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Equivariant Chow groups and multiplicities Nyenhuis, Michael 1993

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EQUIVARIANT CHOW GROUPS AND MULTIPLICITIESByMichael NyenhuisB.Sc., Dalhousie UniversityA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDoctorate in PhilosophyinTHE FACULTY OF GRADUATE STUDIESdepartment of mathematicsWe accept this thesis as conformingto the required standardthe University of British ColumbiaApril 30© Michael Nyenhuis, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of  111 Ci.t4The University of British ColumbiaVancouver, CanadaDate^trj 36 1 9- )DE-6 (2/88)AbstractWe propose a definition of equivariant Chow groups for schemes with a torus action and developthe intersection theory related to it. The equivariant intersection theories that have beenconsidered in the past have been the Chow groups and the K-theory of the quotient scheme,as well as the equivariant K-groups of the original scheme. The equivariant Chow groups arerelated to all of these. At first glance, we would expect a strong relationship with the equivariantK-groups. As it turns out, the equivariant Chow groups are more closely related to the Chowgroups of the quotient scheme.We chose to restrict to tori since for them the equivariant cycles are of a particularly niceform. For general groups the equivariant cycles are harder to describe, and so the intersectiontheory is far messier, if it even exists. By restricting to tori, we are able to define an equivariantmultiplicity that behaves similarly to the degree in the projective case. In particular, we areable to show that for certain schemes, the equivariant multiplicity of an equivariant cycle inthe equivariant Chow group is defined and is an invariant of that cycle.While much of this work involves generalizing the work of others, in particular the work ofFulton, Rossmann and Borho, Brylinski and Macpherson, our approach is new. The equivariantChow groups have not been considered in the past and relating the equivariant multiplicitiesto the equivariant Chow groups is new as well.iiTable of ContentsAbstractAcknowledgments1^Introduction2^Basic Constructionsiiv152.1 Notation ^ 52.2 Elementary Definitions and Results ^ 62.3 Cones and Blowing-Up ^ 82.4 Equivariant Bundles 113 Formal Characters and Equivariant Multiplicity 133.1 Characters^ 133.2 Equivariant Multiplicity ^ 224 Equivariant Chow Groups 264.1 Equivariant Chow Groups ^ 264.2 Proper Pushforward 274.3 Cycle Associated to a Scheme ^ 294.4 Alternate Definition of T-Rational Equivalence ^ 294.5 Flat Pullback ^ 324.6 An Exact Sequence 334.7 Chow Schemes and Changing Tori ^ 344.8 Affine T-Bundles ^ 39iii4.9 Notes ^  415 Multiplicities on Varieties^ 425.1 Multiplicity on Projective Varieties ^  425.2 Equivariant Multiplicities on Varieties  526 Intersections of T-line bundles^ 566.1 T-Cartier Divisors ^  566.2 T-pseudo Divisors  596.3 Intersecting T-pseudo Divisors ^  626.4 Commut at ivity ^  646.5 Intersection with T-line bundles ^  726.6 Notes ^  747 Intersections with T-Vector Bundles^ 767.1 T-Segre classes ^  767.2 T-Chern Classes  847.3 Notes ^  938 Applications and Relations^ 948.1 A(X I IT) ^  948.2 KT(X)  988.3 4 (X)^  99Bibliography 101ivAcknowledgmentsThis thesis is dedicated to my parents, Mieke and Tjeerd Nyenhuis whose support and encour-agement through the long years of my Ph.D was invaluable. Special thanks go to my advisor,Dr. James Carrell, who provided the initial impetus for this work, and without whose guidancethis thesis would never have been completed.vChapter 1Int rod uct ionIn this work, we propose a definition of equivariant Chow groups and work out the intersectiontheory associated to it. For varieties with a group action, the equivariant intersection theoriesconsidered in the past have involved the Chow and K-groups of the quotient varieties as wellas equivariant K-theory on the original variety. The Chow group we propose is related to all ofthese. With respect to KT (X), the equivariant Chow groups provide a geometric interpretationsimilar to that existing between the usual Chow and K-groups. Unfortunately, the analogue isnot quite as strong as we would like. As it turns out, the equivariant Chow groups are moreclosely related to the Chow groups of the quotient variety. For these, the main advantage isthat the equivariant Chow groups are defined on the original variety rather than the quotient,and so are easier to calculate. In defining the equivariant Chow groups, we have followed Fulton[6] very closely. By and large, the only difference between his work and ours is that we need tocheck for equivariance.The groups we consider almost exclusively are tori. The reason for this is that cycles stableunder a torus action are of a particularly nice form, and the intersection theory is fairly clean.In the last chapter we do consider briefly the case of reductive groups. The problem with agroup G with maximal torus T is that the G stable are not particularly nice, so that any typeof rational equivalence that respects the group action is quite messy. We also find that thesegroups are subgroups of the equivariant Chow groups with respect to the torus.By restricting to tori, we are also able to avail ourselves of an equivariant multiplicity similarto the degree in the projective case. In particular, for varieties of a certain form we are able toshow that the equivariant multiplicity of a cycle at a fixed point is an invariant of that cycle.1Chapter 1. Introduction^ 2So, along with the Chow groups, we also develop equivariant multiplicities. Given a projectivevariety X stable under a torus action and an equivariant vector bundle E defined on X, we areable to obtain in a purely combinatorial manner a characteristic number formula relating theweights of E over the fixed points to the geometric multiplicity.Since we are proposing a new definition, very little has been done on equivariant Chowgroups. The individual pieces have been covered extensively, though. Quotient varieties havebeen considered in the past by many authors. In particular, for intersection theory, we haveKraft [11] and Knopf who have considered vector bundles on the quotient varieties, Danilov [4]and Ellingsrud and Stromme [5] who have found the Chow groups of various quotient varieties.Equivariant multiplicities have been considered by Rossmann [17] who has calculated them onsubvarieties of smooth varieties with torus action, and by Joseph and Borho, Brylinski andMacpherson [1] who consider the quotient variety X = G/B, where G is a Lie group. Infact, Borho, Brylinski and Macpherson relate the equivariant multiplicity to KT(X). All thesereferences work on the tangent space of the variety though. We, however, define the equivariantmultiplicity on the variety itself. Finally, intersection theory as developed by Fulton [6] playsan important role in our work. In fact, except for the equivariant multiplicity results, we haveessentially translated his results from the non-equivariant setting to the equivariant one. Withrespect to KT(X), we have Nielsen [16] and Iversen and Nielsen [9]. The former has shown thelocalization theorem for non-singular projective varieties, and the latter has arrived at a formulavalid for 1-dimensional tori involving the Chern classes of a vector bundle and the equivariantmultiplicity of a zero cycle. A similar result has been shown by Brion [3].In the second chapter, we introduce the notation, definitions, results, and constructions weshall be using throughout. These are the same as those of usual intersection theory, but theyare done in the equivariant setting.We define the notion of equivariant Chow group in the third chapter. We show that theproperties Fulton considers in [6] Chapter 1 hold in the equivariant case. We also consider howthe equivariant Chow groups change as we change tori. This allows us to relate the equivariantChapter 1. Introduction^ 3Chow groups for different tori.In the fourth chapter, we consider formal characters and equivariant multiplicities of mod-ules. Since we need equivariant multiplicity in fairly great generality, we consider a definition ofthe formal characters that is valid only under certain conditions, but which is valid independentof the sign (or zero-ness) of the weights. This allows us to define the equivariant multiplicityunder certain conditions independent of what the weights may be.In the fifth chapter, we consider equivariant multiplicities on varieties. We show that forprojective varieties this is defined for T-stable cycles and is an invariant of the cycle in theequivariant Chow groups. We also see how multiplicities behave with respect to various maps,and we derive a few equations concerning multiplicities on vector bundles. For non-projectivevarieties of a particular form, this allows us to show that the equivariant multiplicity is also aninvariant of the cycle in the equivariant Chow group.We define equivariant intersection with an equivariant line