A differential algebra and the homotopy type of the complement of a toric arrangement
aa r X i v : . [ m a t h . A T ] J u l A DIFFERENTIAL ALGEBRA AND THE HOMOTOPY TYPE OF THECOMPLEMENT OF A TORIC ARRANGEMENT
CORRADO DE CONCINI, GIOVANNI GAIFFI
Abstract.
We show that the rational homotopy type of the complement of a toric ar-rangement is completely determined by two sets of combinatorial data. This is obtained byintroducing a differential graded algebra over Q whose minimal model is equivalent to theSullivan minimal model of A . Introduction
Let T ≃ G nm be a complex n dimensional algebraic torus and let us denote by X ∗ ( T ) ≃ Z n its character group.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 homomorphism.A toric arrangement A is given by a 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 .We will denote by M ( A ) the complement T − S i K i of the arrangement. We notice that ifwe consider the saturation ˜ A of A , i.e. the arrangement consisting of all the layers which areobtained as connected components of intersections of layers in A , we have M ( A ) = M ( ˜ A ) .The purpose of this note is to show that the rational homotopy type of M ( A ) is completelydetermined by(1) The partially ordered set ˜ A ordered by reverse inclusion.(2) The set of lattices Γ ⊂ X ∗ ( T ) for K Γ ,φ ∈ ˜ A .We will call these data the combinatorial data of A .This is obtained by introducing an object that may be of independent interest: a differentialgraded algebra over Q , defined using the combinatorial data of A , whose minimal model isequivalent to the Sullivan minimal model of A . In particular we have that its cohomology isisomorphic to the rational cohomology of M ( A ) .Before giving a sketch of our construction, we recall some previous results on this subject.As far as we are aware, the results regarding the (rational) homotopy of M ( A ) have beenobtained in the divisorial case. In this case, in [9] De Concini and Procesi determined thegenerators of the rational cohomology modules of M ( A ) , as well as the ring structure in thecase of totally unimodular arrangements. By a rather general approach, Dupont in [10] provedthe rational formality of M ( A ) . In turn, in [2], it was shown extending the results in [3, 4]and [19] that the data needed in order to state the presentation of the rational cohomologyring of M ( A ) is fully encoded in the partially ordered set ˜ A . It follows that the combinatoricsof the poset ˜ A determines the rational homotopy of M ( A ) .Our approach to the study of the rational homotopy type and of the cohomology of M ( A ) in full generality involves the construction of projective wonderful models for M ( A ) . Results analogous to those in this note were previously obtained in [8] for arrangements oflinear subspaces in a projective space using the construction of wonderful models for subspacearrangements and the fundamental results of Morgan in [18]. In particular Morgan introduced,in the case of a compactification V of a complex algebraic variety X such that V \ X = D is a divisor with normal crossing, a differential graded algebra (which we are going to calla Morgan differential algebra) whose minimal model is equivalent to the Sullivan minimalmodel of X .In the toric case the idea is to do similar considerations using the projective wonderfulmodels of M ( A ) we constructed in [5] and the presentation by generators and relations of theinteger cohomology rings of these models and of the strata in their boundary given in [6].Indeed in [16] for each of these models these presentations were used to describe its Morgandifferential algebra which determines the rational homotopy type of M ( A ) .However the projective models described in [5] do not depend only on the combinatorialdata of the toric arrangement A but also on some extra choices in the construction processand indeed the differential Morgan algebras one obtains also depend on these choices.To overcome this problem we are going to construct a new differential graded algebra as adirect limit of the differential Morgan algebras of the projective wonderful models describedin [5]. This algebra, based on the notion of the ring of conditions of T , is rather large and itis not the Morgan algebra of any compactification of M ( A ) .However we show in Proposition 5.4, that it is quasi isomorphic to any of the Morganalgebras of the projective wonderful models of M ( A ) and it has a simple presentation whichdepends only on the combinatorial data of A .Let us describe more in detail the structure of this paper. First (in Section 2) we brieflyprovide a self contained presentation of the ring of conditions C ( T ) of the torus T . We recallthat it was shown by Fulton and Sturmfels in [11] that this ring over Q is isomorphic to theMcMullen polytope algebra (see [15], and [17] for a similar construction) and is the directlimit of the rational Chow rings of all the compactifications of T . For other descriptions ofthe ring of conditions of the torus the reader can see for instance [1], [13] (and [7] where ringsof conditions appeared in a more general setting).In Section 2 we first introduce an equivariant version B T ( T ) of the ring of conditions asfollows. Let us consider the lattice of one parameter subgroups X ∗ ( T ) = hom( X ∗ ( T ) , Z ) andthe vector space V = X ∗ ( T ) ⊗ Z R . We denote by Σ the space of the continuous functions f on V such that f ( X ∗ ( T )) ⊂ Q and there exists a smooth projective fan such that f restricted toevery face of this fan is linear. Then we consider the algebra B T ( T ) of continuous functions V generated by Σ and finally we obtain C ( T ) as the quotient of B T ( T ) modulo the idealgenerated by a basis of X ∗ ( T ) .Then in Section 3 we construct a differential graded algebra C whose degree 0 term is C ( T ) .This algebra is the direct limit of all the differential graded Morgan algebras associated tothe compactifications of T .So far the toric arrangement A has not been taken into account. It appears in Section4, where we first recall from [5] the construction of the projective wonderful model Y ( X F ) associated to ˜ A and to a suitable smooth projective fan F . Then we recall from [6] thepresentation of the cohomology of Y ( X F ) and of its strata in the boundary and we construct,following Moci and Pagaria (see [16]), the Morgan differential algebra N F for Y ( X F ) .Finally, in Section 5 we introduce the differential graded algebra N as a direct limit ofthe algebras N F and we present it by generators and relations (see Theorem 5.2) startingfrom C ⊗ B , where C is the limit algebra mentioned above and B is a quotient of a Weyl HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 3 algebra. We immediately obtain Proposition 5.4 (the minimal model of N is isomorphic tothe minimal model of M ( A ) ) and, since the generators and relations of N depend only onthe combinatorial data of A , we deduce Theorem 5.5 on the rational homotopy type.2. The ring of conditions, recollections
