The equivariant cohomology of weighted projective space

THE EQUIV ARIANT COH OMOLOGY OF WEIGHTED PR OJECTIVE SP A CE ANTHONY BAHRI, MA TTHIAS FRANZ AND NIGEL RA Y Abstrac t. W e describe the inte gral equiv ar iant cohomology ring of a weigh ted pro j ectiv e space in terms of pi ecewise polynomials , and thence b y generators and relations. W e deduce that the r i ng is a perf ect inv ari an t, and prov e a Chern class formula for weigh ted pro j ectiv e bundles. 1. Introduction Let χ = ( χ 0 , . . . , χ n ) be a vector of pos itiv e natural num b er s. Th e asso ciated weighte d pr oje ctive s p ac e is the quotient (1.1) P ( χ ) = S 2 n +1 /S 1 h χ 0 , . . . , χ n i , where the num b ers χ i indicate the weigh ts with whic h S 1 acts o n the unit spher e S 2 n +1 ⊂ C n +1 , by (1.2) g · ( x 0 , . . . , x n ) = ( g χ 0 x 0 , . . . , g χ n x n ) . So P (1 , . . . , 1 ) is the standard pro jectiv e space CP n . Note that P ( χ ) is equipped with an action o f the n -dimensional torus (1.3) T = ( S 1 ) n +1 /S 1 h χ 0 , . . . , χ n i , where the quotient is defined by analog y with (1.1). W e give an explicit exa mple of such an action in (4.1), for n = 3 . Different w eight v ector s may also give equiv alen t weigh ted pro jectiv e spaces; we will elab or ate on this asp ect in Sectio n 5. Kaw a saki [K w] has computed the ordinar y cohomology ring of P ( χ ) with in teger co efficients. A dditively , the cohomology is is omorphic t o that of CP n , but the m ultiplication is differen t. Mor e precisely , if c 1 is a generator o f the g roup H 2 ( P ( χ )) , then H ∗ ( P ( χ )) is generated as a ring b y elements c m , where 1 ≤ m ≤ n and (1.4) c m 1 = lcm( χ 0 , . . . , χ n ) m lcm { Q i ∈ I χ i : | I | = m } c m in H 2 m ( P ( χ )) . The m ultiplication is induced according ly . In this note we study H ∗ T ( P ( χ )) , the T -equiv aria nt cohomolo gy of P ( χ ) with in teger co efficients; it is defined as the o r dinary coho mology H ∗ ( P ( χ ) T ) o f the Borel constructio n P ( χ ) T = E T × T P ( χ ) . Our ma in result, Theorem 3.7, describ es H ∗ T ( P ( χ )) in terms of generator s and relations . W e give tw o applications. Firstly , w e sho w ho w to recov er the weigh t vector χ from H ∗ T ( P ( χ )) , thereby establishing that different weigh ted pro jectiv e spaces hav e different integral equiv ar iant coho mology rings. Secondly , we consider weigh ted pro jectiv e bundles. F or any direct sum D = L 1 ⊕ · · · ⊕ L n of c o mplex line bundles ov er a ba se space X , the c o homology ring H ∗ ( P ( D )) o f the pro jectivisation is a mo dule ov er H ∗ ( X ) . Its algebra structure is determined by the single r elation (1.5) n Y i =0  ξ + c 1 ( L i )  = 0 , 2000 M athematics Subje ct Classific ation. 55N91 (primary); 13F55, 14M25 (secondary) . 1 2 ANTHONY BAHRI, M A TTHIAS FRANZ AND NIGEL RA Y where c 1 ( L i ) ∈ H 2 ( X ) and − ξ ∈ H 2 ( P ( D )) denote the Chern clas ses of L i and of the canonica l complex line bundle resp ectively . Al Amrani [A] has stated a generalisa tio n of (1.5) to weigh ted pro jectiv e bundles a nd prov ed it in a specia l case. Theorem 6.2 establishes his relation in general. W e consider our calcula tio ns a s lying in the realm of tor ic topolo gy , and will elab- orate on this theme in a subsequent document. Readers who require ba ckground information on equiv aria nt topology may consult [AP], or the survey articles in [M]. A cknow le dgements. The first a uthor thanks Rider Univ ersity for the a ward o f Re- search Leav e. All three authors ar e gr ateful to the Manc hes ter Institute of Ma th- ematical Sciences (MIMS) for ongo ing supp ort that has help ed to sustain their collab ora tio n. 2. From equiv ariant cohomology to piecewise pol ynomials By a ring we always mea n a graded commu tative ring with unit element. All rings we co ns ider happ en to b e concentrated in even degrees, so that they a re commut ative in the ordina ry sense. Remark 2.1. There are several ways to des crib e the divisibilit y of the powers c m 1 in H ∗ ( P ( χ )) . Kaw a saki lo oks at the p -co nt ents of the weigh ts for each prime p separately . If