m is a G factor of H i this would imply G s t = G m a contradiction.We deduce that G s j * G m for all s j ∈ A , and furthermore M / ∈ G ,since otherwise G m = M ⊂ G s j .A G factor of M ′ is contained in at least one of the G s j , s j ∈ A . Inparticular none of these G factors contains G m . Furthermore since G m is a G factor of H i it is also a G factor of M .It follows that we can apply Corollary 3.1.2 and we conclude that M is the transversal intersection of M ′ and G m as desired. Thus,reasoning as above, we observe that P M ′ G si is a valid choice for P MH i soˇ F ( s i , A ) ∼ F ( s i , A ). Therefore ˇ F ( s i , A ) ∈ J m − and t m ˇ F ( s i , A ) ∈ J m .(5) The products t m F (0 , A ) = t m ( Q s i ∈ A t s i ) for A ⊂ { s , ..., s u } suchthat ∩ s i ∈ A H i = ∅ . In this case G m ∩ ( \ i ∈ A G s i ) = \ i ∈ A H i = ∅ so that t m Q i ∈ A t s i = F (0 , A ∪ { m } ) ∈ J m .Putting everything together we have shown that L ⊂ J m so that L = J m and our claim is proved. (cid:3) A way to choose the polynomials P MG Let us use the same notations ( A , ∆ , X = X ∆ , ... ) as in the precedingsections. We want to show an explicit choice of the polynomials P MG ∈ H ∗ ( X, Z )[ t ] = B [ t ], and therefore of the polynomials F ( i, A ) that appear inTheorem 7.1Let us consider two elements G, M ∈ L ′ with G ⊂ M . We can choose abasis B Λ G = { β , ..., β s } of Λ G such that the first k elements ( k < s ) are abasis of Λ M .We recall that the irreducible divisors in the boundary of X are in corre-spondence with the rays of the fan ∆.In particular, let us consider a maximal cone σ in ∆, whose one di-mensional faces are generated by the rays r , ..., r n (a basis of the lattice hom Z ( X ∗ ( T ) , Z )), and let us denote as usual their corresponding divisorsby D r , ..., D r n .The subvariety G = K Λ G ,φ of X has the following local defining equationsin the chart associated to σ : z h β ,r i · · · z h β ,r n i n = φ ( β ) z h β ,r i · · · z h β ,r n i n = φ ( β ) ......z h β s ,r i · · · z h β s ,r n i n = φ ( β s ) Therefore the subvariety G is described as the intersection of s divisors.The divisor D ( β j ) corresponding to β j has a local function with poles oforder − min (0 , h β j , r i i ) along the divisor D r i , for every i = 1 , ..., n . Thisimplies that in P ic ( X ) we have the following relation:(2) [ D ( β j )] + X r min (0 , h β j , r i )[ D r ] = 0where r varies in the set R of all the rays of ∆.Therefore the polynomial in H ∗ ( X, Z )[ t ] = B [ t ] P XG = s Y j =1 ( t − X r ∈R min (0 , h β j , r i ) c r )where c r is the class of the divisor D r , is a good lifting of the Chern poly-nomial of N G X .At the same way, the polynomial in B [ t ] P XM = k Y j =1 ( t − X r ∈R min (0 , h β j , r i ) c r )is a good lifting of the Chern polynomial of N M X . This implies that thepolynomial P XG P XM = s Y j = k +1 ( t − X r ∈R min (0 , h β j , r i ) c r )restricted to H ∗ ( M, Z )[ t ] is a good lifting of the Chern polynomial of N G ( M ),i.e. it is a choice for the polynomial P MG as requested in Section 6.9. The cohomology of the strata