Let T ≃ ( C ∗ ) n be a complex n dimensional algebraic torus. Denote by X ∗ ( T ) ≃ Z n itscharacter group and by X ∗ ( T ) = hom( X ∗ ( T ) , Z ) its lattice of one parameter subgroups. Weset V = X ∗ ( T ) ⊗ Z R .We take a rational smooth projective fan F in V and let Γ F = { c , . . . c N } be vertices, thatis the set of primitive vectors in the one dimensional cones (rays) of F . Any cone C ∈ F isof the form C = C ( c i , . . . c i k ) = { v = k X r =1 a r c i r | a r ≥ } with c i , . . . c i k the basis of a split direct summand in X ∗ ( T ) .Let us take variables x c , c ∈ Γ F , and in the polynomial ring Q [ x c ] | c ∈ Γ F take the ideal I F generated by the monomials m J = Q c ∈ J x c for all subsets of J ⊂ Γ F for which C ( c j ) j ∈ J isnot a cone in F . Take the algebra A F := Q [ x c , . . . x c N ] /I F . A F is the Stanley-Reisner ring of F and it is the equivariant cohomology ring of the toric variety X F corresponding to F (see[1] Corollary 1.3 and Proposition 2.2). The algebra A F inherits a grading from the gradingof Q [ x c , . . . x c N ] in which deg x c = 2 for all c ∈ Γ F . The degree 2 part is spanned by theclasses of the elements x c and we denote it by S F . We may associate to each x c the function s c on V defined as follows. We set s c ( d ) = δ c,d . Then for v ∈ V there exist a unique cone C = C ( c i , . . . c i k ) ∈ F such that v lies in the relative interior of C , so v = P kr =1 a r c i r , a r > .We then set s c ( v ) = ( if c = c i r ∀ ra r if c = c i r Notice that s c is continuous so that sending x c to s c we get a homomorphism ρ F : Q [ x c , . . . x c N ] → C ( V ) , where C ( V ) is the algebra of continuous functions on V .The functions s c are linearly independent in C ( V ) . Their span will be identified with S F and denoted by the same letter.The following result is well known and we prove it for completeness. Proposition 2.1. (1) The space S F ⊂ C ( V ) is the space of continuous functions on V with the property that their restriction to each cone of F is linear.(2) The ideal I F is the kernel of ρ F . In particular we obtain an inclusion µ F : A F → C ( V ) . (3) Let G be a smooth refinement of F , that is every cone in F is subdivided by cones in G . We know that there is a map γ FG : A F → A G . Then (2.1) µ F = µ G γ FG . CORRADO DE CONCINI, GIOVANNI GAIFFI
Proof.
The first claim is clear since it is immediate to check that any function f ∈ C ( V ) whose restriction to each cone of F is linear can written as f = X c f ( c ) s c . To see the second claim recall that for any cone C ∈ F , its star S ( C ) consists of conesin F having C as a face. The function s c is clearly supported on S ( c ) (for brevity we write S ( c ) instead than S ( C ( c )) ). From this it follows that for any monomial m = x h c i · · · x h is c is thesupport of ρ F ( m ) = s h c i · · · s h is c is is S ( C ) if C = C ( c i , . . . , c i s ) ∈ F while ρ F ( m ) = 0 otherwise.We deduce that I F ⊂ ker( ρ F ) . In particular we obtain a homomorphism µ F : A F → C ( V ) . Thus for a monomial m = x h c i · · · x h is c is with C = C ( c i , . . . , c i s ) ∈ F , if C ′ = C ( c j , . . . , c j n ) is a cone of maximal dimension, µ F ( mx c j · · · x c jn ) is supported on C ′ if C is a face of C ′ ,while µ F ( mx c j · · · x c jn ) = 0 otherwise.Take now a polynomial P ( x c j , · · · , x c jn ) . The restriction of µ F ( P ( x c j , · · · , x c jn )) to C ′ isjust the evaluation of P ( x c j , · · · , x c jn ) hence it is zero if and only if P ( x c j , · · · , x c jn ) ≡ .Take a ∈ ker µ F . If a = 0 there is an n − dimensional cone C = C ( c i , . . . , c i n ) such that b = ax c i · · · x c in = 0 . Then µ F ( b ) is the restriction of a polynomial P ( x c i , · · · , x c in ) to C .Hence it is zero if and only if b = 0 . A contradiction.The last statement follows since, if we denote by y d the variable corresponding to a vertex d of G , for any vertex c of F (2.2) γ FG ( x c ) = X d vertex of G ( ρ F ( x c )( d )) y d . as the reader can easily verify. (cid:3) We now take a suitable algebra of continuous functions on V Definition 2.1. (1) The space Σ consists of the functions f ∈ C ( V ) such that(a) If λ ∈ X ∗ ( T ) , f ( λ ) ∈ Q .(b) There exists a rational smooth projective fan F such that for any C ∈ F therestriction of f to C is linear.(2) The equivariant ring of conditions B T ( T ) is the Q subalgebra of the ring of continuousfunctions on V generated by Σ .In this paper we will always consider rational fans, so from now on the adjective ‘rational’will be omitted. Since any two smooth projective fans admit a common refinement which isstill smooth and projective, it is clear that Σ is a Q -vector space.Let us order the set of F of smooth projective fans using refinement. We get a directedsystem ( A F , γ FG ) . By Proposition 2.1 we deduce that B T ( T ) is the union of the images of thehomomorphisms µ F and we deduce (see [1]): Proposition 2.2. B T ( T ) = lim −→ F A F . Each element ℓ ∈ X ∗ ( T ) is a linear function on V taking integral values on X ∗ ( T ) which,in terms of classes of the elements x c i , is the class of N X i =1 h ℓ, c i i x c i . HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 5 (we are taking into account the identification of S F with ρ F ( S F ) ). In this way if we take abasis ξ , . . . , ξ n of X ∗ ( T ) and set R := Q [ ξ , . . . , ξ n ] , A F is a free R module (see [1] Corollary1.3 and Proposition 2.2.) and we may consider the quotient algebra B F := A F / ( ξ , . . . , ξ n ) ≃ H ∗ ( X F , Q ) . It is clear that the γ FG induces an algebra homomorphism γ FG : B F → B G . Definition 2.2.