bundle in the sixth chapter. Inthe third chapter, the generalization of Fulton [6] was fairly straightforward. In this chapter,the differences between equivariant and usual intersection theory begin to show. We define theintersection as in Fulton [6]: to a line bundle we associate a section, a Cartier divisor, and aWeil divisor. We have several possibilities of definitions of equivariant intersection, dependingon what conditions we place on the section. We chose to demand that it be equivariant. Theadvantage of this choice is that equivariant intersections then have equivariant properties. Inparticular, we are able to find the equivariant multiplicity of the intersection in terms of theweights of the line bundle and the equivariant multiplicity of the cycle. The problem we haveis that an equivariant section need not exist. This means that the properties that Fulton showsin [6] need not hold in all generality. We show they do hold, subject to the existence of sectionson the appropriate varieties.In the seventh chapter, we consider intersections with equivariant vector bundles. As in theprevious chapter, we have problems with the existence of sections. This forces us to restrictthe vector bundles we can intersect with. Because of the conditions we must place on theseChapter 1. Introduction^ 4bundles, it does not seem worthwhile to define an equivariant intersection of cycles on a varietywith T-action. We also consider equivariant multiplicities with respect to these intersections.We derive some characteristic number formulae for these.In the eighth chapter, we consider previous work that relates to our work. We show that ifthe action is free, the Chow group of the quotient is the equivariant Chow group, after tensoringwih Q.We also consider how the equivariant Chow group is related to the equivariant K-groups.This shows that equivariant multiplicity is also an invariant of the quotient variety.Chapter 2Basic ConstructionsWe collect the basic definitions, results and constructions we shall be using throughout thiswork. These are the usual ones of algebraic geometry, but we do them in the equivariantsetting. The constructions are cones, blowing-up and vector bundles. These have all been donein Fulton [6] appendices A and B.2.1 NotationNotation:k is an algebraically closed fieldT is an r-dimensional torus with Lie algebra TX : T Gm is a character of TX(T) is the set of charactersR(T) = Z[X(T)] is the representation ringA =dx:r-4kis the differential of x, called the weight of xIf T= EK 19,72 , we will occasionally write A as (Ai, A2,^, A r )•A(T) is the lattice of weightsIn general, if f is a weight vector of a module M with T-action, we willwrite its weight as Af, and character as xf.The main schemes we will be considering are subschemes of Pn and A.Let Pn be the projectivization of An+ 1 = Spec(k[xo, , xd), where xi is a weight vector ofweight —Ai. Let Pi = (0 : : 0 : 1 : : 0) be the point in Pn with a 1 in the i-th position.Let Ui = Uzi . We also assume that x E (pn)T is Po.5Chapter 2. Basic Constructions^ 6If P 1 has the T-action defined by the weight A, we will write it as P. Unless otherwisestated, we will be assuming that PI = Proj (k[xo, xl]), where xo has weight 0 and x1 has weightA. Unless otherwise stated, P 1 will be P6.If An has weights Ai = (Ali, ... , A ni), we will occasionally write these as/ An Al2 Aln A21 A22 A2n(2.1)•••\ Arl Ar2 Arn •While the form of the matrix may seem odd, it turns out that the kernel of the linear transfor-mation defined by this matrix is the cycles T-rationally equivalent to 0 in An .2.2 Elementary Definitions and ResultsDefinition: X is a T-scheme if X has a T-action defined on it. We write the T-action asa:TxX—>X^ (2.2)(t, x)i— t • x.In general, we assume that all varieties are defined over the field k and are irreducible andreduced. We will also be assuming that there is a cover of X by open affine T-subsets U suchthat U is a T-subscheme of Am for some m. We make this assumption so that Ou has aT-grading. Note that if X is normal, then Sumihiro's Theorem shows that X is of this form.Remark: The T-action on P 1 is defined by a character. So, the set of T-actions on P 1 is in1-1 correspondence with X(T).Definition: f : X —0' is a T-morphism if f(t • x).-- t•f(x) in Y. We occasionally call suchmorphisms equivariant morphisms.Remark: Note that the set of weight vectors f E R(X)* of weight A is in 1-1 correspondenceChapter 2. Basic Constructions^ 7with the maps f : X -+ Pi. In fact, if f* : Opi^Ox is the morphism on structure sheaves,then f E R(X) * is defined by f * (xi/x0)•Proposition 2.2.1 If X is a T-scheme, then its normalization is a T-scheme.Proof: This is a local assertion. Suppose that X is an affine T-scheme and A = Ox. Let Kbe the quotient field of A and let A' be the integral closure of A in K. Since A has a T-action,K does as well. We want to show that A' is invariant under the T-action. Suppose that r E A'solves the equationxn + an_ixn -1 + • • + ao = 0.^ (2.3)Since(t • r)n + t • an_1(t • r)n -1 -1- • • t • ao = t • (rn + an-irn-1 + • • + ao) = 0,^(2.4)t-r E K solves the equationxn t • an_ ixn- 1 +...  + t • ao = O.^ (2.5)So, t • r E A', and A' is T invariant.^ 0Lemma 2.2.2 Let f E R(X)* be a weight vector of T. If U is an open affine T-subset of X,there are weight vectors a, b E Ox (U) such that f = alb.Proof: We proceed by induction. Note that for the trivial torus, the result is obvious. Supposethe result is true for dim T'^n - 1, and T =^x Gm. Then, locally f = alb wherea, b E Ox(U) are weight vectors of T'. Gm induces a grading on Ox(U), so a = E iEzb = E iEz bi, and all but a finite number of the ai and bi are zero. Let io and jo be the smallestintegers such that ai o 0 0 and bio 0 0. Then,a^aio + E i>jo ai(2.6)b^4-Ei>jobj.Chapter 2. Basic Constructions^ 8So, if f has weight m with respect to Gm ,t • -1; = en =  . °^=^°^a^a ticiai + E tiai ai + Eb tlobjo + E Obi^bio + E ti—job;^(2.7)So,tin (abjo + a E^= tio—jo (baio^ai).^(2.8)Since Ox(U)[t] has the usual Z-grading, we have tm =^and abjo = baio , or alb = aio lbjo •The reason we demand that X be locally a subscheme of affine space is so that this lemmaholds. If we could ensure that f was always locally a ratio of weight vectors of Ou, we wouldnot need the assumption.We have the following basic lemma on T-schemes:Lemma 2.2.3 If X is a T -scheme with components Xi, then the Xi are T invariant.Proof: This is just the statement that the minimal associated primes of a module with T-actionare T stable. For a more geometric proof, we consider the map,^  : T x X1— X^ (2.9)(t,x)1-4t • x.Since T x Xi is a variety, a(T x Xi) is as well. Since Xi is a maximal subvariety of X andC a(T x Xi), we have a(T x Xi) = Xi.^ ^2.3 Cones and Blowing -UpThe constructions of this section are found in Fulton [6] appendix B.5 and B.6. Let S =be a graded Ox-module with a T-action such that1. So is T-isomorphic to Ox,2. Si is locally generated by a finite set of elements^• • • , ft.} such that t • fi E Si,Chapter 2. Basic Constructions^ 93. Si generates S as an Ox-algebra.We associate to S its cone and its projective cone,C(S) = Spec(S,7_0S7,),^ (2.10)P(C(S)) = Pro.i(e,7—oSn). (2.11)Since S has a T-action, these schemes are T-schemes.The cones we are particularly interested in are those defined by T-subvarieties and blow-ups.Let X be a T-subscheme of a T-scheme Y. Ox is defined by a sheaf of T invariant ideals Iof Oy. The cone to X in Y is defined asCx (Y) = SpeC(Ccto/n/p+1 ).^ (2.12)Cx (Y) is a T-scheme. Since the map Ox 10/I is a T-isomorphism, the induced morphismon schemes p : Cx (Y) X is a T-morphism.If Y is a T-subscheme of An for some n, then the generators of I can be chosen to be weightvectors. These weight vectors determine the action on Cx(Y). When Y is locally a subspaceof An , this construction glues together to define the cone.Note that if Y is nonsingular, x E YT and mx = (h, ..., fn.) is the maximal ideal definingx where the h are weight vectors, thenCx (Y) = Tx (Y) = Spec(Symm(mx /mD v )^(2.13)andOTT(y) = SYmm(mxim!) = 0 Nt (Y)•^ (2.14)With the same notation, the projective cone isP(Cx(Y)) = Pro.i(e,70In fin+1).^ (2.15)Chapter 2. Basic Constructions^ 10Again, this is a T-scheme and the morphism p : P(Cx(Y)) -4 X induced by Ox c I° II is aT-morphism.If Y is a T-subscheme of An for some n E Z, we can construct P(Cx(Y)) more explicitly.Let I = (fi,...,fr ) where the fi are weight vectors of weight -Ai. P(Cx(Y)) is a T-subschemeof X x pr-1 where pr-i Proj(k[fi, • • • , id) = Proj(k[xl,..• , x,.]). Glueing this local con-struction together, we get P(Cx(Y)) for the non-affine case.Note that if we set the weight of xi to be -Ai - A for all i and for some A E X(T), then theaction induced on P(Cx(Y)) is the same as the original one.P(Cx(Y)) has a canonical line bundle defined on it. Locally, we can define this as the pullback of 0(1) from Pr-1 to P(Cx(Y)).We can also close a cone off in a projective space. Let S[z] be the graded algebra withgraded piecesskin = Sn ® (sn_1 ® z) ® (sn_2 ® z 2) e ® zn.