q 0 ( p ) , . . . , q n ( p ) are their p -conten ts, in increa sing o r der, then (2.1) c m 1 = Y p q n ( p ) m q n ( p ) · · · q n − m +1 ( p ) c m . Kaw a saki also considers sets I o f size m + 1 instead of m and writes (2.2) c m 1 = lcm( χ 0 , . . . , χ n ) m lcm { gcd { χ i : i ∈ I } − 1 Q i ∈ I χ i : | I | = m + 1 } c m [Kw , p.248], as do es Al Amrani [A, Sec. I.5]. T aking pro ducts of m + 1 weights a nd dividing by their gre a test common diviso r, as in the denominator of (2.2), r emov es the smallest p -conten t for each prime p . Co mputing the least common m ultiple ov er all such terms then giv es the product of the m largest p - conten ts of all weigh ts, in accordance with the deno minators o f (1 .4 ) and (2.1). Now let ι : P ( χ ) → P ( χ ) T be the inclusion of a fibre into the Borel c o nstruction. Lemma 2.2. As an H ∗ ( B T ) -mo dule, H ∗ T ( P ( χ )) is fr e e of r ank n + 1 : as a ring, it is gener ate d by the image of H 2 ( B T ) in H 2 T ( P ( χ )) , to gether with any su b gr oup A ∗ < H ∗ T ( P ( χ )) that surje cts onto H > 0 ( P ( χ )) u nder ι ∗ . Pr o of. By K aw a s aki, H ∗ ( P ( χ )) is free over Z and concentrated in even degrees. So the Serr e spectr al sequence of the fibra tion P ( χ ) → P ( χ ) T → B T degenera tes at the E 2 level, and H ∗ T ( P ( χ )) is isomor phic as H ∗ ( B T ) -mo dules to H ∗ ( P ( χ )) ⊗ H ∗ ( B T ) b y the Leray–Hirsch theorem. The isomorphism is induced by any additive section to ι ∗ , which we may assume takes v alues in A ∗ . This prov es the cla im.  The equiv ar iant co homology of CP n is well-known, a nd may b e describ ed co nv e- nien tly in the context of toric v arieties. Indeed, P ( χ ) is an n -dimensional pro jectiv e toric v ar iety for every χ , and may b e co nstructed from any complete simplicial fan Σ whose rays v 0 , . . . , v n ∈ N = Z n span N , and satisfy the relation (2.3) χ 0 v 0 + · · · + χ n v n = 0 . Every such Σ is the normal fan o f a lattice n - s implex in N ⊗ Z R , a nd is p olytopa l; we refer to F ulton [F u, §2.2], or the nice overview in [Ks, §4.1], for a full discuss ion. In pa rticular, H ∗ T ( CP n ) is isomorphic to the in tegral Stanley–Reisner algebra (2.4) Z [Σ] = Z [ a 0 , . . . , a n ] / ( a 0 · · · a n ) THE EQUIV ARIANT COHOMOLOGY OF WEIGHTED PROJECTIVE SP ACE 3 of the appro priate Σ , where each genera tor a i corresp o nds to v i , a nd has degree 2 . In other words, the only relation amongst the generators is (2.5) n Y i =0 a i = 0 . The situation for singular v arieties is less straig ht forward, a nd the P ( χ ) offer a natural family of test cases. Our aim is to gener alise (2.4), and express H ∗ T ( P ( χ )) in terms of ge ner ators a nd relations for a rbitrary χ . W e use the langua ge of piecewise po lynomials, to whic h we now turn. A function f : N → Z is called pie c ewise p olynomial on Σ if it coincides with some g lobally defined p olynomial g ∈ Z [ N ] on each co ne σ . Suc h functions are closed under p oint wise addition and multiplication, a nd form an algebra P P [Σ] ov er the ring of glo bal p olyno mials Z [ N ] . W e g rade bo th P P [Σ] and Z [ N ] by twice the degree of ho mogeneous elements, a nd use the Cher n classes o f the n ca nonical line bundles to iden tify Z [ N ] with H ∗ ( B T ) . Prop ositio n 2.3. L et Σ b e a p olytop al fan in N , and X Σ the asso ciate d c omp act pr oje ctive toric variety: if H ∗ ( X Σ ) is c onc entr ate d in even de gr e es, then H ∗ T ( X Σ ) is isomorphic to P P [Σ] as an algebr a over H ∗ ( B T ) . Pr o of. Set X = X Σ , a nd deno te the orbit space by X /T . The latter may b e iden tified with a conv ex p olytop e P Σ , whos e normal fan is Σ . F o llowing Gor esky and MacPhers on [GP], we deduce tha t X is homeomorphic to a quo tient space of T × P Σ , a nd may therefore b e e x pressed as a finite T -CW complex [M] with co n- nected iso tropy groups. As in the pro of of Lemma 2.2, the Ser re spec tral seq uence for the fibra tion X → X T → B T degenera tes at the E 2 level b ecause H ∗ ( X ) is concentrated in e ven degrees. Hence, by a result of F ranz–P uppe [FP], the Chang – Skjelbred sequence (2.6) 0 − → H ∗ T ( X ) j ∗ − → H ∗ T ( X T ) δ − → H ∗ +1 T ( X 1 , X T ) is exact for in tegral co efficients. Here X T denotes the T -fixed po int s, X 1 the union of X T and all 1 -dimensional orbits, j the inclusion X T → X and δ the differential of the long exact cohomolog y seq uence fo r the pair ( X 1 , X T ) . W e may identify the kernel of δ with the algebra P P [Σ] , as follows. W rite O σ for the orbit under the co mplexification T C of T corres po nding to the cone σ ∈ Σ , a nd Z [ σ ] for the p olyno mia ls with in teger co e fficients on the linear hull of σ . Any such polyno mial is uniquely determined by its re s triction to σ . When σ is n -dimensiona l, we have that (2.7) H ∗ T ( O σ ) = H ∗ ( B T ) = Z [ σ ] . F or ( n − 1) -dimensional τ , we deno te the isotropy group of O τ b y T τ . Then the action of the circle T /T τ on the clo sure ¯ O τ is isomorphic to the standar d action of S 1 on CP 1 , whose fixed p oints we write as 0 and ∞ . W e o btain (2.8) H ∗ T ( ¯ O τ , ∂ O τ ) = H ∗ ( B T τ ) ⊗ H ∗ T /T τ ( ¯ O τ , ∂ O τ ) and (2.9) H ∗ T /T τ ( ¯ O τ , ∂ O τ ) ∼ = H ∗ S 1 ( CP 1 , { 0 , ∞} ) ∼ = Z [+1]; hence (2.10) H ∗ +1 T ( ¯ O τ , ∂ O τ ) ∼ = Z [ τ ] . Moreov er, whenever τ ⊂ σ is a facet, the differential (2.11) H ∗ T ( O σ ) − → H ∗ +1 T ( ¯ O τ , ∂ O τ ) 4 ANTHONY BAHRI, M A TTHIAS FRANZ AND NIGEL RA Y is the ca nonical r estriction Z [ σ ] → Z [ τ ] , multiplied by ± 1 a c c o rding to the orienta- tion of the in terv al O τ /T ≈ (0 , ∞ ) . It follows that the differen tial δ of (2.6) is a signed sum of restr ictions (2.12) δ : M σ ∈ Σ n Z [ σ ] − → M τ ∈ Σ n − 1 Z [ τ ] of sums of p oly no mial algebra s, whose comp onent into Z [ τ ] is the difference of the r estrictions of the p olynomials o n the tw o n -dimensiona l cones having τ as their common fa c e t. So the kernel consists of those collections o f p olynomials on n - dimensional cones whic h may b e glued alo ng their common facets. This corr esp onds to the requir ement that the p olynomials agre e on any in tersection τ = σ ∩ σ ′ , beca use σ a nd σ ′ are co nnected by a seq uence of n -dimensional co nes, ea ch of which contains τ and shares a facet with the next. (In other words, Σ is a her e ditary fan [BR].) Th us δ has kernel P P [ σ ] , as require d. Finally , let f b e the piecewise polynomial corresp onding to a cla ss α ∈ H ∗ T ( X ) . By construction, the po lynomial coinciding with f on an n -dimensional cone σ is the image of α under the restriction (2.13) H ∗ T ( X ) → H ∗ T ( O σ ) = H ∗ ( B T ) = Z [ N ] . Since the comp osition H ∗ ( B T ) → H ∗ T ( X ) → H ∗ T ( O σ ) = H ∗ ( B T ) is the iden tit y , it follows that the map H ∗ ( B T ) → H ∗ T ( X ) co r resp onds to the inclusion of the algebra of g lobal p olynomials.  Remark 2.4 . The integral e q uiv ariant cohomolog y of any smo oth, not necessarily compact toric v ariety X Σ is given by the Stanley–Reisner algebra Z [Σ] [BDP], [DJ]; or equiv a lent ly , by P P [Σ] [Br]. A canonical isomorphism b etw een the tw o is defined b y assig ning the Coura nt function a ρ of the ray ρ to the Stanley–Reisner genera tor corresp o nding to ρ . The function a ρ is piecewise linear on Σ , a nd assumes the v alue 1 on the g enerator of ρ and 0 on all o ther rays. It is well-defined b eca use the smo othness of X Σ implies that the r ays of any cone may b e completed to a basis of the lattice N . Brion w as probably the first to note the relationship betw een piecewise po ly nomials and the Chang–Skjelbred sequence. Similarly , when Σ is simplicial the rational equiv ariant coho mology H ∗ T ( X Σ ; Q ) is the Stanley–Reis ner algebr a Q [Σ] ; or equiv alently , P P [Σ] ⊗ Q [F u, p.107]. In particular, there is an is omorphism (2.14) H ∗ T ( P ( χ ); Q ) ∼ = Q [ a 0 , . . . , a n ] / ( a 0 . . . a n ) . Pa y ne [P] has s hown that P P [Σ] is isomor phic to the equiv ar ia nt Chow ring of X Σ , for any fan Σ . F urther dev elopments ar e documented in [KP]. Propo si- tion 2.3 is also v alid for more genera l fans, but we hav e b een unable to lo cate a reference identif ying X Σ as a T -CW complex; so w e pro ceed b y as s uming appro- priate cohomolog ica l finiteness conditions, following [FP, Remark 1.4]. 