The ring of conditions for T is the algebra C ( T ) = lim −→ F B F = B T ( T ) / ( ξ , . . . , ξ n ) . Two differential graded algebras
We now want to define some differential graded algebras (DGA). Again we take a projectivesmooth fan F in V .We start with the algebra Q [ x c ] ⊗ V ( τ c ) , c ∈ Γ F . We define a bigrading on this algebra bysetting deg x c = (2 , , deg τ c = (0 , . In general all the differential graded algebras we aregoing to consider will be easily seen to be bigraded, so we will often omit to specify how theirbigrading is defined. Definition 3.1. (1) The algebra D F is the quotient of the algebra Q [ x c ] ⊗ V ( τ c ) modulothe bigraded ideal J F generated by “the square free monomials" x c i · · · x c ih τ c j · · · τ c jk for each sequence of vertices c i , . . . , c i h , c j , . . . , c j k not spanning a cone in F .(2) The differential d is the unique derivation on D F defined by d ( x c ) = 0 , d ( τ c ) = x c . Remark that d preserves the relations in D F and hence it is well defined and of degree 1.It is easily seen that if for any cone C = C ( c i , . . . , c i k ) ∈ F , we take the algebra A C, F := Q [ x c , . . . x c N ] / [ I F : ( x c i · · · x c ik )] , we have D F = ⊕ C ∈F A C, F τ c i · · · τ c ik . and setting for each m = 0 , . . . , nD m, F = ⊕ C = C ( c i ,...,c im ) A C, F τ c i · · · τ c im , the decomposition D F = ⊕ nm =0 D m, F . Proposition 3.1. H i ( D F , d ) = ( Q if i = 00 if i > . Proof.
Let us consider the complex ( D + F , d ) of elements of positive degree. We need to provethat this complex is exact. Let us define define a map of degree − S : D + F → D + F and show that Sd + dS is the identity. This will give our claim.For this let fix a total order c , . . . , c N of the vertices of F . CORRADO DE CONCINI, GIOVANNI GAIFFI
Below, for brevity, we will write x i s , τ i s instead than x c is , τ c is .Let m = x h i · · · x h s i s τ j · · · τ j r ∈ D + F , with j < j · · · < j r and if s > , h i > . If s = 0 (resp. r = 0 ) we set i = ∞ (resp. j = ∞ ). Notice that r + s > and set f = min( i , j ) . Wedefine S ( m ) = ( if f = j x h − i · · · x h N N τ i τ j · · · τ j r if f = i < j Now let us compute ( dS + Sd )( m ) . If f = j we have dS ( m ) = 0 and Sd ( m ) = S ( x j x h i · · · x h s i s τ j · · · τ j r + r X ℓ =2 ( − ℓ +1 x j ℓ x h i · · · x h s i s τ j · · · ˇ τ j ℓ · · · τ j r ) = m. If f = i < j , one easily sees that dS ( m ) = m − r X ℓ =1 ( − ℓ +1 x j ℓ x h − i x h i · · · x h s i s τ i τ j · · · ˇ τ j ℓ · · · τ j r = m − Sd ( m ) and everything follows. (cid:3) When the fan G is a (smooth, projective) refinement of F , we want to compare the algebras D F and D G .As before in order to avoid confusion for d ∈ Γ G we denote by y d and υ d the correspondingeven and odd variables. We define a homomorphism ξ FG : Q [ x c ] ⊗ ^ ( τ c ) → Q [ y d ] ⊗ ^ ( υ d ) by setting(3.1) ξ FG ( x c ) = X d vertex of G ( ρ F ( x c )( d )) y d . (3.2) ξ FG ( τ c ) = X d vertex of G ( ρ F ( x c )( d )) υ d . We then set ξ FG = q ◦ ξ FG : Q [ x c ] ⊗ ^ ( τ c ) → D G ,q being the quotient modulo J G . We then have Proposition 3.2. ξ FG ( J F ) = 0 . It follows that ξ FG factors through a homomorphism of dif-ferential graded algebras ζ FG : D F → D G . Proof.