^(2.16)We setP(S 1) = Proj(S[z]).^ (2.17)If we set A = 0, then the open set defined by inverting z is T-isomorphic to the cone C(S).If X y Y is a T-subscheme of Y, the blow-up of Y along X is defined byBlx(Y) = Proj(C°_0In ).^ (2.18)The map it : B1 (Y) Y induced by Oy I° is a T-morphism, and it is proper, birational,and surjective.As before, if Y is a T-subscheme of An , I =^, fr ) where the fi are weight vec-tors of weight -Ai. Blx(Y) is a subspace of Y x Pr -1 where Pr -1 =^fr]) =Proj(k[xi,... ,x,•]). If we set the weight of xi to be -Ai - A, then the action on Blx(Y) isunchanged. As before, this glues together to give B1 (Y) in the non affine case.Chapter 2. Basic Constructions^ 11Let E be the pullback of X to Blx(Y). This is an effective Cartier divisor and is definedlocally by pairs ((la , fa), where fa E Otra is a weight vector. In fact, E is the Cartier divisorassociated to the canonical line bundle on Blx(Y) obtained by pulling back 0(1) from Pr -1 .2.4 Equivariant BundlesThe contents of this section are to be found mostly in Fulton [6] apppendix B 3.A T-vector bundle p : E^X of rank e on a T-scheme X is defined by the followingconditions:1. There is a collection of open T-subsets {Ui of X such that coi : E 1u4^U, x Fi, Wiis a T-morphism and Fi is a representation space for T of dimension e,2.So1 ocoi 1 :^Ui F^Ui 11 Ui F'^ (2.19)is the identity when restricted to Ou,nu, x 0 and is linear on the F's. By this we meanthat on the structure sheaves, the 99i 0 6 1 are defined by e x e invertible matrices withcoefficients in Ou, nu.7 . We denote these maps by gii 0 Lulu, x OF -4 OUinU, X OF'•E can be determined by either the (pi, or theNote that the actions on F and F' need not be the same. So, while the yoi o 6 1 areequivariant, the gii need not be given by T invariant matrices.Definition: We define a weight section s of E of weight A = dx to be a morphism s : X -4 Esuch that p o s = id, and t -1 • s(t • x) = X(t)s(x). s is a T-section if A = 0.A weight section is defined by a collection of weight vectors fi E R(Ui) * such that fi = gii fi.Since we will be especially concerned with T-line bundles L, we consider this case explicitly.Chapter 2. Basic Constructions^ 12If the Fi are spanned by the weight vectors Xi, then F1 = Spec(k[xi]) are 1-dimensionalrepresentation spaces, so the action defined on the Fi is defined by a single character Xi withassociated weight Ai. The gi•  are invertible functions in Ouinui . Since the 99i o 6 1 are equivari-ant, the gib are weight vectors of weight Ai - Ai. A weight section s of L of weight A determinesweight vectors fi E R(Ui) * of weight -Ai - A. To see this, let s : OL Ox be the morphismon structure sheaves. Locally, we require that on the scheme, t - s(t-1 • xi) = x(t- i)s(xi). Thistranslates as t • xi(t- i)s(xi) = x(t -1 )s(xi) on the structure sheaves. So, s(xi) E R(Ui)* is aweight vector of weight -Ai - A.Definition: We say a T-line bundle has weight A at x E XT if the weight of the representationspace in Lx is A.To projectivize T-vector bundles, we consider them as equivariant Ox-modules and takeP(E) = Proj(SymmEv ).^ (2.20)P(E) is a T-scheme. If E = U x F locally where F has a basis of weight vectors X1, • • • , Xe,with related functions xl, , x e , then on the open subset U x Uk of P(U x F) defined byinverting xk, the xi/xk have weight -A + Ak, and the associated vectors Xi/Xk have weightA3 - Ak. If we set the weight of Xi to be Ai + A for all i and some A, then the action on P(E)is unchanged.P(E) has a canonical T-line bundle 0(1). On U x Uk it is defined by the free Ou x uk -modulegenerated by 1/xk. So, over U, 0(1) has weight Ak.Chapter 3Formal Characters and Equivariant MultiplicityWe define the formal character and equivariant multiplicity associated to a finitely generatedmodule over a polynomial ring, both with T-action, and develop some of its properties. Whilemost of the material in this chapter can be found elsewhere, it seems that none of it canbe found in a single reference. In particular, the results concerning changes of torus seemto have only been used implicitly. In the first section, we define the formal character of afinitely generated module with T-action for non-zero weights and then extend this definitionunder certain circumstances to the arbitrary weight case. In the second section, we use formalcharacters to define the notion of equivariant multiplicity. References for the material in thischapter are Borho, Brylinski, and Macpherson [1] and Rossmann [17] . We follow mainly thepresentation of Rossmann.3.1 CharactersIn the past, formal characters have been defined mainly for the "un-mixed" weight case. Sincewe need characters for arbitrary weights, we follow the presentation of Rossmann [17] and definethem indirectly for the non-zero weight case. We then show how the formal characters behaveas we change tori. Finally, we consider bi-graded modules and extend the definition of formalcharacters to this case. Using the change of torus property, we can then define the formalcharacter for the "mixed" and zero weight cases.Let R = k[xi,x2,...,x n ] be a polynomial ring on which T acts diagonally. So, t•xi = xi(t)xifor some Xi E X(T), for all i.Definition: M is a R, T module if M is a finitely generated R-module with T-action such13^Chapter 3. Formal Characters and Equivariant Multiplicity^ 14that it has a finite set of generators^fm that are also weight vectors. Recall that ifM has a T-action, then t • (rm) = (t • r)(t • m) for t E T, r E R and m E M.If M is a R, T module, let x E X(T) and dx = A.Mx = MA ={f E M:t•f=x(t)f for all t}.^ (3.1)We haveM = eAEA(T)MA•^ (3.2)If all the MA are finite dimensional, the usual character as defined by Borho, Brylinski andMacpherson [1], say, ischT(M)^E (dim MA)e A^(3.3).EA(T)chT(M) E S-1R(X(T)), where S is the multiplicative set generated by the 1 - eA for allA E A(T).We wish to consider the more general case where dim MA may be infinite To do this, wehave to define the character indirectly. Rossmann [17] has shownTheorem 3.1.1 If M is a finitely generated R, T module such that Ai 0 0 for any i, thenchT(M) E S-1R(X(T)) is defined uniquely by:I. If M is finite dimensional as a k-module, then^chT(M) = E (dim MA )e A^(3.4)AEA(T)2. If0 -4 M' M M" -> 0^ (3.5)is an exact sequence of R, T modules, thenchT(M) = chT(0)chT(M").^ (3.6)Chapter 3. Formal Characters and Equivariant Multiplicity^ 153. If F is a finite dimensional T-module, thenchT(M F) = chT(M)chT(F).^ (3.7)Furthermore, chT(M) is of the form,chT(M) = AT(R) -1 E aAe A^(3.8)AEA(T)where AT(R) =^eA0, as E Z, all but a finite number of the a), are zero, and if as 0 0,then A is a sum of the Ai and the Afi .Proof: The proof is contained in Rossmann [17]. The method of proof is to resolve M intofree R, T modules and then take the characters of these. AT (R) comes from setting Rik[xi,^, xn] and considering the exact sequence0^xiRi --+^Ri+i --÷ 0.^ (3.9)We have chT(Ri+i) = (1 — eAs)chT(Ri), and using induction then yieldsChT(R) = 11(1 —^)-1. (3.10)i=1Remark: If dim MA < oo , for all A, Borho, Brylinski and Macpherson show that the usualcharacter r EA (T) (dimMA)eA satisfies all three properties. So chT(M) is really an extension ofthe usual character.Note that if T = Gm and all xi have weight 1, then we get the usual Poincare seriesEZ)=0 dim Mn t" for M.The theorem has the following corollary:Corollary 3.1.2 If = k[yi,... ,yk] with the yi as weight vectors, then M R' is a R 0 ,T-module, andchT(M 0 R') = chT(M)chT(R').^ (3.11)Chapter 3. Formal Characters and Equivariant Multiplicity^ 16Proof: If M has resolution0 MN--+ ••Mo M -4 0,then M R' has resolution0 -4 MN 0 /7' -4 • • • -4 MO Rt M RI —> O.The result then comes from chT(R R') = AT(R) -1 AT (RI ) - 1(3.12)(3.13)0As with R-modules, R, T modules have composition series of R, T submodules. To showthis, we start with the following lemma:Lemma 3.1.3 If v is a weight vector of M, then its annihilator Ann(v) is a T invariant ideal.Proof: Note thatav = 0 •:=> xv(t)(av) = 0 <#. a(t • v) = O.^ (3.14)So, Ann(v) = Ann(t • v). Also,av = 0 e#, (t • a)(t • v) = t • (av) = O.^ (3.15)So, Ann(t • v) = t • Ann(v).^ ^Theorem 3.1.4 I. If M is a R, T module, then M has a composition series of R, T modules:0 = Mo C M1 C • • • C Mk = M^ (3.16)where Mil M;_1 is T -isomorphic to the 1-dimensional R, T module RIPi • vi, where Pi is a Tinvariant prime ideal of R and vi E M is a weight vector of weight pi .2.chT(m) = EeP'chT(RIPi).^ (3.17)Chapter 3. Formal Characters and Equivariant Multiplicity^ 17Proof: For 1, the proof is exactly as in Lang [12]. All that needs to be remarked is that inchoosing a vector in M, we can choose it to be a weight vector, so that its annihilator is a Tinvariant ideal. Note that if all the weights are of one sign, this is just be the decompositioninto highest weight modules.2. This follows from the exactness property of chT( — )•^ ^We will also need to know how chT(M) behaves as T changes. As far as we know, theseresults have only been used implicitly, but have never been proved explicitly.Suppose co :^T is a (homo)-morphism of tori. Let t' E T' . If T eli' iGm , we candecompose w into the characters cc :^—+ Gm . So, w = (w1, , w r ). We setdyo = (dwidt' ,^dco,.dt').