3. Genera tors of the ring of piecewise pol ynomial s F or i = 0 , . . . , n we will write σ i ∈ Σ for the full-dimensional cone spanned b y a ll rays ex c e pt v i . This cone is simplicia l and full-dimensional b ecause the set { v j : j 6 = i } is a bas is of the Q -vector space N ⊗ Z Q for ea ch i . Moreover, given a piecewise polyno mial f , w e will denote the unique po lynomial which co incides with f on σ i b y f ( i ) . W e call a piecewise polynomia l r e duc e d if it is not divisible in P P [Σ] by any rationa l prime. Let b ij , where i 6 = j , b e the r educed linea r function that assumes a p ositive v a lue on v i and v anishes on all v k , for i 6 = k 6 = j . THE EQUIV ARIANT COHOMOLOGY OF WEIGHTED PROJECTIVE SP ACE 5 Lemma 3.1. W e have that (3.1) b ij ( v i ) = χ j gcd( χ i , χ j ) = lcm( χ i , χ j ) χ i . Pr o of. Applying b ij to the relation (2.3) yields (3.2) χ i b ij ( v i ) = − χ j b ij ( v j ) . Since b ij is reduced a nd v i and v j span N / ker b ij ∼ = Z , the v a lues b ij ( v i ) and b ij ( v j ) m ust b e coprime. This implies the claimed formula.  Prop ositio n 3.2. The b ij gener ate N ∨ , t he lattic e dual to N . Pr o of. Multiplying all w eig hts b y a constant facto r changes neither the fan Σ nor the functions b ij , so we may as sume that the greatest common divisor o f the weigh ts equals 1 . F or given j , let N j be the span of the linearly independent s et V j = { v i : i 6 = j } and N ∨ j its dual. By Lemma 3 .1, the restrictio n of ea ch b ij with i 6 = j to N j is divisible by a diviso r of χ j , and the quotient is an element of the basis dual to V j . Let M j < N ∨ denote the sublattice generated b y those b ij for which i 6 = j . Our goal is then to show that M = N ∨ , where M is g enerated by the M j . W e hav e that (3.3) N ∨ j / N ∨ = ( N ∨ j / M j )  ( N ∨ / M j ) . Therefore, the o rder of N ∨ / M j divides that o f N ∨ j / M j , which itself divides χ n j from ab ov e. Hence, the order of N ∨ / M j also divides χ n j , a nd the same applies to N ∨ / M beca use (3.4) N ∨ / M = ( N ∨ / M j )  ( M / M j ) . This implies that the o rder of N ∨ / M divides the gr eatest common diviso r of a ll χ n j , which we assumed to be 1 .  Let a i denote the Cour ant function cor resp onding to v i , for 0 ≤ i ≤ n . By this we mean the reduced piecewise linear function that ass umes a p os itive v alue on v i and v anishes on all v j for j 6 = i . Each σ j is simplicial, so a i is w ell-defined. Lemma 3.3. T o gether with the line ar functions, e ach a i gener ates the pie c ewise line ar functions in P P [Σ] . Pr o of. Let f b e piecewise linea r. Then f − f ( i ) v anishes on σ i , and is therefore a m ultiple of a i .  Lemma 3.4. W e have that (3.5) a i ( v i ) = lcm( χ 0 , . . . , χ n ) χ i and a ( j ) i = lcm( χ 0 , . . . , χ n ) lcm( χ i , χ j ) b ij in P P [Σ] , for al l i 6 = j . Pr o of. By Lemma 3.1 we can define a piecewise linear function f on Σ by setting f ( i ) = 0 , and f ( j ) equal to the given formula for j 6 = i . Now let χ k be a weigh t with ma ximal p - conten t, for so me prime p . If k = i , then p cannot divide f ( j ) for any j 6 = k ; and if k 6 = i , then p cannot divide f ( k ) . So f is r educed, and f = a i .  Lemma 3.5. In P P [Σ] , the r elation (3.6) b ij = lcm( χ i , χ j ) lcm( χ 0 , . . . , χ n ) ( a i − a j ) holds for al l i 6 = j . 6 ANTHONY BAHRI, M A TTHIAS FRANZ AND NIGEL RA Y Pr o of. By Lemma 3 .4, we hav e that (3.7) a ( j ) i = − a ( i ) j = lcm( χ 0 , . . . , χ n ) lcm( χ i , χ j ) b ij , and a ( i ) i = − a ( j ) j = 0 ; so ( a i − a j ) ( i ) = a ( j ) i = ( a i − a j ) ( j ) . F urthermore, by Lemma 3 .3, a i − a j may be written as ma k + r fo r some int eger m and linear function r . But ev er y a k restricts to distinct linear functions o n maximal cones σ i and σ j , so m = 0 . Hence a i − a j is linear, and divisible as claimed.  