Let us take a monomial x c i · · · x c ih ∈ J F that is c i , . . . , c i h do not span a cone in F .We know by Proposition 2.1 that ξ FG ( x c i · · · x c ih ) = 0 .Now notice that ξ FG ( x c ) is a linear combination with non negative coefficients of the y d .We deduce that ξ FG ( x c i · · · x c ih ) is a linear combination with non negative coefficients ofmonomials in the y d . Thus each monomial appearing with non zero coefficient has to lie in J G . Necessarily if in any such monomial we substitute some of the y d ’s with the corresponding υ d ’s we also get a relation in D G . HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 7
This immediately implies that for any h > , ξ FG ( τ c i · · · τ c ih ) ∈ J G and for any ≤ ℓ ≤ h , ξ FG ( x c i · · · x c iℓ τ c iℓ +1 · · · τ c ih ) ∈ J G .The fact that d ◦ ζ FG = ζ FG ◦ d then follows from the definitions. (cid:3) Notice now that it is clear that ζ FG is a quasi isomorphism so that setting ( D, d ) = lim −→ F ( D F , d ) we deduce that H i ( D, d ) = ( Q if i = 00 if i > . Finally, let us remark that ζ FG ( D m, F ) ⊂ D m, G for each m = 0 , . . . n . It follows that, takingthe limit, D m = lim −→ F ( D m, F ) we get a direct sum decomposition D = M m D m . In particular for m = 0 , D = B T ( T ) . We denote by ν F : D F → D the natural morphism.We want to give a more explicit description of the algebra D . In order to do so let us takethe exterior algebra V (Σ) . Define the algebra E = B T ( T ) ⊗ V (Σ) /H , where H is the idealgenerated by the elements s · · · s t ⊗ σ · · · σ r , with s i , σ j ∈ Σ , such that s · · · s t σ · · · σ r = 0 in B T ( T ) . Notice that the natural differential d on B T ( T ) ⊗ V (Σ) defined by setting d ( a ⊗ s ) = as ⊗ ∈ B T ( T ) and extended as an algebra derivation, clearly preserves H . It follow that weget a differential on E. We claim,
Proposition 3.3. E ≃ D as differential graded algebras.Proof. For our usual smooth projective fan F , we have already defined a map µ F : A F → B T ( T ) , with the property that µ F ( x c ) = s c for any ray c of F , which gives an inclusion of S F into Σ and hence a map V ( S F ) → V (Σ) . Tensoring, we obtain a map A F ⊗ ^ ( S F ) → B T ( T ) ⊗ ^ (Σ) and composing with the quotient, a map A F ⊗ ^ ( S F ) → E By the very definition of D F we deduce that this map factors through a map ν F : D F → E Passing to the limit and recalling that Σ is spanned by the functions s c for some ray in asuitable fan, we get a surjective map µ : D → E On the other hand, if we take an element s · · · s t ⊗ σ · · · σ r , we can find a fan F with therequired properties, such that each s i = µ F ( x i ) and each σ j = µ F ( τ j ) with x i , τ j ∈ S F \ { } .Thus we can map s · · · s t ⊗ σ · · · σ r to ν F ( x · · · x t ⊗ τ · · · τ r ) . In order to see that this mapis well defined we just have to show that if the function s · · · s t σ · · · σ r = 0 in C ( V ) , theelement x · · · x t τ · · · τ r = 0 in A F .If we write each x i and each τ j as a linear combination of the basis elements x c , c a ray of F , of S F we get that x · · · x t τ · · · τ r is the image of a polynomial P ( x c ) ∈ Q [ x c ] which is aproduct of non zero linear functions and hence non zero. We deduce that if we write P ( x c ) CORRADO DE CONCINI, GIOVANNI GAIFFI as a linear combination of monomials and we compute it as a function on V , we get if andonly if each monomial appearing with non zero coefficient in P ( x c ) is zero in A F hence theelement x · · · x t ⊗ τ · · · τ r = 0 in A F .It follows that we get a map E → D and it is immediate to check that this map is theinverse of µ . (cid:3) For every cone C ∈ F , A C, F is a R -module. Again by Corollary 1.3 and Proposition 2.2.in [1], A C, F is free of rank equal to the number of n dimensional cones in the star S ( C ) of C .Furthermore A C, F is isomorphic to the T -equivariant cohomology of the closure X C, F of the T -orbit associated to the cone C ∈ F and the quotient algebra B C, F := A F / ( ξ , . . . , ξ n ) ≃ H ∗ ( X F , Q ) . From this we deduce in particular that D F is a free R module and, setting by abuse ofnotation, ξ i := ξ i ⊗ , for each i = 1 , . . . , n , we may consider the quotient algebra C F := D F / ( ξ , . . . , ξ n ) . Since each element ξ j is a cocycle, we deduce that the ideal ( ξ , . . . , ξ n ) is preserved by thedifferential d and we have an induced differential on C F which we shall denote by the sameletter.Notice that if we set in D F ψ j = N X i =1 h ℓ j , c i i τ c i . in C F we get that d ( ψ j ) = 0 so that we obtain an inclusion of the exterior algebra V ( ψ , . . . , ψ n ) into the subalgebra Z ( C F ) of cocycles and a degree preserving homomorphism j F : H ∗ ( T ) → H ∗ ( C F ) , defined by setting j F ( ℓ j ) = ψ j .We have Proposition 3.4.