^ (3.18)Suppose x : T -4 Gm is a character. Considering x o yo : —4 Gm as a function, we see thatd(x o cc)^(dr) o (4).^ (3.19)Theorem 3.1.5 Let co :^T be a morphism of tori, and define the T'-action on R byt' • x = w(e) • x for all x E R. Let M be a R, T and a R, T' module such thatso01r M^T 0 M\M(3.20)commutes. Then chT,(M) = chT(M) o dw.Proof: We prove the results by considering the properties of chT(M)odw. Suppose M is finitedimensional as a k-module. If A' is a weight of T', then MA, is generated by T' weight vectorsvi, , vi. Let dx' = A'. Since each vi has a decomposition into T weight vectors, we get,MA' C 84=1/1/4^ (3.21)Chapter 3. Formal Characters and Equivariant Multiplicity^ 18for some set of weights Ak E A(T). We assume this set is minimal. Now, ifvi = wi + • • • + Wk,t' • vi = X (e)vi = X1 ° C5'(e)wl + " • + Xk ° 60 (e)wk,and Xi 0 co = x' for all j. So, MA, C MA, for all j, andkvu'j=1 Mai = MA'.So, if M is finite dimensional as a k-module,kchr (M) = E (E dim MA, )ear = E (dim mA )dodw = chT(M) o dcp.VEA(r) j=1^AEA(T)(3.22)(3.23)(3.24)(3.25)The other two properties, exactness and multiplicativity, follow from the exactness and multi-plicativity properties of chT(M). 0This restriction property allow us to define chT(M) under certain conditions when some ofthe Ai are zero. Suppose R has its usual grading by degree and M is a graded R,T module.We will call such modules bi-graded R,T modules. The usual grading determines a G m-actionon R and on M, which we label by T1. So, if M is bi-graded, then it is a R, T x T1 module. IfA2 0 0 for any i, letcoc, : T -+ T x^ (3.26)t^(t, 1 ).Then,chT(M) = chTxTi(M) o dcpand since ATxTi (R) o dcp = AT(R),AT (R)chT (M) — (AT xi\ (R)chT xTi (M)) dcp.Since dcp = 1 ® 0, we write this asAT(R)chT(M) = (tTxT1(R)chTxT1(M)) IT(3.27)(3.28)(3.29)Chapter 3. Formal Characters and Equivariant Multiplicity^ 19Definition: Suppose M is a bi-graded R, T module. Then^AT(R)ChT(M) ATxTi (R)ChTxTi (M) IT •^(3.30)Remark: We could have extended the formal character in several ways: for the multiplicity,we really only need the numerator of the formal character, so we could have ignored the de-nominator even when some of the Ai were 0. Alternatively, we could have taken a residue. Theformer has the problem that its formal properties are not easy to prove, while the latter hasthe problem that the restriction property of tori becomes hard to state. Since all modules weconsider are bi-graded R, T modules, we decided to use the restriction property even thoughgreater generality could have been obtained using residues.In future, if M is a bi-graded R, T module, and we do not specify Ai 0 0 for all i, we shallbe using this definition. For completeness, we list its properties.Proposition 3.1.6 Suppose that M is a bi-graded R, T module.1. If Ai 0 0 for all i then the two definitions of L.T(R)chT(M) agree.2. If0 -4 M' M M" 0^ (3.31)is an exact sequence of bi-graded R, T modules, thenAT (R)chT(M) = AT (R)chT (0) + T (R)chT (M).^(3.32)3. If F is finite dimensional as a k-module, and is a bi-graded R, T module, then^AT(R)chT(M F) = AT(R)chT(M)chT(F).^(3.33)where chT(F) is the usual character of F.Chapter 3. Formal Characters and Equivariant Multiplicity^ 204.AT(R)chT(M) = > ^ (3.34)AEA(T)where a), E Z, and all but a finite number of them are zero. Also, as 0 0 implies A is a sum ofthe Ai and the AA.5. If M is a bi-graded R, T module, R' =^,y,.n] has a T-action, thenAT (R ® R')chT(M ® R') = AT (R)ChT (M)•^ (3.35)6. Let co :^-4 T be a morphism of tori, and define the action of T' on R by t' • x yo(e)•x.If M is a bi-graded R, T module, and‘oxiT'xM --+ TxM\^(3.36)Mcommutes, then(AT(R) o cl(P)AP(R)chT'(M) = (ATI (R))(6•T(R)chT(M)) o dw.^(3.37)7. If f E R is a weight vector and is homogeneous in the usual sense and f acts as anon-zero divisor on M, thenAT(R)chT(M f M) = (1 — eAf )AT(R)chT(M).^(3.38)8. If M is a bi-graded R, T module, then M has a composition series0=M0-4Mi-4•••—>Mk=M^ (3.39)of bi-graded R, T modules such thatRI Pi • vi^ (3.40)Chapter 3. Formal Characters and Equivariant Multiplicity^ 21for Pi a T x T1 invariant prime ideal of R and vi a T x T1 weight vector of M. Under theseconditions,k(R)ChT (Al) =^eAvi AT (R)chT(R/ Pi).i=1Proof: Only 6 is not a direct consequence of previous results. For 5, recall thatchTxz (M 0^= chTxT, (M)ATxTi (RI ) - 1 •and AT x Ti (R 0 R') = AT x71(R) AT xTi(-RI) . So,ATxTi (R -Fe)chTxTi (M 0 R') LTXTi (R)ChTxTi (M)Restriction to T now yields the result.For 6,(3.41)(3.42)(3.43)(AT^(R) o (dyo x id))AT' x T1 (R') h^(M) A^(pp\ (A^(pm^114.\ ^id))k C..71 X T1^/^—T' xTi k-^XT1 k.. 1./CrbTXTikl."1 ) kwp x 'tun(3.44)and on restricting,AT (R) 0 d(6°)(AT, (R)ChT'( 1)) AT' (R)(AT(R)chT(M) o^(3.45)For 7, while we have not stated it explicitly for non bi-graded modules before, we have usedit in Theoren 3.1.1 to show chT(R) = (R)-1 0Remark: Note that 6 does not yield much if Ai = 0 for some i. In this case, all it yields is= 0. Also note that if M and R have trivial T-action, but are Z graded, then chT xTi (M) isthe usual Poincare series, E(dim MOO.Definition: If M is a bi-graded R,T module, or R has no Ai = 0 and M is a R,T module, wewill say that M is quasi bi-graded.Chapter 3. Formal Characters and Equivariant Multiplicity^ 223.2 Equivariant MultiplicityUsing the formal character defined for quasi bi-graded modules, we define the equivariant mul-tiplicity of a quasi bi-graded module. While equivariant multiplicities have been considered forthe non-zero weight case, the zero weight case has not to have been treated in the past.Definition: Suppose M is quasi bi-graded. If we consider AT(R)chT(M) as a function inA(T), where ea = 1 + A + a2 +•• • , then we define the equivariant multiplicity multT(M, R) ofM as the first non-identically zero term in LT(R)chT(M).From the properties of AT (R)ChT(M) we get:Proposition 3.2.1 Suppose M is a quasi bi-graded module.1.multT(M, R) = N— E aAAN1^! AEA(T)^ (3.46)for some N E Z, as E Z, all but a finite number of the as are zero, and as 0 0 implies that Ais a sum of the Ai and the Afi .2. If R' is a polynomial ring with T-action, thenmultT(M 0 R' , R R') = multT(M, R).^ (3.47)3. Let cp^T be a morphism of tori, and define the^action on R by t' • x yo(e) • xfor all E R. If^T i X M^sox ]^7' X^M^\1 (3.48)Mcommutes, then([J(—A 1 ) o dOmuitr (M,^Y)(multT(M , R) o d(p)^(3.49)Chapter 3. Formal Characters and Equivariant Multiplicity^ 23.1. Suppose f E R is a weight vector which is homogeneous in the usual sense if R has someAi = 0. If f acts as a non-zero divisor on M, thenmultT(M/fM R) = —A fmultT(M , R).^ (3.50)Furthermore, if d(M) is the Krull dimension of M, thend(M/ f M) = d(M) — 1.^ (3.51)5. M has a filtration by quasi bi-graded modules with quotients isomorphic to RI Pi • vi wherethe Pi are equivariant prime ideals of R and the vi are weight vectors of M. With this notation,if J is the set of i where AT(R)chT(RI Pi) has its leading term of minimal degree, thenmultT(M, = E multT (R I Pi, R)^ (3.52)iE,7Proof: Except for 5, these are all consequences of Proposition 3.1.6. The second part of 4 is aproperty of Krull dimension. For 5, M has a composition series with factor modules R/Pi. Sok(R)ChT (M) = E ell' AT (R)chT (R Pi, R)i=i(3.53)for some weights pi. Let J be the set of i where the leading term of AT(R)ehT(R/Pi) is ofminimal degree. Since ePi does not affect the multiplicity, we havemultT(M, R) = E malty. (RI Pi, R).^ (3.54)0Note that 2 and 4 have the following corollary:Corollary 3.2.2 Suppose that R' = k[y] ..yd, where the yi are weight vectors of weight piand M is a quasi bi-graded module. We can consider M as a quasi bi-graded R® R', T module,where (1®y)m = 0 for all y E R' and m E M. Then M MORV( 1 0Y1,• • • , 1 0N-1)(MOR'),and identifying M with this module,multT(M, R 0 R') = II( — ili)multT(M , R).^ (3.55)i= 1Chapter 3. Formal Characters and Equivariant Multiplicity^ 24Proof: 1 0 yi acts as a non-zero divisor of M R7(1 0 yl, • • • ,1 0 yi-i)(Al R')^^We also haveLemma 3.2.3 If M is a zero-dimensional quasi bi-graded R, T module, then^multT(M, R) =^Ai ) dim M.Proof: M is a finite dimensional k-vector space, so,ChTxTi (M)^E^dim M(A x Ai)`'(AxA0EA(TxTi))and(3.56)(3.57))eAxAi.dim M(A x Ain11( ,_ eAixi,ATxTi(R)chTxTi(m).^E^(3.58)i=1 (AxAoen(TxTi)On restricting to T, the first term of (1 — eA, x 1) IT1 is --Ai. On restricting, since the e' in thesum do not affect the first term, we get the result.