W e now consider higher-deg r ee analog ues of the Co urant functions a i . Lemma 3.6. F or any n onempty subset I ⊂ { 0 , . . . , n } , the function Q i ∈ I a i is divisib le by (3.8) Y i ∈ I lcm( χ 0 , . . . , χ n ) lcm { χ i , χ j : j / ∈ I } in P P [Σ] . Pr o of. W e lo ok a t each prime p separately . If the maximal p -conten t o ccurs in χ j for some j / ∈ I , then there is nothing to prov e b eca use the p -con tent o f (3.8) is 1 . W e can therefore assume that it o ccurs in χ k for some k ∈ I . Cho ose a n i ∈ I and denote the p -co nt ents of χ i and χ k b y q i and q k resp ectively . Then a ll a ( j ) i with j / ∈ I a r e divisible by q k /q i , w hich is gre a ter than o r equal to the p -conten t of (3.9) lcm( χ 0 , . . . , χ n ) lcm { χ i , χ j : j / ∈ I } . T aking the pro duct ov er a ll i ∈ I finishes the pro of.  Hence, for I ⊂ { 0 , . . . , n } we may define the piecewise p olynomial (3.10) a I =  Y i ∈ I lcm( χ 0 , . . . , χ n ) lcm { χ i , χ j : j / ∈ I }  − 1 Y i ∈ I a i in the 2 | I | -dimensional compo nent o f P P [Σ] . Theorem 3. 7. The ring H ∗ T ( P ( χ )) is gener ate d by the functions a I and b ij , wher e 1 ≤ | I | ≤ n and i 6 = j r esp e ct ively. The only re lations ar e (2.5), (3.6 ) and (3.10). Pr o of. F rom (2.14), H ∗ T ( P ( χ ); Q ) is genera ted by the a i sub ject only to the r elation (2.5). The re lations (3.6) and (3.10) s how that the a I with | I | > 1 and the b ij are redundant ov er Q , so adding these g e nerators and rela tions gives an isomo rphic ring. Since there a re no more rela tions b et ween the a I and b ij in H ∗ T ( P ( χ ); Q ) , the same is true in H ∗ T ( P ( χ )) ; for the latter is free over Z , and injects int o H ∗ T ( P ( χ ); Q ) . It remains to show that these elemen ts are ring generato r s. By P rop osition 3.2, the b ij generate the linear functions, which are the image of H 2 ( B T ) in H 2 T ( P ( χ )) . Hence, b y Lemma 2 .2, it suffices to show that the subgroup generated b y the a I surjects onto H ∗ ( P ( χ )) . In other words, we have to show that c m lies in the span of { ι ∗ ( a I ) : | I | = m } for each 1 ≤ m ≤ n . F or m = 1 , this is true by Lemma 3 .3 b ecause we know ι ∗ itself to b e surjective. Moreov er, Lemma 3.5 implies that a ll elements a i are mappe d to the same element of H 2 ( P ( χ )) . This must neces sarily b e a generator , which we may assume to b e c 1 . THE EQUIV ARIANT COHOMOLOGY OF WEIGHTED PROJECTIVE SP ACE 7 F or 1 < m ≤ n , we obtain ι ∗ ( a I ) =  Y i ∈ I lcm( χ 0 , . . . , χ n ) lcm { χ i , χ j : j / ∈ I }  − 1 Y i ∈ I ι ∗ ( a i ) (3.11a) =  Y i ∈ I lcm( χ 0 , . . . , χ n ) lcm { χ i , χ j : j / ∈ I }  − 1 c m 1 (3.11b) = Q i ∈ I lcm { χ i , χ j : j / ∈ I } lcm { Q j ∈ J χ j : | J | = m } c m b y (1.4) . (3.11c) W e must show that these multiples of c m generate H 2 m ( P ( χ )) . Once more, we consider each prime p separately , and let I be the set of indices (which need not be unique) tha t corresp o nd to m weigh ts with gr eatest p - conten t. Since this is also the set J whic h max imises the p - conten t of the denominator o f (3.11c), we conclude that for ea ch p ther e app ears a m ultiple of c m whose p -co nten t is 1 . In other w ords, the greatest common divisor of a ll multi ples is 1 , as required.  4. An example W e illustra te the results o f the preceding section in the ca se χ = (1 , 2 , 3 , 4) , which confirms tha t the element s b ij cannot b e omitted from the statement o f Theorem 3.7. W e choo se v 0 = ( − 2 , − 3 , − 4) , v 1 = (1 , 0 , 0) , v 2 = (0 , 1 , 0) , and v 3 = (0 , 0 , 1) . So is omorphisms ( S 1 ) 4 /S 1 h 1 , 2 , 3 , 4 i ↔ ( S 1 ) 3 iden tifying the torus T of (1.3) a re induced by ( t 0 , t 1 , t 2 , t 3 ) 7→ ( t − 2 0 t 1 , t − 3 0 t 2 , t − 4 0 t 3 ) and ( u 1 , u 2 , u 3 ) 7→ (1 , u 1 , u 2 , u 3 ) resp ectively , and its action on P (1 , 2 , 3 , 4) is equiv alent to that of ( S 1 ) 3 , g iven by (4.1) ( u 1 , u 2 , u 3 ) · [ x 0 , x 1 , x 2 , x 3 ] = [ x 0 , u 1 x 1 , u 2 x 2 , u 3 x 3 ] on homogeneous co ordinates. W riting an element f of P P [Σ] as f = ( f (0) , f (1) , f (2) , f (3) ) , and the canonica l basis of N ∨ as ( x, y , z ) , we deduce a 0 = (0 , − 6 x, − 4 y , − 3 z ) , (4.2a) a 1 = (6 x, 0 , 6 x − 4 y , 6 x − 3 z ) , (4.2b) a 2 = (4 y , − 6 x + 4 y , 0 , 4 y − 3 z ) , (4.2c) a 3 = (3 