The homomorphism j F is an isomorphism.Proof. We shall deduce this by induction from a slightly more general fact. For any ≤ h ≤ n consider C ( h ) F = ( D F if h = 0 D F / ( ξ , . . . , ξ h ) if h > . Our claim is that for every h , H ∗ ( C ( h ) F ) ≃ V ( ψ , . . . , ψ h ) .For h = 0 , C ( h ) F = D F and our claim is Proposition 3.1.We proceed by induction on h and assume the claim proved for h − . By reasoning asabove we deduce that ψ h is a cocycle in Z ( C ( h ) F ) and hence gives a class in H ( C ( h ) F ) .Since clearly ξ h is a non zero divisor in C ( h − F we take the exact sequence(3.3) → C ( h − F [ − ◦ ξ h −−−−→ C ( h − F −−−−→ C ( h ) F → . the corresponding long exact sequence in cohomology and using the ξ h being a coboundaryinduces the trivial homomorphism in cohomology, we deduce the exact sequence(3.4) → H ∗ ( C ( h − F ) → H ∗ ( C ( h ) F ) → H ∗− ( C ( h − F ) ψ h → . Since by induction H ∗ ( C ( h − F ) ≃ V ( ψ , . . . , ψ h − ) , this implies our claim. (cid:3) HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 9
We finish by remarking that clearly the map ζ FG is a map of R -modules, so it induces amap χ FG : C F → C G and we may also consider(3.5) C = D/ ( ξ , . . . , ξ n ) = lim −→ F C F . It is immediate to see that C inherits a direct sum decomposition C = M m C m , with C m = lim −→ F C m, F = lim −→ F D m, F / ( ξ , . . . , ξ n ) . In particular for m = 0 C = C ( T ) = B T ( T ) / ( ξ , . . . , ξ n ) is the ring of conditions of the torus T .Furthermore the χ FG are quasi isomorphism and we deduce that also for C we have H ∗ ( C , d ) ≃ ^ ( ψ , . . . , ψ n ) ≃ H ∗ ( T ) . Toric arrangements
Let us now recall from the Introduction the definition of a toric arrangement. 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 ) ∼ = Z n and φ : Γ → C ∗ is a homomorphism.A toric arrangement A is given by finite set of layers A = {K , ..., K m } in T .In [5] it is shown how to construct projective wonderful models for the complement M ( A ) = T − S i K i .A projective wonderful model is a smooth projective variety containing M ( A ) as an openset and such that the complement of M ( A ) is a divisor with normal crossings and smoothirreducible components. As we mentioned in the Introduction, we have M ( A ) = M ( ˜ A ) ,where ˜ A is the saturation of A , i.e. the arrangement consisting of all the layers which areobtained as connected components of intersections of layers in A .Therefore from now on, for brevity of notation, we are going to assume A = ˜ A .Let us put V = hom Z ( X ∗ ( T ) , R ) = X ∗ ( T ) ⊗ Z R . A layer K Γ ,φ is a coset with respect tothe torus T Γ = ∩ χ ∈ Γ Ker ( e πıχ ) , and we can consider the subspace V Γ = { v ∈ V | h χ, v i = 0 , ∀ χ ∈ Γ } . Since X ∗ ( T Γ ) = X ∗ ( T ) / Γ , V Γ is naturally isomorphic to hom Z ( X ∗ ( T Γ ) , R ) = X ∗ ( T Γ ) ⊗ Z R . Definition 4.1.
Let F be a fan in V . A finite set { χ , . . . , χ s } of vectors in X ∗ ( T ) is said tohave equal sign with respect to F if for each i = 1 , . . . , s and each cone C ∈ F , the function h χ i , −i has constant sign on C , i.e. it is either non negative or non positive on C .In [5] (see Proposition 6.1) it was shown how to construct a projective smooth T -embedding X F whose fan F in V has the following property. For every Γ i there is an integral basis of Γ i , χ , . . . , χ s , which has equal sign with respect to F . The basis χ , . . . , χ s is called an equalsign basis for Γ i .In fact by the same proof one can even show that one can construct F such that for anypair of layers K Γ ,φ ⊂ K Γ ′ ,ψ ∈ A , there is an equal sign basis for Γ whose intersection with Γ ′ is an equal sign basis for Γ ′ . In view of this we define
Definition 4.2.
Let A be a toric toric arrangement. A smooth projective fan F is compatiblewith A if for any pair of layers K Γ ,φ ⊂ K Γ ′ ,ψ ∈ A , there is an equal sign basis for Γ whoseintersection with Γ ′ is an equal sign basis for Γ ′ .In what follows we are always going to consider fans F compatible with A . Once such F has been constructed, the strategy used in [5] is to first embed the torus T in X F .In such a toric variety X F consider the closure K Γ ,φ of a layer. This closure turns out tobe a toric variety, whose explicit description is provided by [5]. Theorem 4.1 (Proposition 3.1 and Theorem 3.1 in [5]) . For every layer K Γ ,φ , let T Γ be thecorresponding subtorus and let V Γ = { v ∈ V | h χ, v i = 0 , ∀ χ ∈ Γ } . Then,(1) For every cone C ∈ F , its relative interior is either entirely contained in V Γ or disjointfrom V Γ .(2) The collection of cones C ∈ F which are contained in V Γ is a smooth fan F Γ .(3) K Γ ,φ is a smooth T Γ -variety whose fan is F Γ .