^ ^By Proposition 3.2.1 part 1, we can consider multT(M, R) as a polynomial in A(T) of degreeN. If M is a quasi bi-graded R, T factor module of R, then multT(M, R) is a polynomial in nvariables of degree N.we can determine N precisely:Theorem 3.2.4 In Proposition 3.2.1 part 1, N = n — d(M).Proof: The proof is a slight variation of that in Borho, Brylinski and Macpherson [1]. Weprove this by induction on d(M). Our hypothesis is that for d(M') < d(M), multT(M, R) is apolynomial of degree N' = n — d(M'), and (-1)N'InuitT(M i, R) as a polynomial is positive onpositive weights.If d(M) = 0, then the previous lemma shows that (-1) Nmu/tT(M, R) is positive on positiveweights. Suppose d(M) > 0. Let f E R — Pi be a non-constant weight vector. If L = RI Pi,multT (L f L, = (— f)multT(L, R)^ (3.59)Chapter 3. Formal Characters and Equivariant Multiplicity^ 25andd(L/ f L) = d(L) —1.^ (3.60)Since (-1)N+ 1 MUitT (L I f L, R) is positive on positive weights, (-1)NMUitT(L,R) is as well.Since multT(M, R) is a sum of the mu/tT(L,R) with positive coefficients, (-1)NmuitT(M,R)is also positive on positive weights, and is a polynomial in A(T) of degree N. ^Note that the theorem implies that while muitT(M,R) may be zero, considering it as apolynomial on A(T), it is non-zero.If T = Gm, and all the xi have weight 1, then multT(M, R) is the usual multiplicity of amodule over a polynomial ring.We also have the following Bezout theorem:Proposition 3.2.5 If f E R is a weight vector acting as a non-zero divisor on a quasi bi-gradedmodule M, then Mlf M has a composition series with factor modules RIPi. Let J be the setof i with d(RIPi) maximal, and mi the number of times each RI Pi occurs as a factor module.ThenmultT(M f^= (—A f)multT (M , R) = E mimultT(R I Pi, R).^(3.61)iEJProof: This is a direct consequence of Proposition 3.2.1 parts 4 and 5 and Theorem 3.2.4. ^Chapter 4Equivariant Chow GroupsIf X is defined over an algebraically closed field k, has a torus action defined on it and islocally isomorphic to a representation space, we define the equivariant Chow groups AT(X),and prove some of the basic results of AT(X). The results we are particularly interested in arethose considered in Fulton [6] and the change of torus properties. Most of this chapter concernsdetermining the former. In doing this, we have followed Fulton very closely. In fact, the resultswe present are really rephrasings of those of Fulton in the equivariant setting. For the changeof torus properties, we need to develop the theory of Chow schemes. We do this in section 4.7.The results of that section are mainly extensions of results of Brion [3].Throughout the rest of this work, we will be assuming that all T-schemes X can be coveredby open affine T-subsets that are T-isomorphic to T-subschemes of T-representation spaces.4.1 Equivariant Chow GroupsWe define the equivariant Chow groups. The definitions are the same as in Fulton [6], withthe exception that we demand that the subvarieties be T invariant and that the functions beweight vectors of weight 0.Definition: We denote the free group on the set of k-dimensional T-subvarieties by Zr(X).Definition: Suppose X is an n-dimensional affine T-scheme and f E Ox is a weight vector.We definedivf = E ordw(f)[W],^ (4.1)w26Chapter 4. Equivariant Chow Groups^ 27where the sum is over all codimension 1 subvarieties of X and ordw (f ) = iow,x (0w,x/f Ow,x)•When X is not affine this local definition glues together to give a global definition of divf .Note that if a E Ox (U) is a weight vector, the closure of the scheme defined by a hasT-varieties as components. So, the set for which ordvf 0 0 consists of T-subvarieties anddivf E Z4:_1(X).If X is an affine T-scheme and if f = a/b E R(X)*, we setdivf = diva — divb.^ (4.2)When X is not affine, this local definition glues together to give a global definition. Since thereare weight vectors a' and b' in Ox such that f = lb', diva' and divb are T-cycles, divf is aT-cycle as well.Definition: a E ZT(X) is T-rationally equivalent to 0 if there exists a collection of k 1-dimensional T-subvarieties , Vn of X and weight vectors fi E R(Vi)* of weight 0 suchthata = Edivfi.^ (4.3):=1We define AT (X) as 4(X) modulo this relation.4.2 Proper PushforwardLet p : X -+ Y be a proper T-morphism of T-schemes. We define p. : ZIT (X) —* 4(Y). If Vis a k-dimensional T-subvariety of X, let d = [R(V) : R(p(V))]. We set0^if dimp(V) < dim Vp.(V) =1 d[p(V)] if dimp(V) = dim V.(4.4)We extend this definition by linearity to p. : 4(X) —> ZIT (Y).Chapter 4. Equivariant Chow Groups^ 28Definition: Let L be a finite field extension of the field K. We consider L as a finite vectorspace over K. For f E L, we define the norm Nk(f) of f to be the determinant of the K-linearmorphism defined by multiplication by f in L.This definition agrees with the definition of the norm as a product of conjugates, as foundin, say, Lang [12]. This is shown in Godement [7] chapter 26, exercises 4 and 5.Theorem 4.2.1 Let p : X^Y be a proper T-morphism of T-schemes. If a E 4 (X) isT-rationally equivalent to 0, then p * (a) is T-rationally equivalent to 0 in 4 (Y).Proof: We can restrict to the case where X is a k + 1-dimensional T-variety, Y = p(X),f E R(X) * is a weight vector of weight 0 and a = divf. The result is then a consequence ofthe following proposition.Proposition 4.2.2 Let p : X —> Y be a proper surjective T-morphism of T-varieties, andf E R(X)* be a weight vector. Thenp* (divf ) = {0^if dim X > dim Y^[divNk(f)] if dim X = dim Y^ (4.5)where K = R(Y) and L = R(X).Proof: The proof is as in Fulton, Proposition 1.4. The only thing we have to check is thatif X is a T-variety, then its normalization is also a T-variety. This, however, was shown inProposition 2.2.1. ^To complete the proof of Theorem 4.2.1, note that if f is a weight vector in R(X)*, thenNk(t • f) = Nk(xi(t)f) = xf(t) dNk(f),^(4.6)where d = [R(Y) : R(X)]. So, if Af = 0, Nk(f) is a weight vector in R(p(V))* of weight 0. ^Chapter 4. Equivariant Chow Groups^ 294.3 Cycle Associated to a SchemeIf X is a T-scheme with components Xi, let mi = /0x. ,x (Oxi ,x) and set^[X]. Emi [Xi] E Z,T(X).^ (4.7)If X is a T-subscheme of a T-scheme Y, then [X] E ZT(X) C Z,T(Y), and we write [X] for theasociated cycle in ZT(Y) as well.Note that the subscheme defined by a single function f E Oy has [divf] as its associatedcycle. This is a local result. If Y is an affine T-scheme, let A = Oy and let the prime idealdefining Xi be P. If we identify the image of f in AP with f and if we identify the image of Pin A/ fA with P, then Ox iy = Ap, Ox,,x (Al f A)p and AP/ f AP (A/ f A)p. This yields,^ordx 4 f = lAp(AP f AP) = 1 (Al fA)p((A1^f A)P) = mi.^(4.8)Example: This is Example 1.5.1 of Fulton [6]. Suppose f : X Pi is a dominant T-morphismwhere X is a k + 1-dimensional T-variety. f defines a weight vector in R(X)* of weight A. Wedenote this weight vector by f as well. We have,^[1 -1 (0 )]— If -1 (c0)] = [divf].^ (4.9)If 131 has the trivial action, then the weight vector f has weight 0.4.4 Alternate Definition of T-Rational EquivalenceSuppose X is a T-scheme. Let V be a T-subvariety of X x PI such that the T-morphisminduced by the projection onto PlA is dominant Label this map f : V P1. f defines a weightvector f E R(V) * of weight A. Let p : X x Pi —> X be the projection onto X. If P E P lAT ,then f -1 (P) is mapped isomorphically by p onto a T-subscheme of X which we call V(P). So,[V(P)] P*V -1 (P)1,[f_1(0)]— [f -l (00)]= [divf]^(4.10)Chapter 4. Equivariant Chow Groups^ 30in 4(X) and,[V(0)] — [V(oo)] = p* [divf]^ (4.11)in 4(X).If P 1 has trivial T-action, then f E R(V)* has weight 0. So, [V(0)] — [V(oo)] is T-rationallyequivalent to 0.We claim that all cycles T-rationally equivalent to 0 arise in this way.Proposition 4.4.1 Let a E^(X). a is T-rationally equivalent to 0 if and only if there existsa collection of k+l-dimensionalT -subvarieties, V1, ...^of X x P 1 such that the T-morphismsf :^P1 induced by the projection onto P 1 are dominant andnDV=(0 )] — [ 3/4 (00 )] = a.^ (4.12)i=1Proof: The proof is as in Fulton. We have shown the backward implication. For the otherdirection, we need only consider one T-subvariety W of X and a single weight zero vectorg E R(W)* with a = divg. g defines a T-morphism which we label g : W --+ P 1 . Let V be theclosure of the image of the graph morphism,gr:W--+XxP l^ (4.13)w H (w,g(w)).Let f : V —> P 1 be the dominant T-morphism induced by the projection X x Pl P 1 . fdefines a weight vector f E R(V) * of weight 0. From the example of the previous section, wehave[f - 1^- [f -1 (00)] =^ (4.14)The T-morphism p : V -4 W induced by the projection X x P 1 X is birational, so,[V(0)] — [V(oo)] = p* [divf] = [divNR(w") (f)] = [divg].