z , − 6 x + 3 z , − 4 y + 3 z , 0 ) . (4.2d) Each a i is reduced, althoug h individual comp onents a ( j ) i may have non-trivial divi- sors. The situation changes for pro ducts Q i ∈ I a i , b ecause comp o nents a ( j ) i with j ∈ I a r e multipl ied by 0 ; this is the essence of Lemma 3.6. Since lc m { 1 , 2 , 3 , 4 } = 12 , we obtain a 01 = a 0 a 1 , a 02 = a 0 a 2 / 3 , a 03 = a 0 a 3 / 2 , (4.3a) a 12 = a 1 a 2 / 3 , a 13 = a 1 a 3 / 4 , a 23 = a 2 a 3 / 6 , (4.3b) a 012 = a 0 a 1 a 2 / 9 , a 013 = a 0 a 1 a 3 / 8 , (4.3c) a 023 = a 0 a 2 a 3 / 36 , a 123 = a 1 a 2 a 3 / 72 . (4.3d) An y globally linear function obtained from the a i is a linear combination of a 1 − a 0 = 6 x, a 2 − a 0 = 4 y , a 3 − a 0 = 3 z , (4.4a) a 2 − a 1 = 4 y − 6 x, a 3 − a 1 = 3 z − 6 x, a 3 − a 2 = 3 z − 4 y , (4.4b) b y Pr op osition 3.2 and Le mma 3.5. The functions b 10 = x , b 20 = y and b 30 = z are ob v iously not in the span of the a i . O n the other hand, giv en x , y and z , the 8 ANTHONY BAHRI, M A TTHIAS FRANZ AND NIGEL RA Y remaining b ij are redundant, as are most of the a I . In fa c t we may write (4.5) H ∗ T ( P (1 , 2 , 3 , 4)) ∼ = Z  x, y , z , a 3 , a 23 , a 123  ( a 0 a 1 a 2 a 3 ) b y Lemma 2.2, sub ject to the r elations ab ov e. But we k now o f no canonica l choice for a minimal set of generators . F or examples in w hich χ i 6 = 1 for any i , the cons truction of a fan Σ a nd an explicit T -action on P ( χ ) is more difficult, and amounts to completing χ/ gcd( χ 0 , . . . , χ n ) to a lattice basis. F urther details may b e found in [Ks, §4.2]. 5. Recovering the weights It is clear from the definition (1.1) that P ( χ ) do es not change if a ll w eights are m ultiplied by the s ame factor. In particular , we may always divide the weigh ts b y their greatest co mmon divisor , as in the pro of of Pro p o sition 3.2. Moreo ver, if every w e ig ht except χ i is divisible by some prime p , then P ( χ ) is equiv a r iantly isomorphic to P ( χ ′ ) , where χ ′ = ( χ 0 /p, . . . , χ i − 1 /p, χ i , χ i +1 /p, . . . , χ n /p ) [D]. This may be seen from the tor ic viewp oint: for (2 .3) implies that v i is also divisible by p , and co ntin ues to hold when v i and χ are r e placed by v i /p a nd χ ′ resp ectively . So the fan Σ is unch anged, and the corres po nding toric v ar ieties ar e isomo r phic. By rep eating these simplifications, we can always ensure that tw o or more weigh ts are not divisible b y p , for each prime p . The r esulting weight vector is uniquely defined b y χ (up to order), and the corresp onding weigh ts are called normalise d . Theorem 5.1. The gr ade d ring H ∗ T ( P ( χ )) determines the normalise d weights. Pr o of. The length n + 1 of χ is given by the rank of the free ab elian group H 2 T ( P ( χ )) . So we may in terpret H ∗ T ( P ( χ )) as a ring o f piecewise p olynomials P P [Σ] , where Σ has cones σ i and Courant functions a i , for 0 ≤ i ≤ n . A ccording to r elation (2 .5 ), we may choose piecewise linear functions f 0 , . . . , f n in H 2 T ( P ( χ )) that are re duced, no n-zero, a nd s a tisfy f 0 . . . f n = 0 . On each co ne σ j , some f i m ust therefore v anish; but it cannot v anish on σ k for a ny other k 6 = j , or e lse it w ould also v anish o n ev ery ray of Σ and b e identically z e ro. Because f i is reduced, it follows that f i = ± a j . So we may assume that f i = ± a i for every 0 ≤ i ≤ n , by p er m uting the cones as necessary . Given any i 6 = j , we may no w r e a d off the p -conten t q ij of a i − a j from the functions f i , as fo llows. Since f i = ± a i and f j = ± a j , we k now that q ij is the p -conten t of either f i − f j or f i + f j ; in fact it is the lar ger of the tw o (and the smaller is the p -conten t of a i + a j ). This is b ecause a j restricts to 0 on σ j , whence ( a i − a j ) ( j ) = ( a i + a j ) ( j ) . But a i − a j is g lobally linea r by Lemma 3 .5, so its p - conten t is unaltered by restriction to σ j , wher eas that of a i + a j may increase. Appealing to Lemma 3.5 once more, we find that lcm( χ 0 , . . . , χ n ) / lcm( χ i , χ j ) has p -conten t q ij . Mor eov er , there exist int egers j and k for which lcm( χ j , χ k ) is no t divisible by p , since the weigh ts are norma lised. So for i 6 = j or k , the p - conten t o f χ i is precisely q j k /q ik , a nd the weigh ts ar e completely determined up to order.  