(4) Let O be a T orbit in X := X F and let C O ∈ F 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 T Γ orbit in K Γ ,φ corresponding to C O ∈ F Γ . Let us denote by Q ′ (resp. Q ) the set whose elements are the subvarieties K Γ i ,φ i of X F (resp. the subvarieties K Γ i ,φ i and the irreducible components of the complement X F − T ).We then denote by L ′ (resp. L ) the poset made by all the connected components of all theintersections of some of the elements of Q ′ (resp. Q ). In [5] (Theorem 7.1) we have shownthat the family L is an arrangement of subvarieties in X F in the sense of Li’s paper [14]. Asa consequence also L ′ , being contained in L and closed under intersection, is an arrangementof subvarieties.Let L ′ = { G , ..., G m } , ordered in such a way that if G i ( G j then i < j . Thus for each i = 1 , . . . , m we have G i = K Γ i ,φ i for a suitable pair (Γ i , φ i ) .A this point, following Li’s construction for L ′ we construct the variety Y ( X F ) , which isa projective wonderful model for M ( A ) = X F − S A ∈L A. This means that Y ( X F ) contains M ( A ) as a dense open set whose complement is a divisor with smooth irreducible componentshaving transversal intersections.More in detail we choose L ′ as a building set (see Definition 2.5 in [6]). Then we obtain Y ( X F ) starting from X F and blowing up the elements of L ′ (after the first step, their trans-forms) in any order such that if G i ⊂ G i we blow up (the transform of ) G i before (thetransform of ) G i . In particular we notice that the ordering we chose in L ′ is one of theadmissible orderings to perform these blowups.In Proposition 5.2 of [6] we observed that Y ( X F ) is isomorphic to the variety Y + ( X F ) obtained by choosing as a building set the set L + = L ′ ∪ { D c i } i =1 ,...,N where for every vertex c i of F , D c i is the associated irreducible divisor in the boundary of X F . The isomorphism isan immediate consequence of the fact that the D c i ’s (and hence their transforms in X F ) aredivisors.From Theorem 1.2 in [14] it follows that Y + ( X F ) \ M ( A ) is a divisor with normal crossingswhose irreducible components are smooth and indexed by L + . For j = 1 , . . . m , we denote by D G j the component of Y + ( X F ) \ M ( A ) corresponding to G i and, by abuse of notation, westill denote by D c i its transform in Y + ( X F ) , so that Y + ( X F ) \ M ( A ) = ( ∪ mj =1 D G j ) ∪ ( ∪ c i D c i ) . HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 11
It follows from the theory of torus embeddings that for a collection of rays c i , . . . c i t theintersection ∩ th =1 D c ih is non empty if and only if C = C ( c i , . . . , c i t ) is a cone in F .Furthermore from the general definition of nested set (see Definition 5.6 of [14] and alsoDefinition 2.7 in [6]), one can easily check that in our special situation, if we take a subset G = { G j , G j , . . . , G j s } of L ′ and a cone C = C ( c i , . . . c i t ) , the intersection Y ( G,C ) := ( ∩ sk =1 D G jk ) ∩ ( ∩ th =1 D c ih ) is non empty if and only if G j ( G j ( · · · ( G j s and C ⊂ V G j . In this case Y ( G,C ) issmooth and irreducible. Remark 4.2.
From now on we will identify Y ( X F ) and Y + ( X F ) . In [6] we have described the cohomology ring H ∗ ( Y ( X F ) , Z ) by generators and relationsin a greater generality. Here we shall illustrate this result under our assumption, leaving thestraightforward translation to the reader. We refer to [6] for the geometric explanation of ourrelations.To simplify notation we are going to add to L ′ the element G m +1 := X F . We need tointroduce certain polynomials in B F [ t , . . . , t m ] .Take a pair ( i, j ) with i ∈ { , ..., m } , and j ∈ { , ..., m + 1 } in such a way that G i ( G j .Consider the set B i = { h | G h ⊆ G i } .Take an equal sign basis χ of Γ i whose intersection with Γ j (if j = m + 1 , Γ m +1 = { } ) isa basis of Γ j . We then set P G j G i ( t ) := Y χ ∈ χ \ ( χ ∩ Γ j ) ( t − χ −F ) ∈ B F [ t ] with χ −F = X c ray min(0 , h χ, c i ) x c , and F ( i, j ) = P G j G i ( X h ∈ B i − t h ) t j , with t m +1 := 1 .From [6] we easily get Theorem 4.3 (Proposition 6.3 and Theorem 7.1 in [6]) . Let I be the ideal in B F [ t , . . . , t m ] generated by(1) the products t i x c for every ray c ∈ F that does not belong to V Γ i .(2) the products t s t r if G s and G r are not comparable.(3) the polynomials F ( i, j ) , for G i ( G j .Then(i) I does not depend on the choice of the polynomials F ( i, j ) .(ii) The cohomology ring H ∗ ( Y ( X F ) , Q ) is isomorphic to B F [ t , . . . , t m ] /I . More generally one can compute the cohomology algebra of every stratum Y ( G,C ) of Y ( X F ) as follows (in fact in Theorem 9.1 of [6] one of the relations, the relation (1), was stated in aincorrect way; this was corrected in Theorem 4.3 of [16]).First of all one shows that the restriction map r ( G,C ) : H ∗ ( Y ( X F ) , Q ) → H ∗ ( Y ( G,C ) , Q ) is surjective.If a ray c is such that c / ∈ V Γ j ,(4.1) r ( G,C ) ( x c ) = 0 . If G i is such that C is not contained in V Γ i or G ∪ { G i } cannot be reordered into a flag wehave(4.2) r ( G,C ) ( t i ) = 0 . Let us take a pair ( i, j ) with i ∈ { , ..., m } , and j ∈ { , ..., m + 1 } in such a way that G i ( G j and set S i = { s | G i s ∈ G, G i s ) G i } .Let us start with a pair ( i, m + 1) . If S i = ∅ one has the relation F ( i, m + 1) = P G m +1 G i ( X h ∈ B i − t h ) . which already holds in H ∗ ( Y ( X F )) .Otherwise, set k = min( s | s ∈ S i ) . One has the relation(4.3) F G ( i, m + 1) = P G ik G i ( X h ∈ B i − t h ) in H ∗ ( Y ( G,C ) , Q ) .Now let us consider the case of a pair ( i, j ) with j ≤ m . If G ∪ { G j } cannot be reorderedinto a flag we already know from the relation (4.2) that r ( G,C ) ( t j ) = 0 .Assume now that G ∪ { G j } can be reordered in a flag. Then also S i ∪ { G j } is a flag andlet H be its smallest element.If H = G j and G j / ∈ S i , we get the relation F ( i, j ) = P G j G i ( X h ∈ B i − t h ) t j . which already holds in H ∗ ( Y ( X F )) .If H = G i k ∈ S i we get the relation F G ( i, j ) = P G ik G i ( P h ∈ B i − t h ) t j , which is a consequenceof (4.3). Theorem 4.4.