^(4.15)0Chapter 4. Equivariant Chow Groups^ 31This proposition allows us to move a subvariety with respect to a different torus action. Thefollowing has been proved in the case of a connected solvable group and T' = 1 in a differentmanner by Brion [3] in part 1 of the Theorem of section 1.3.Proposition 4.4.2 Suppose that w : T' --4 T is a morphism of tori. Let X be a T and aT'-scheme such thatsox1T'xX -4 TxX\1X(4.16)commutes. If a E 4' (X), then a is r -rationally equivalent to a cycle 0 E ZIT (X). Inparticular, a E Zk(X) is rationally equivalent to a cycle /3 E ZT(X).Proof: We prove this by induction. First, we can restrict to the case where a = [V] E Zr (X).If w(r) = T, there is nothing to prove. Suppose that T = w(r) x Gm . We move V withrespect to the Gm-action. Consider the graph morphism,a: Gm x V -4 (Gm x V) x X (4.17)(t, v)i-+ (t, v, t • v).Let T' act on Gm x V by t' • (t, v) = (t, t' • v), and on Gm x V x X by t' • (t, v, x) = (t, t' • v , t' • x).If Gm acts on Gm x V by t' • (t, v) = (et, v), and on Gm x V x X by t' • (t, v, x) = (et,v,(et).x),then a is a T-morphism. Consider the projection p13 onto the first and third factors. Let A = 1and let r act trivially on P. We inject p13(0- (Grn x V)) into P1 x X and close it to get aT-subvariety W of P1 x X. The projection f : W -4 Pi is a dominant T-morphism. Now,the cycle [W(oo)] is T'-rationally equivalent to [V] and is Gm invariant. Since T = T' x Gm ,[W (co)] is T invariant as well. ^Note that the method of this proposition moves an effective cycle in zr (X ) to an effectiveChapter 4. Equivariant Chow Groups^ 32cycle in ZiT (X).4.5 Flat PullbackSuppose that f : X -+ Y is a flat T-morphism of relative dimension n. For V a k-dimensionalT-subvariety of Y, letfly] = [f -1 (v)]^(4.18)in g_n (X).This extends by linearity to give a morphism,^f* : 4(Y) -4 4+7,(X).^ (4.19)The proofs of the next three results are contained in Fulton [6].Lemma 4.5.1 If f : X —0 1.  is a flat T-morphism, then for any T-subscheme Z of Y,rili = [f -i (Z)].^ (4.20)^Proof: The proof is exactly as in Fulton [6] Lemma 1.7.1. ^Lemma 4.5.2 IfX' --9-^ xIIj f (4.21)Y' ---> Y9is a fibre square with f a proper T-morphism and g a flat T-morphism, then f' is a proper T -morphism, and g' is a flat T-morphism. If a E ZIT (X), then g* Ma) = f,,' gi* (a) in ZT+n (Y').Proof: Again, the lemma is as in Fulton [6], Proposition 1.7.^ ^Chapter 4. Equivariant Chow Groups^ 33Theorem 4.5.3 If f : X -4 I' is a flat T-morphism of relative dimension n and a E ZT (Y)is T-rationally equivalent to 0, then f*(a) is T-rationally equivalent to 0 in Zr+n (X).Proof: The proof is in Fulton [6], Theorem 1.7. The proof depends on the alternate definitionof T-rational equivalence and uses the two lemmas above, as well as the results of section 4.3.^4.6 An Exact SequenceTheorem 4.6.1 Let X be a closed T-subscheme of a T -scheme Y. Let U = Y — X . ThenAT (x)^-4 4(u) -> 0^(4.22)is an exact sequence, where i and j are the inclusion morphisms.Proof: The proof is exactly as in Fulton [6]. First, note thatZr(X) 4 Z (Y ) I t+ Z (U) 0^ (4.23)is exact. Let a E 4(Y). Suppose j* a is T-rationally equivalent to 0 in 4(U). Then, therein a set of k + 1-dimensional T-subvarieties V1, , Vn of U and weight vectors It E R(Vi)*of weight 0 such that a = a_ i [div(fi)]. Let Vi be the closure of j(V) in Y. j induces anisomorphism R([ t) R(R). Let fz be the function in R(Vi) associated to fi. Then, f, is aweight vector of weight 0, and j* (a — E[div:6)] = 0 in 4(U). So,a — E[divn =^ (4.24)for some E 4(X). On passing to T-rational equivalence, we geta =^ (4.25)in gk'(Y).^ ^Chapter 4. Equivariant Chow Groups^ 344.7 Chow Schemes and Changing ToriWe want to know how 4(X) changes as T changes. For this, we need to develop the theory ofChow schemes. The theory is contained entirely in Samuel [18]. Our main result relating theChow groups for different tori is a re-writing of a result of Brion [3]. We only give an outline ofthe construction of the Chow scheme. In this section, we do not assume that k is algebraicallyclosed.Let V C Pn be an r-dimensional subvariety (not necessarily a T-subvariety) with fieldof definition k. We consider the generic projections f : Pn --+ Pr+ 1 . These are defined byequations^ cijXj =Y^(4.26)i=owhere (Xo, . , Xn ) is a point in P" and (Yo, , Yr+1) is a point in Pr+ 1 , and the cii arealgebraically independent over k. Kerf is then an (n — r — 2)-dimensional linear subspace ofPn . Since the cii are algebraically independent over k, ker f and V do not intersect. So, f (V)is an r-dimensional subvariety of Pr+ 1 . f (V) is defined by a single equation Gv(Yi, cij) Ek[Y0, • • • , Y•-I-1] depending only on the cii. We call this the Chow form of V. The coefficients ofGv(Yi,cii) form the Chow coordinate.If a = E nv [V] E Zk(X) is effective, we define its Chow form to be= H^cii)nv^ (4.27)The coefficients of the Chow form of a form the Chow coordinate of a.The set of all Chow coordinates of effective cycles in Zk(X) form the Chow scheme Chowk (X)of X . If a E Zk(X), we will label its Chow coordinate by a E Chowk(X).The Chow scheme has an addition defined on it. If a,^Zk(X), a + /3 E Chowk(X) isthe Chow coordinate of a + E Zk(X).Chapter 4. Equivariant Chow Groups^ 35The T-action on Pri induces a T-action on the Chow scheme. Suppose T acts on Pn byt • Xi = X3(t)XT. We define (t • f) byxocijx, Yi. (4.28)Since t • - :V->t•Vis an isomorphism, the field of definition of t • V is k. So, the Xi (t)cii arealgebraically independent over k, Gt•v(Yi, Xj(t)cij) = Gv(Yi, cif) and the Chow form of t -V isGt.v (Yi cii) = Gv (Yi, Xj(t-i )cii). This induces the T-action on Chowk (X).Rational equivalence of cycles can also be defined in the Chow scheme. The statement ofthe following is in Fulton [6], Example 1.6.3.Theorem 4.7.1 Let X E Pn. If a, a' E Zk(X) are two rationally equivalent effective cyclesin X, then there exists an effective cycle E Zk(X) and a map g : P 1 -> Chowk(X) such thatg(0) = a + /3 and g(oo) = a' + /3.Proof: As usual, we need only consider the case of a single k + 1-dimensional variety V anda single function f E R(V) * such that divf = a - a'. If the zero cycle of divf is 7, let= -y - a. The pole cycle of divf is then a' + f defines a function f : V P 1 , witha - a' = [divf] = [f -1 (0)] - [f (oo)]. We consider 1 -1 (P) for P E P 1 . We associate the Chowpoint of [f -1 (P)] to P. This defines the morphism g : P 1 Chowk(X) with g(0) = a +p andg(oo) = +For the opposite direction, let g : P 1 Chowk(X) with g(0) = a + /3, and g(oo) = a' + /3.If P E P 1 , we associate the scheme in Pn given by the cycle associated to the Chow point f (P).We take the diagonal morphism 1 x g : Pl -* P 1 x X. Let W = (1 x g)(P 1 ). The projectionP 1 x X -4 P 1 induces a morphism f :W PI which is a dominant, and 1 -1 (0) = g(0) = a+/3,f (oo) = g(oo) = a' +We now consider what happens if we demand that the cycles be T invariant. If V is Tinvariant, then its Chow point is as well. In the above, if the k + 1-dimensional subvariety Vis T invariant and f E R(V) * is a weight vector of weight 0, then the morphism f : V -+ P1Chapter 4. Equivariant Chow Groups^ 36defined by f is equivariant where P 1 has the trivial action. So, each f -1 (P) is T invariant andthe map g : Pl -4 Chowk(X) is a T-map.Note that the curve g (P 1 ) is pointwise T invariant. So, g(P 1 ) C Chowk (X)T .Proposition 4.7.2 Two effective cycles a, a' E ZiT(X) are T -rationally equivalent if there isa T -map g : P 1 -+ Chow k(X) , g(0) = a + /3 and g(oo) = a' + /3.Proof: This is a result of the Theorem and discussion above.^ ^The following has been proved by Brion [3] in part 2 of the Theorem of section 1.3 in thecase of solvable connected groups, where T' = 1. The proof we use is a re-writing of his proof.Theorem 4.7.3 Suppose w : T' T is a morphism of tori, Pn has a T and a T' -action definedon it andsox1r x Pn ---4 T X P n\1Pn(4.29)commutes. Suppose that X C P T' is a normal T -scheme and hence a normal r -scheme. Thenthe set of effective cycles in Zr (X) that are r -rationally equivalent to 0 is generated by thecycles [div(f)] where V is a T invariant k + 1-dimensional subvariety of X and f E R(V)* isa T weight vector of weight 0 with respect to r .Proof: By Proposition 4.4.2, we can assume that a, a' E 4(X) and are T' rationally equiva-lent. So, there is a /3' E zr (X) such that in Chowk(X) there is a T'-map g : P 1 --) Chowk(X)with g'(0) = a+/3' and g' (oo) = a' + /3' . Using Proposition 4.4.2, we can connect /3' to a T invari-ant point 0 by a T' invariant curve. So, there is a /3 E 4(X) and a T-map g : P 1 -4 Chowk (X)such that g(0) = a +/3 and g(oo) = a' +0. We can move this with respect to T to a T invariantChapter 4. Equivariant Chow Groups^ 37curve. Since g(P 1 ) E Chowk(X)T' and since the actions of T and T' commute, we move thecurve within Chowk(X) Ti and the new curve is also pointwise T' invariant. Each componentof the curve has a T-action, and has at least two fixed points. We normalize each componentto get a collection of curves isomorphic to P 1 . The action on each component extends to oneon its normalization, and the action on Pl is by a character X(t) with x o yo(e) = 1. So fromthe proposition above, the two cycles are T'-rationally equivalent and the T'-rational functionsinvolved are weight vectors of T of weight zero with respect to r . ^Corollary 4.7.4 If X is a normal quasi-projective variety and iC C Pn , with notation as above,the obvious map AT (X) AT (X) is surjective with kernel generated by the cycles describedabove. In particular, if T' = 1, lik(X) is generated by T invariant cycles, and those rationallyequivalent to 0 can be obtained as described above.Proof: If X is quasi-projective, we can close it in Pn to get