Let us consider, with the same notation as before ( A , ∆ , R , X = X ∆ , L , L ′ ),a well connected building set G = { G , ..., G m } for L ′ . As we know fromSection 5, the models Y ( X, G ) and Y ( X, G + ) are isomorphic. As in Propo-sition 5.1, we set G + = G ∪ { D r } r ∈R , and for any G ∈ G + we denote by D G its corresponding divisor in Y = Y ( X, G + ).In this section we are going to generalize our main result and explain howto compute the cohomology ring for any variety Y S = T G ∈S D G for anysubset S ∈ G + . Notice that if S is not ( G + )-nested, Y S = ∅ , so that we aregoing to assume that S is nested.We set T S = S ∩ G and D S = S ∩ { D r } r ∈R , so that S is the disjoint unionof T S and D S . Remark that, since S is nested, the rays R S = { r | D r ∈ D S } span a cone in the fan ∆.Fix a pair ( i, A ) with i ∈ { , ..., m } , and A ⊂ { , ..., m } such that if j ∈ A then G i ( G j . Set S i = { h | G h ∈ S and G h ) G i } and consider the set B i = { h | G h ⊆ G i } . Denote by M = M S the unique connected componentof T j ∈ A ∪S i G j that contains G i (if A ∪ S i = ∅ we put M = X ). Then, after OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 23 choosing all the polynomials P MG i as explained in the previous sections, weset: F S ( i, A ) = P MG i ( X h ∈ B i − t h ) Y j ∈ A t j We also set F S (0 , A ) = F (0 , A ). We have Theorem 9.1.
For any nested set
S ⊂ G + , the cohomology ring H ∗ ( Y S , Z ) is isomorphic to the polynomial ring B [ t , . . . , t m ] modulo the ideal J m ( S ) generated by the following elements:1. The classes c r ∈ B for any ray r such that { r } ∪ R S does not span a conein the fan ∆ .2. The products t i c r , with i ∈ { , ..., m } , for every ray r ∈ R that does notbelong to V Λ Gi .3. The polynomials F S ( i, A ) , for every pair ( i, A ) with i ∈ { , ..., m } and A ⊂ { , ..., m } such that if j ∈ A then G i ( G j , and for the pairs (0 , A ) where A is such that \ j ∈ A G j ∩ \ H ∈S H ! = ∅ The image in H ∗ ( Y S , Z ) of the classes c r and t j is just the restriction of thecorresponding classes in H ∗ ( Y ( X, G + ) , Z ) . Proof.
As in the proof of Theorem 7.1 we proceed by induction on m . Thecase m = 0 follows from the well know computation of the cohomology ofstable subvarieties in a complete smooth toric variety ([10]). So we take G + m − = G + \ G m which, by Theorem 5.1, is a building set. Furthermore weremark that the nested sets in G + m − coincide with the nested sets in G + notcontaining G m .We set for any G ∈ G + m − , D ′ G equal to the divisor corresponding to G in Y ′ = Y ( G m − , X ) and for S nested in G + m − , Y ′S = ∩ G ∈S D ′ G .Let us take a nested set S for G + and, as usual, put G m = Z .Assume G m / ∈ S . If S ∪ { G m } is not nested, then Y ′S ∩ ˜ Z = ∅ , sothat Y ′S = Y S . In particular t m is in the kernel of the restriction map H ∗ ( Y ( G , X )) → H ∗ ( Y S ). Now t m = F (0 , { m } ) is one of our relations andall the other relations different from F S ( m, ∅ ) either are divisible by t m orthey already appear among the relations for H ∗ ( Y ′S ). As for F S ( m, ∅ ), thiscoincides with F ( m, ∅ ), therefore it is already equal to 0 in H ∗ ( Y ( X, G )) sothere is nothing to prove.If S ∪ { G m } is nested then the intersection N = Y ′S ∩ ˜ Z is transversal,so that Y S is the blow up of Y ′S along N . Now N is just the transversalintersection of the divisors D ′ G i ∩ ˜ Z in ˜ Z then again we can use our inductivehypothesis exactly as in the proof of Theorem 7.1.Finally if G m ∈ S , setting S ′ = S \ { G m } , we deduce that Y S is theblow up of Y ′S ′ along the (necessarily transversal) intersection Y ′S ′ ∩ ˜ Z . Thus again everything follows from our inductive assumption and the nature ofthe relations. Remark 9.1.