For any pair ( G, C ) with C ⊂ V G i , the cohomology ring H ∗ ( Y ( G,C ) , Q ) is thequotient of the polynomial ring B C, F [ t , . . . , t m ] modulo the ideal generated by(1) the image of the ideal I modulo the quotient homomorphism π : B F [ t , . . . , t m ] → B C, F [ t , . . . , t m ] . (2) The relations (4.1), (4.2) and (4.3). We can now apply this to give a presentation of the differential graded algebra associatedto Y ( X F ) and the divisor with normal crossings Y ( X F ) \ M ( A ) following [18]. Recall thatin our case this algebra is the direct sum M F = ⊕ ( G,C ) H ∗ ( Y ( G,C ) , Q )[ − n ( G,C ) ] with n ( G,C ) equal to dim C + | G | which is the codimension of Y ( G,C ) .In order to do so, we take the algebra B = Q [ t , . . . , t m ] ⊗ V ( κ , . . . , κ m ) /K where K is theideal generated by the products t i t j , t i κ j , κ i κ j whenever G i and G j are not comparable. HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 13
We grade B by setting deg t j = 2 and deg κ j = 1 and we remark that the usual differentialon B = Q [ t , . . . , t m ] ⊗ V ( κ , . . . , κ m ) given by d ( κ j ) = t j preserves K so that B inherits adegree 1 differential d B .We can then consider the algebra C F ⊗ B with differential d C F ⊗ ⊗ d B .Remark that B F is the subalgebra of C F consisting of element of bidegree (2 n, , n ≥ and so the polynomial ring B F [ t , . . . t m ] is a subalgebra of C F ⊗ B . Using this remark we cantake the ideal Θ F in C F ⊗ B generated by the elements(1) x c t j , τ c t j , x c κ j , τ c κ j , c / ∈ V G j .(2) F ( i, j ) , for G j ) G i , i = 1 , . . . m , j = 1 , . . . , m + 1 .(3) P G j G i ( P h ∈ B i − t h ) κ j , with i ∈ { , ..., m } , and j ∈ { , ..., m } in such a way that G i ( G j .Observe that Θ F is preserved by the differential d C F ⊗ ⊗ d B . It follows that we get aninduced differential d N F on the algebra N F = C F ⊗ B / Θ F .We know that Θ F is a graded ideal so that N F is also graded and the differential d N F is ofdegree 1. Theorem 4.5.
The differential graded algebra ( N F , d N F ) is isomorphic to the Morgan algebra ( M F , d M F ) .Proof. The proof given in [16] can be applied verbatim in this more special case. (cid:3) A limit and the rational homotopy type of M ( A ) Before we start, let us briefly discuss the generators of Θ F . Remark that c / ∈ V Γ j if andonly if the function s c = ρ F ( x c ) vanishes on V Γ j . Indeed the support of s c is the interior of S ( c ) the star of c and such interior intersects V Γ j if and only if c ∈ V Γ j .Thus the first relations can be written as(5.1) xt j , τ t j , xκ j , τ κ j , for x, τ ∈ { f ∈ S F | ρ F ( f ) ≡ on V Γ j } .As for the elements χ −F = X c ray min(0 , h χ, c i ) x c , appearing in the definition of P G j G i ( t ) := Q χ ∈ χ \ ( χ ∩ Γ j ) ( t − χ −F ) , we remark that ρ F ( χ −F ) = χ − where χ − ( v ) = min(0 , χ ( v )) for each v ∈ V .Notice that if G is a refinement of F an equal sign linear function relative to F is also equalsign relative to G , so if F is compatible with A also G is compatible with A .The first consequence of this fact is Proposition 5.1.
Let F be a fan compatible with A . Let G be a refinement of F and ψ FG : X G → X F the unique T -equivariant projective morphism extending the identity on T .Then, for each layer K Γ ,φ of A , the preimage of the closure of K Γ ,φ in X F is the closureof K Γ ,φ in X G . Proof.