X. The result is true for effectivecycles on X, and from the exact sequence of section 4.6, the result is true for effective cycleson X . Since the effective cycles generate AT(X), the result is true in general. ^Example: We find 4(F) where F is an n-dimensional representation space for T.We assume the T-action is diagonal. Let F = Spec(k[x , x n]), and let the vector inF related to xi be Xi. Let be the subspace of F spanned by Xi„ , Xyk . Let= and let (ti, .. • , tn) • Xi = tiXi for all i. The T'-subvarieties of F are theand the weight vectors of R(F)* are of the form fi^for ni E Z. The weight zero vectors arejust the constants. So,TA 'k (F) EB2Z (4.30)is the free group generated by theSuppose T^. Then T C T' . Consider the cycles generated by restricting the f =that are weight vectors of weight 0 with respect to T to the F( 11 ,..., ik+1 ). Let B be the submoduleof AT (F) generated by all these divf. Then,AT (F)^(F) I B^ (4.31)Chapter 4. Equivariant Chow Groups^ 38We can also describe this using linear equations. Recall that if dimT = r and the T-actionon F was defined by weights A1, , A n where Ai = (Ali, , Ari) we can represent the actionbyAn Al2(Ar1 Ar2• • •^Al• • •^Arn •(4.32)To get a T' weight vector of R(F( ii ,..., ik+i ))* that is of weight 0 with respect to T, we replaceall but the , ik+ i columns by columns of 0's to get a matrix M. Let Y be a solution ofMY = 0 where Y E Zn and Yii = 0 for 1 < j < k. Since MY = 0 if and only if n is a weightzero vector with respect to T these Y correspond to the T' weight vectors in R(F(ii ,..., ik ))* that()are of weight zero with respect to T. Let B be the subgroup of Cli Z generated by the ker Mfor all choices of^,(=)4 (F) = Epi1 Z/B. (4.33)Remark: If X = Pn has a T-action, the same method of this example shows thatAT (Pn ) = EDPI . Z/B,^ (4.34)where we define B as above and we also require that Y be such that E^0.Example: Let F" be the semi-simple points of F. Then F88 is F less a few linear subspaceseach of which is spanned by a subset of the Xi. Let B' be the B of the previous exampleunion the free group on the set of k-dimensional T'-subspaces contained in the deleted linearsubspaces. Then,Ar(X) = Z(:)93'.^ (4.35)ik+ 1 as above. Then,Note that this is the group Ellingsrud and Stromme [5] have calculated for Ak_ r (F"fiT) whenthis is non-singular.Chapter 4. Equivariant Chow Groups^ 39Example: We work out a concrete example of the above. This example also shows that theequivariant Chow group and the usual Chow group are not necessarily the same.Let F = A3 have the T = G,n 0 Gm-action defined byC 01 —1 1)—2 3) •The T-vectors of weight zero correspond to the solutions ofY1( —1 1 —1)Y2 = MY = 0.0 2 —3)Y3ker M = (1, 3,2)Z, and(F) = Z31(1,3,2)ZA2 (F,33) = Z3/((1, 3, 2), (0, 1, 0)) = Z (DZ.(4.36)(4.37)(4.38)4.8 Affine T-BundlesTheorem 4.8.1 Suppose X is a normal quasi-projective scheme. If E X x F where F is aT-representation space, then there is a surjective mape5=0 (4._i (X) e A T (F)) —> A if (E).^ (4.39)In particular, if X has the trivial T-action, then the map is an isomorphism.Proof: We assume that T acts diagonally on F = Spec(k[x , , x e]). Let T' = ED7=1Gm . LetT' act trivially on X, and let (t1, te )• xi = tr lxi for all i. We consider E with T xr actionwhere T acts trivially on F. The k-dimensional T x r invariant subvarieties of E are of theform W = V x where V is a k — j-dimensional T-subvariety of X. So,4_i (x) ®^(F) = Z17:_''iTs (X) e Z.TxTi^zt'xT' (E).^(4.40)We need to show that the map is defined with respect to T-rational equivalence and issurjective. Since any T-cycle in 4(F) and ZT(E) is T-rationally equivalent to a V-cycle, weChapter 4. Equivariant Chow Groups^ 40need only consider r-cycles. If W = V x the T x T' weight vectors of R(W)* are ofthe form f ru=l x7s, where f E R(V)* and ni E Z. If the weight of xi with respect to T is —Ai,we need Af = E niAi. Since the weight vectors with Af = 0 and E niAi = 0 are in this set, themap passes to T-rational equivalence and it is a surjection.To get the isomorphism in the case of a trivial T-action on X, let W = V x andlet f m i x=' E R(W) * be a weight zero vector. Since Af = 0 for all f E R(V) * , we haveE niAi = 0, and the cycle diva -U=1 x7i) in ZT(F) must have been T-rationally equivalent tozero. So, we have an injection and hence an isomorphism. ^Remark: In the theorem, if T is the 1-dimensional torus, X has trivial T-action and theweights of F are all 1, then we getED.;=0 Ar_i (X) •.-• AT (E).^ (4.41)So, in this case, AT (E) is the same as Ak(P(E)).Proposition 4.8.2 Supppose p : X -+ Y is a T-morphism of T-schemes, Y has trivial T-action, and there is a covering of Y by open sets U such that X = U x T. If dimT = r,thenp* : Ak(Y) Ar+r(X) (4.42)exists, and is an isomorphism.Proof: Since p is a flat T-morphism, we have existence.The T-subvarieties of X are of the form W = p-1 (V) where V is a subvariety of X. Tosee this, note that the result is certainly true locally, and glueing together gives the globalresult. This shows that p* : Zk(Y) -4 Zr+r (X) is a surjection. So, on passing to T-rationalequivalence, p* : Ak(Y) --+ 4+,(X) is a surjection as well.For injectivity, T weight vectors g E R(W) * of weight 0 are locally of the form f 0 1 ER(U x T)* for f E R(U)*. However, [div(f 0 1)] = pldivf]. So, kerp* = 0 in Ak(X) and p* isinjective. ^Chapter 4. Equivariant Chow Groups^ 414.9 NotesWe could consider different types of equivariant Chow groups. For example, we could demandthat the fi be weight vectors, but not necessarily of weight 0. For this form of equivariantChow group, all the properties we have considered in this section hold. In particular, properpushforward, flat pullback, the alternate definition of rational equivalence where Pl is replacedby Pi for some unspecified A and the exact sequence result hold. However, as we have seenin the change of torus section, for normal projective varieties, this form of equivariant Chowgroup is just the usual Chow group. We could also define A r (X) as g(X) modulo rationalequivalence. The problem with this group is that its properties are too hard to determine.In particular, pushforward and pullback become hard to show. Also, for normal projectivevarieties this is only the usual Chow group.Chapter 5Multiplicities on VarietiesWe consider the results of the last two chapters applied to varieties. We start by consideringprojective varieties. On a projective variety X, it is possible to define equivariant degrees andmultiplicities that are invariants of AT (X). As such, the equivariant degree is related to theusual degree. The relationship between the equivariant multiplicity and the usual multiplicityis not quite as strong. We consider these two equivariant objects in the first section. In thesecond section, we consider the usual equivariant multiplicity as defined by Rossmann [17], orBorho, Brylinski and Macpherson [1] in a particular case.Since there seems to be some confusion in the literature concerning signs of characters, westate the following convention explicitly:Convention:If An is the vector space with basis X1,^X, then its structure sheafis OAk = Spec(k[xi, , x n]). We assume the Xi are weight vectors ofweight Ai. This means that the related functions xi have weight —Ai.The net effect of this is that the equivariant multiplicity is no longeralternating in sign.5.1 Multiplicity on Projective VarietiesIn this section we develop the basic properties of equivariant multiplicities of projective T-varieties. We define an equivariant degree as well as an equivariant multiplicity. Unfortunately,this multiplicity is not defined in the generality we would like. A lot of this section consists ofshowing that we can extend the equivariant multiplicity to all cases. We accomplish this by42Chapter 5. Multiplicities on Varieties^ 43using the equivariant version of a result of Mumford. We also show that these are invariants of4(X). We end this section with a useful result involving the equivariant multiplicities of allthe fixed points of P. The definitions and results of this section are new.Unless otherwise stated we are assuming that x = Po.Definition: If X is a T-scheme in P", it is defined by a homogeneous T invariant idealI C k[so,... , xn]. Let degT(X, P") be the equivariant multiplicity of the origin in the affinecone defined by I in An+ 1 .Proposition 5.1.1 I. degT(X, P") is a polynomial over Q in A0, ... , An of degreecodim(X, Pn ).Proof: this results from Proposition 3.2.1 and Theorem 3.2.4.^ ^Definition: Let x be a fixed point of P", Uo be the open affine T-subset of Pn definedby inverting xo and let X be a T-subscheme of P. X 1u0 is defined by a T invariant idealJ C k[xi/x0,...x n /x0]. If x is an isolated fixed point, letmulti p,. (x, X) = multT(k[x1/x0, ••• , xn / x01/ J, k[x1/x0, ••• , xn / x0]) •^(5.1)If x is not isolated, we consider degT(X, Pn ) as a polynomial and definemultTR.