We notice that the arguments used in the proof above and inthe proof of Theorem 7.1 can be applied almost verbatim to the case of pro-jective wonderful models of a subspace arrangement in P ( C n ) . Everything inthis case is simpler: all the building sets are well connected, the polynomials P MG ( t ) are powers of t and the initial projective variety is P ( C n ) . One finallygets, with a shorter proof, the same presentation by generators and relationsof Theorem 5.2 in [5] . (cid:3) References [1]
Adiprasito, K., Huh, J., and Katz, E.
Hodge theory for combinatorial geometries. arXiv:1511.02888 (2015).[2]
Callegaro, F., Gaiffi, G., and Lochak, P.
Divisorial inertia and central elementsin braid groups.
Journal of Algebra 457 (2016), 26 – 44.[3]
De Concini, C., and Gaiffi, G.
Projective wonderful models for toric arrangements.
Advances in Mathematics https://doi.org/10.1016/j.aim.2017.06.019 (2017).[4]
De Concini, C., and Procesi, C.
Hyperplane arrangements and holonomy equa-tions.
Selecta Mathematica 1 (1995), 495–535.[5]
De Concini, C., and Procesi, C.
Wonderful models of subspace arrangements.
Selecta Mathematica 1 (1995), 459–494.[6]
Denham, G. C., and Suciu, A. I.
Local systems on arrangements of smooth, com-plex algebraic hypersurfaces.
ArXiv e-prints (June 2017).[7]
Etingof, P., Henriques, A., Kamnitzer, J., and Rains, E.
The cohomology ringof the real locus of the moduli space of stable curves of genus 0 with marked points.
Annals of Math. 171 (2010), 731–777.[8]
Feichtner, E.
De Concini-Procesi arrangement models - a discrete geometer’s pointof view.
Combinatorial and Computational Geometry, J.E. Goodman, J. Pach, E.Welzl, eds; MSRI Publications 52, Cambridge University Press (2005), 333–360.[9]
Feichtner, E., and Yuzvinsky, S.
Chow rings of toric varieties defined by atomiclattices.
Invent. math. 155 , 3 (2004), 515–536.[10]
Fulton, W.
Introduction to toric varieties , vol. 131 of
Annals of Mathematics Stud-ies . Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lecturesin Geometry.[11]
Fulton, W.
Intersection theory , second ed., vol. 2 of
Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Re-sults in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys inMathematics] . Springer-Verlag, Berlin, 1998.[12]
Gaiffi, G.
Blow ups and cohomology bases for De Concini-Procesi models of sub-space arrangements.
Selecta Mathematica 3 (1997), 315–333.[13]
Gaiffi, G.
Permutonestohedra.
Journal of Algebraic Combinatorics 41 (2015), 125–155.[14]
Henderson, A.
Representations of wreath products on cohomology of De Concini-Procesi compactifications.
Int. Math. Res. Not. , 20 (2004), 983–1021.[15]
Keel, S.
Intersection theory of moduli space of stable n -pointed curves of genus zero. Trans. Amer. Math. Soc. 330 , 2 (1992), 545–574.[16]
Li, L.
Chow motive of Fulton-MacPherson configuration spaces and wonderful com-pactifications.
Michigan Math. J. 58 , 2 (2009), 565–598.
OHOMOLOGY RINGS OF COMPACTIFICATIONS OF TORIC ARRANGEMENTS 25 [17]
Li, L.
Wonderful compactification of an arrangement of subvarieties.
Michigan Math.J. 58 (2009), 535–563.[18]
Moci, L.
Wonderful models for toric arrangements.
Int. Math. Res. Not. , 1 (2012),213–238.[19]
Rains, E.
The homology of real subspace arrangements.
J Topology 3 (4) (2010),786–818.[20]
Yuzvinsky, S.
Cohomology bases for De Concini-Procesi models of hyperplane ar-rangements and sums over trees.