Fix the layer K Γ ,φ of A . Clearly we can identify K Γ ,φ with the torus T Γ = ∩ χ ∈ Γ ker e πıχ and the restriction ψ FG to K Γ ,φ is the identity.We know by the compatibility of F and G with A , that a cone C in F or in G has eitherrelative interior disjoint from V Γ or it is contained in V Γ . The cones contained in V Γ define smooth projective fans F Γ and G Γ respectively and G Γ isa refinement of F Γ .We can then identify the closure of Z F (resp. Z G ) of K Γ ,φ in X F (resp. X G ) with the T Γ variety associated to the fan F Γ (resp. G Γ ) and the restriction of ψ FG to Z G with the unique T Γ equivariant morphism extending the identity on K Γ ,φ (identified with T Γ ).We need to prove that ( ψ FG ) − ( Z F ) = Z G . In order to see this let us take a cone C ∈ F .Consider the new T -orbit O C ⊂ X F corresponding to C . We know that ( ψ FG ) − ( O C ) is theunion of the T -orbits O C ′ ⊂ X G corresponding to the cones C ′ ∈ G whose relative interior iscontained in the relative interior of C .Also as a O C ≃ T /T Γ C , were T Γ C = ∩ χ ∈ X ∗ ( T ) , h χ,C i =0 ker e πıχ , the restriction of the map ψ FG to a T -orbit O C ′ ⊂ X G corresponding to a cone C ′ ∈ G whoserelative interior is contained in the relative interior of C can be then identified with theprojection T /T Γ C ′ → T /T Γ C whose fiber is the torus T Γ C /T Γ C ′ .Having recalled these facts let us examine the preimage of Z F . Fix a cone C ∈ F .If C is not contained in V Γ then O C ∩ Z F = ∅ , so no orbit O C ′ ⊂ X G corresponding to a cone C ′ ∈ G whose relative interior is contained in the relative interior of C intersects ( ψ FG ) − ( Z F ) .If C is contained in V Γ then also every cone C ′ ∈ G whose relative interior is contained inthe relative interior of C is contained in V Γ . Furthermore we have inclusions T Γ C ′ ⊂ T Γ C ⊂ T Γ . From this and the description of the restriction of the map ψ FG to O C ′ , it follows that ( ψ FG ) − ( Z F ) ∩ O C ′ ⊂ Z G , proving our claim. (cid:3) At this point we observe that the universal property of Blowing up (see, [12], pp.164-165)implies that we get a morphism ν FG : Y ( X G ) → Y ( X F ) extending the identity of M ( A ) .The map ν FG then induces a homomorphism Φ FG : N F → N G of differential graded algebras having the following properties: Φ FG coincides with χ FG on C F and one has Φ FG ( t i ) = t i , Φ FG ( κ i ) = κ i for every i = 1 , . . . , m .Let us now remark that the function χ − depends only on χ and not on the choice of anyparticular fan. This allows us to define the polynomial P G j G i ( t ) := Q χ ∈ χ \ ( χ ∩ Γ j ) ( t − χ − ) ∈ C [ t ] We can then consider the algebra
C ⊗ B with differential d C ⊗ ⊗ d B .For any Γ j we consider the subspace S Γ j which is the image modulo V ∗ of the space offunctions whose restriction to V Γ j is linear.Now remark that the space Σ surjects on both C (2 , and C (0 , . So, given υ ∈ Σ , we maytake the corresponding elements s υ ∈ C (2 , and σ υ ∈ C (0 , .In C ⊗ B we then take the ideal Θ generated by the elements(1) s υ t j , σ υ t j , s υ κ j , σ υ κ j , for υ ∈ S Γ j . (2) F ( i, j ) , for G j ) G i , i = 1 , . . . m , j = 1 , . . . , m + 1 .(3) P G j G i ( P h ∈ B i − t h ) κ j , with i ∈ { , ..., m } , and j ∈ { , ..., m } in such a way that G i ( G j .and set N = C ⊗ B / Θ .Observe that Θ is preserved by the differential d C ⊗ ⊗ d B . It follows that we get aninduced differential d N on N . HE HOMOTOPY TYPE OF THE COMPLEMENT OF A TORIC ARRANGEMENT 15
We know that Θ is a graded ideal so that N is also graded and the differential d N is ofdegree 1.From (3.5) and the above observations we then deduce Theorem 5.2. ( N, d N ) = lim −→ F ( N F , d N F ) . We now want to remark a few facts about the algebras ( N F , d N F ) and ( N, d N ) . First of allwe have seen that although the set of generators of the defining ideal of N F depends on thechoice of equal sign bases, by Theorem 4.3(i) the algebra itself is independent on the choiceof these equal sign bases.Assume now that we have given two smooth projective fans F and F ′ each equipped witha choice of equal sign bases for our arrangement. We know that we can find a commonrefinement G of F and F ′ . We have already remarked that the two sets of equal sign basesare both choices of equal sign bases for G , so that the algebra ( N G , d N G ) is independent fromthese choices.We deduce Proposition 5.3.
The differential graded algebra ( N, d N ) is independent from any choice ofequal sign bases. Let us now recall that according to [18], for smooth projective fans F as above the mini-mal model of the differential graded algebra ( N F , d N F ) is isomorphic to the minimal model of M ( A ) so it determines the rational homotopy type of M ( A ) and in particular its cohomologyis the cohomology ring H ∗ ( M ( A ) , Q ) . Furthermore as we have already seen the homomor-phism Φ FG is induced by the morphism ν FG : Y ( X G ) → Y ( X F ) which is the identity whenrestricted to M ( A ) , so Φ FG is a quasi isomorphism of differential graded algebras.Since. taking homology of a chain complex is an exact functor we deduce that also thecohomology of the differential graded algebra ( N, d N ) is H ∗ ( M ( A ) , Q ) and that the naturalmaps ( N F , d N F ) → ( N, d N ) are quasi isomorphisms.We deduce Proposition 5.4.
The minimal model of the differential graded algebra ( N, d N ) is isomorphicto the minimal model of M ( A ) . To state our final result in its more general form, let us now consider an arbitrary toricarrangement A (i.e. we drop the assumption A = ˜ A ). We recall that, according to thedefinition in the Introduction, the combinatorial data of A are provided by the following twosets:(1) The partially ordered set ˜ A ordered by reverse inclusion.(2) The set of lattices Γ for K Γ ,φ ∈ ˜ A .We have: Theorem 5.5.
The rational homotopy type of the complement M ( A ) depends only on thecombinatorial data of A .Proof. By Proposition 5.4 the rational homotopy type of the complement M ( A ) depends onlyby the differential graded algebra ( N, d N ) . In turn we have that this algebra is defined onlyin terms of the combinatorial data of A . Hence our claim follows. (cid:3) References [1]
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