Remark 5.2. The analogue of Theorem 5.1 is false for ordinary co homology , be- cause the divisibilit y rule (1.4) do es not take into a ccount the relationship b etw een the distribution of the p - conten ts of the w eights for different pr imes p . F or ex a mple, the g r aded rings H ∗ ( P (1 , 2 , 3 )) and H ∗ ( P (1 , 1 , 6 )) are iso morphic, since c 2 1 = 6 c 2 in bo th cases . Ho wev er , P (1 , 2 , 3) and P (1 , 1 , 6) ca nnot be homeomo r phic, b e c ause the former has tw o s ing ular p oints, and the latter only o ne; nevertheless, they are b oth homotopy equiv a lent to the 2 -cell complex S 2 ∪ 6 η e 4 , where η genera tes π 3 ( S 2 ) ∼ = Z . THE EQUIV ARIANT COHOMOLOGY OF WEIGHTED PROJECTIVE SP ACE 9 F rom the toric viewp oint, b oth quo tien t spaces P (1 , 2 , 3) /T a nd P (1 , 1 , 6) /T may be identified with the 2 -simplex. On the other hand, it fo llows from Theorem 5.1 that the corresp o nding homotopy quotients cannot even b e homotopy equiv a lent . 6. Weighted proje ctive bundles Suppos e given complex line bundles L i ov er a base space X for 0 ≤ i ≤ n , and denote their direct sum by D = L 0 ⊕ · · · ⊕ L n . The torus T ′ = ( S 1 ) n +1 acts on D , and o n the cor resp onding sphere bundle S ( D ) , in canonical fashion. The asso ciated weighte d pr oje ctive bu nd le ov er X has fibre P ( χ ) , and total space the quotient (6.1) P ( D , χ ) = S ( D ) / S 1 h χ 0 , . . . , χ n i . The universal exa mple is given b y E = E T ′ × T ′ C n +1 ov er B T ′ . The rea soning o f Lemma 2.2 shows that H ∗ ( P ( E , χ )) is a free H ∗ ( B T ′ ) -mo dule o f rank n + 1 , and the naturality of the Serre sp ectral sequence implies the same result for arbitrar y D . If all weigh ts ar e equal to 1 , then P ( D , χ ) is an ordinary pro jectiv e bundle, and H ∗ ( P ( D , χ )) is gener a ted by the Cher n classes c 1 ( L i ) and − ξ , s ub ject only to the relation (1.5). In [A, Ch.I I I], Al Amrani generalised (1.5) to those weight ed pro jective bundles whose χ i form a divisor c hain. In a ll such cases, he proved that (6.2) n Y i =0  ξ + lcm( χ 0 , . . . , χ n ) χ i c 1 ( L i )  = 0 for a certain ξ ∈ H 2 ( P ( D , χ )) , which restric ts to c 1 ∈ H 2 ( P ( χ )) on fibr es. By natu- rality , it suffices to verify (6.2) for the universal case; in other words, in H ∗ T ′ ( P ( χ )) , where T ′ acts on P ( χ ) via the pro jection (6.3) T ′ − → T = T ′ /S 1 h χ 0 , . . . , χ n i . W e may pro fita bly describ e H ∗ T ′ ( P ( χ )) in terms of piecewise p olyno mials, as follows. Let π : N ′ = Z n +1 → N b e the epimorphism defined o n ca no nical ba sis vectors b y π ( e i ) = v i , for 0 ≤ i ≤ n . Its k er nel is the subgroup g enerated by χ 0 e 0 + · · · + χ n e n , which we a bbr eviate to u . The pull-back Σ ′ = { π − 1 ( σ ) : σ ∈ Σ } of Σ is a gener alise d fan [Br] b ecause every cone co ntains the line through u , and π induces a monomorphism π ∗ : P P [Σ] → P P [Σ ′ ] o f piecewise po lynomial rings. Lemma 6.1. A s H ∗ ( B T ′ ) -algebr as, H ∗ T ′ ( P ( χ )) is natu r al ly isomorphic to P P [Σ ′ ] ; furthermor e, t he pr oje ction T ′ → T induc es the homomorphism π ∗ : H ∗ T ( P ( χ )) → H ∗ T ′ ( P ( χ )) . Pr o of. On coho mo logy rings we hav e the natural monomorphism (6.4) H ∗ T ( P ( χ )) − → H ∗ T ′ ( P ( χ )) = H ∗ T ( P ( χ )) ⊗ H ∗ ( B S 1 ) , beca use T ′ splits a s T × S 1 , where S 1 is the kernel of (6.3) a nd acts trivia lly on P ( χ ) . So the freeness o f H ∗ T ( P ( χ )) ov er H ∗ ( B T ) implies the freeness o f H ∗ T ′ ( P ( χ )) ov er H ∗ ( B T ′ ) . Moreover, every isotr o py subg r oup of the T ′ -action is connected. W e can therefore imitate the reasoning of Pr o p osition 2.3. F or cones σ ′ ∈ Σ ′ that hav e k = co dim σ ′ ≤ 1 (and in fact for all cones σ ′ ), we hav e that (6.5) H ∗ + k T ′ ( ¯ O σ ′ , ∂ O σ ′ ) ∼ = Z [ σ ′ ] , and the b o undary map in cohomolo gy cor resp onds to the restriction o f po lynomials up to sign. Hence the integral Chang–Skjelbre d seq uence is exact, which in turn iden tifies H ∗ T ′ ( P ( χ )) with P P [Σ ′ ] . Since for a ll σ ∈ Σ and σ ′ = π − 1 ( σ ) the map (6.6) H ∗ T ( ¯ O σ , ∂ O σ ) → H ∗ T ′ ( ¯ O σ ′ , ∂ O σ ′ ) 10 ANTHONY BAHRI, M A TTHIAS FRANZ AND NIGEL RA Y corresp o nds to the pull-back o f functions b y π , the same applies to the restriction of e q uiv ariant cohomo logy from T to T ′ .  Because no cone of Σ ′ contains every e i , we can define ξ as the piecewise linea r function on N ′ which takes the v alues ξ ( u ) = − lcm( χ 0 , . . . , χ n ) , and ξ ( e i ) = 0 for all 0 ≤ i ≤ n . Eq uiv alently , (6.7) ξ ( i ) = − lcm( χ 0 , . . . , χ n ) χ i x i for all i , where ( x i ) denotes the basis dual to ( e i ) for N ′ . Theorem 6. 2. A s an element of H ∗ T ′ ( P ( χ )) , the c ohomolo gy class ξ r estricts t o c 1 in H 2 ( P ( χ )) and satisfies e qu ation (6.2 ). Pr o of. As elements o f P P [Σ ′ ] , we hav e that π ∗ ( a i ) ( j ) = lcm( χ 0 , . . . , χ n ) χ i x i − lcm( χ 0 , . . . , χ n ) χ j x j (6.8a) = lcm( χ 0 , . . . , χ n ) χ i x i + ξ ( j ) , (6.8b) beca use the right -hand side v a nishes o n u and assumes the v a lue a i ( v k ) on e k , for all k . Since ξ differs fro m π ∗ ( a i ) by a linear function, it restricts to the same element in H 2 ( P ( χ )) as π ∗ ( a i ) and a i , namely c 1 . Ident ifying c 1 ( L i ) with x i , w e conclude that equation (6.2) is nothing but the pull-back of relation (2 .5 ).  No other relation is required to describ e H ∗ ( P ( E , χ )) ra tionally , nor therefore in tegrally . Reference s [A] A. Al Amrani, Cohomological study of we ight ed pro jective spaces, pp. 1–52 in: S. Sertöz (ed.), A lge br aic ge ometry (A nkar a 1995) , Lect. Notes Pure Appl. M ath. 19 3 , Dekker, N ew Y or k 1997 [AP] C. Al l da y , V. Pupp e, Cohomolo gic al metho ds in t r ansformation gr oups , Cambridge U ni v. Press, Cambridge 1993 [BDP] E. Bifet, C. De Concini, C. Procesi, Cohomology of regular embeddings, A dv. Math. 8 2 (1990), 1–34 [BR] L. J. Bill era, L. L. Rose, M odules of piecewise pol ynomials and their freeness, Math. Z. 20 9 (1992), 485–497 [Br] M. Brion. Piecewise p olynomial functions, conv ex p olytopes and enumerativ e geometry , pp. 25–44 in: P . Pragacz (ed.), Par ameter sp ac es , Banach Cen t. Publ. 3 6 , W arszaw a 1996 [DJ] M. W. D a vis, T. Janu szkiewicz, Conv ex polytopes, Coxeter orbif olds and to rus act ions, Duke Math. J. 62 (1991), 417–451 [D] I. Dolgac hev, W eighted pro jective v ari eties, pp.25–44 in: Gr oup actions and vector fields (V anc ouver BC, 1981) , LNM 956 , Springer 1982 [FP] M. F ranz, V. Pupp e, Exact cohomology sequences with integral co efficient s for torus ac- tions, T r ansformation Gr oups 12 (2007), 65–76 [F u] W. F ulton, Intr o duction to toric variet ies , Princeton Univ. Press, Princeton 1993. [GP] M. Goresky , R. MacPherson, O n the topology of algebraic torus ac tions, pp. 73–90 in: A lge br aic g r oups (Utr e cht 1986) , LNM 127 1 , Springer 1987 [Ks] A. M. K aspr zyk, T or ic F ano v ari eties and conv ex polytop es, PhD thesis, Universit y of Bath 2006, av ail able at http:// www.math.u nb.ca/~kasprzyk/research/pdf/Thesis.pdf [KP] E. Katz, S. P ayne , Piecewise polynomials, Minko wski weigh ts, and l o calization on toric v arieties, preprint arXiv:math/0703672 [Kw] T. Kaw asaki, Cohomology of t wisted pr o j ectiv e spaces and lens complexes, Math. A nn. 206 (1973), 243–248 [M] J. P . May (ed.), Equivariant Homotopy and Cohomolo gy The ory , CBMS Regional Conf. Ser. M ath. 91 , AMS, Pr ov idence RI 1996 [P] S. P ayne , Equiv ariant Chow cohomology of toric v ari eties, Math. R es. L ett. 13 (2006 ), 29–41 THE EQUIV ARIANT COHOMOLOGY OF WEIGHTED PROJECTIVE SP ACE 11 Dep ar tment of Ma thema tics, Rider University, La wrenceville NJ , 0 8648, U.S.A. E-mail addr ess: bahri@rider. edu F a chbereich Ma thema tik, Universit ä t Konst anz, 78457 Konst anz, Germa ny E-mail addr ess: matthias.fra nz@ujf-gr enoble.fr School of Ma thema tics, University of Man chester, Oxfo rd Ro ad, Man chester M13 9PL, England E-mail addr ess: nige@maths.m anchester .ac.uk