(x, X) := degT(0, Ai — Ao, • • • , An — A0)•^ (5.2)Remark: Note that J need not be homogeneous in the usual sense. This is the reason wehave a different definition for non-isolated fixed points. We show later that the definition isconsistent.Also, note that the definition makes sense even if x 0 X. We show later that in this casethe equivariant multiplicity is 0.As for the equivariant degree,Chapter 5. Multiplicities on Varieties^ 44Corollary 5.1.2 multTxn(x, X) is a polynomial over Q in Al — A0,..., An — Ao of degreecodim(X, Ps ).Remark: If T = Gm and we consider the Gm-action on An+ 1 where xi has weight 1 for all i,then degT(X, Pn ) is just the usual degree of a projective scheme. With this action, the weightof the xi/x0 in Uo are all 0. So, mu/tT,pn (x, X) = 0. Needless to say, this differs from theusual multiplicity for a point in a projective scheme. We can obtain the usual one by using theGm-action where xo has weight 0 and xi has weight 1 if i 0 0. If the tangent cone Cx (X) isconsidered as a subscheme of Pn , the usual multiplicity of x in X is then muitT ,Fin (x, C x (X)).From our results on equivariant multiplicities of modules, we easily get:Proposition 5.1.3 1. For 0 < m < n, let Pm. = Proj(k[xo, , x id), and An—mSpec(k[xm+1,...,xd), let X C Pm and let x E (Pni)T . Let f : Pm x An—m -4 Pm be theprojection, and let i be the inclusion Pm y Pn . We have the diagram: f -1 (X)^Pm x An—m C PnfX^Pm.(5.3)Let Y = I-1 (X) C Pn . ThendegT(Y, Pn ) = degT(X,Pm).If i(x) is an isolated fixed point of Pn,muitTR. (i(x), Y) = multi pm (x, X).2. With notation as in 1,(5.4)(5.5)degT(i(X),Pn) = ft AodegT(x, Pm)i=m+1(5.6)Chapter 5. Multiplicities on Varieties^ 45andnmultTR.(i(x),i(X)) = II (Ai - AomuitT,p. (x, X).^(5.7)i=m+13. Let w :^T be a morphism of tori, Pn be T and T' invariant, and let the action ofT' on Pn be defined by the weights AP, ,^If X is a T and a T' -subvariety of Pn andcox].T' X Pn^T X Pn\commutes, theno dw)degT,(X, Pn) = f A(degT(X, Pn ) o dw).^(5.9)If x is an isolated fixed point of Pn, then,H (Ai — Ao) o dcpmuitTp.(x, X) = 11(A — A'0 (muitT,pn (x, X) 0 d(P)•^(5.10)4. Let H be the T hypersurface of Pn defined by the function f of weight —AH. If Hintersects X in codimension one, thendegT(X n H, Pn) = AHdegT(X, Pn).^ (5.11)Suppose x is an isolated fixed point of Pn. If Hlu. is defined by a function of weight —Am u.and H intersects X in codimension one, thenmuuTpn (x, x n H) Amu. muitTp. (x, x).^(5.12)5. Let X be a T-subscheme of Pn with components^. If the geometric multiplicityof Vi in X is mi , thendegT(X,Pn) = EmidegT (Vi,Pn)).i=16. With notation as in 4 and 5,AHdegT(X n H, Pn) = E midegT(X, Pn ).i=1(5.13)(5.14)(5.8)PnChapter 5. Multiplicities on Varieties^ 46Proof: These are all direct consequences of the results in section 3.2.^ ^Note that we have not stated 6 and 7 for mu/tTyn (x, X). These are true, but we need toknow that the definition of muitTp. (x, X) is consistent.We can extend degT(x, Pn) linearly to cycles a E g(Pn ).Corollary 5.1.4 1. If X is a subscheme of Pn, thendegT(X, Pn) = degT([X], Pn )•2. If a E 4:(Pn) is T-rationally equivalent to 0, thendegT(a, Pn) = 0.In particular, degT(cf, Pn) is an invariant of the cycle a E 4(X).(5.15)(5.16)Proof: The first statement is a rephrasing of 6 above. For the second, a is T-rationallyequivalent to 0 if there is a collection of k + 1-dimensional T-subvarieties V and weight vectorsE R(Vi)* of weight 0 such that a = E divfi. Since the f2 are quotients of weight vectors ofox, 4 yields the result.^ ^We now relate degT(X, Pn) to mUitT ,pn (x, X).Proposition 5.1.5 If x E (Pn ) T is isolated, thendegT(X, Pn )(0, Al — Ao, • . • , An — A0) = muitTpn (x, X).^(5.17)Proof: Suppose X luo is defined by the ideal J. We can homogenize the generators of J to getan ideal J' C k[xo,...,xn]. J' defines a scheme X' E Pn , each component of X extends to oneof X' with the same geometric multiplicity, and all other components of X' are contained inPn — U0. Resolving k[xo....,x,j1 J i and k[xilx0,...,x,,Ixo]il yield the same sequences, butChapter 5. Multiplicities on Varieties^ 47with weights Ao, . . . An and Al — A0, ... , An — Ao respectively. So viewing degT as a polynomialin A(T), we get,degT(X' , Pn )(0, Al — Ao,..., A n — A0) = multTRn(x , X).^(5.18)Now, if V C Pn — U0 = Pn -1 we inject V into Pn, and Proposition 5.1.3 part 2 gives us,degT (V, Pn ) = AodegT(V9 pn-1).^ (5.19)So,degT(V, Pn )(0, Ai — Ao, ... A n — Ao) = 0,^ (5.20)anddegT (X, Pn ) (0, Al — Ao, ... , An — A0) = mzdtTpn (x, X).^(5.21)0This show that defining muitTpn(x, X) = degT(X,Pn)(0, Ai — Ao, . . . , An — A0) for non-isolated fixed points is consistent.We can extend the definition of muitTpn (x, X) linearly to cycles a E ZIT (Pn ).Corollary 5.1.6 All the statements of Proposition 5.1.3 hold for multTR.(x, X) even when xis not an isolated fixed point of Pn . In particular, if X is a T-subscheme of Pn , thenmuitTpn (x, X) = multi ,p n (x, [X]) .^ (5.22)If a E ZZ' (Pn) is T-rationally equivalent to 0, thenmultTR. (x, X) = 0.^ (5.23)Note that this implies that unlike the usual multiplicity, muit T,pn (a, X) is an invariant ofa E AT (X).Chapter 5. Multiplicities on Varieties^ 48Consider the Gm-action on Pn where in An+ 1 , xo has weight 0 and xi has weight 1 fori 0. If X is a Gm-scheme, i.e. X = Cx (X), thendegGm (X, Pn)(0, 1,^, 1) multTxn(x,Cx(X))-^(5.24)If T acts on Pn where xi has weight 1 for all i, then since the ideal defining Cx (X) in k[xo,. • • ,xn]has its generators in k[xi, .. • , xn], degT(Cr (X), Pn) degG,n (Cx (X), Pn). So, the usual degreeof Cx (X) in Pn and the usual multiplicity of x in X in Pn agree. We have a similar result forthe equivariant degree and multiplicity. This is the equivariant version of a result of Mumford.Let X be a k-dimensional subscheme of Pn . We consider moving X around with respect toa 1-dimensional torus acting on Pn . Let the graph morphism beo- : X X Gm ,4 pn x pl(x, t) H (t • x, t)^ (5.25)and let P2 : Pn x P 1 -+ P 1 be the projection. If Y = o- (X x Gm ) in Pn x P 1 , let f : Y -4 P 1be the map induced by p2. With notation as in section 4.4, we define^t • X as:X' = lim t • X = Y(oo).^ (5.26)t—>ooSince [X'] = [X] E^(Pn) degT(X, Pn) = degT(X', Pn) and mu/ tT,pn (X, X) =MU/413n (x, X').Suppose we moved X with respect to the torus defined by the weights {0, 1, , 1} in An+ 1 .Theorem 5.1.7 Let muleTR„(x, X) = degT(Cx (X), Pn) and letPx : X — x C^(X) C^(Pn) C Pn x Pn-124Pn-1^(5.27)where Pn-1 = Proj(k[xi,... , xn ]). Then, if k = C,Ao degpxdegT(Px(X — {x}), P n) if X Cx (X)degT(X,Pn) — MU/4 ,13,, (X, X) =^ (5.28)0 if X = Cx (X) .Chapter 5. Multiplicities on Varieties^ 49If k 0 C,degT(X, Pr') — muieTx. (x, X) = 1 AodegT(P211imt_,„0 t • (X — {x})], np —1) if X 0 CC(X)0 if X = Cx (X).(5.29)Proof: We can describe the map px using the action above. We end up moving X to the cyclewith components Cx (X) and a subvariety of Pn that is T-isomorphic to px (X — {x}). ConsiderX —{x} C Pn— {x}. px(X-{x}) is the variety defined by limt_+. t. (X —{x}). We can see this bynoting that Pn — {x} = A x Pn-1 . The weights of the action are {0, 1, ... , 1}, so that on pointslimt_.÷00 t • (v, y) = p2 (v, y) = y. Note that restricting to Uo gives limt—roo t • (X Ivo) = Cx(X)Itio•So,[X] = [tli!'i t • X] = [Cx (X)] + [X'],^ (5.30)where X' = limt—,,,t • (X — {x}) in Pr'. If k 0 C, degT([X'], Pn ) = AodeyT(P2*([Xl, Pn-1 ))and we have the result. Since px is proper, if k = C,[X'] = degPx[Px(X — {x})]^ (5.31)and taking equivariant degrees gives1 Ao degPxdegT(p,(X — {x}), Pn-1)degT(X, Pn ) — MUieTpa (x, X) =Corollary 5.1.8 I.degT (X, Pn )(0, Al — Ao, ... , An — A0) = mu/413n (x, X)(0, Al — Ao, ... , An — Ao)^(5.33)= multTxn (x, X).2. If x 0 X , then multTxn(x, X) = 0.3. If the fixed point component of X containing x does not contain the fixed point componentof Pn containing x, then multTyn(x, X) = 0.0if X 0 Cx(X)if X = Cx(X).(5.32)0Chapter 5. Multiplicities on Varieties^ 50Proof: For 1, the first equality results from the theorem, the second has already been shown.2 results from the theorem. If x fl X, Cr (X) does not exist and moving X as in the theorem,we get [X] = [X'] [limt-÷00 t • X]. degT(X', Pn)(0, Al — Ao, • • • , An — AO) = 0 then yields theresult.For 3, if X does not contain the fixed point component of Pn containing x, then there is apoint y in the same fixed point component of Pn as x, but which is not in X. With no loss ofgenerality, we can assume that x is P1 and y is Po. Moving X as in the theorem, we getdegT (X, Pn) = AodegT(P2*( tliat . X), Pn-1 ).^(5.34)Since Al = A0,degT(X, Pn )(0, 0,^An —^= MUitTR. (X, X) = 0.^(5.35)0Let Gm act on An+1 with weights (0, 1,^, 1). In this case, since the usual multiplicity isgiven by mu/tGm ,pn (x,Cx (X)), the theorem states that if deg(X) and mult(x, X) are the usualdegree and multiplicity then,fdegpxdeg(px (X — {x})) if X Cx (X)deg(X) — mult(x, X) =0^ if X = Cs (X).This is precisely the result that Mumford [14] Theorem 5.11 gets.(5.36)By the example after Corollary 4.7.4 any T-cycle in ZT(Pn) is T-rationally equivalent to acycle whose components are defined by ideals of the form (xi l, xin _ k ). We can also show thismore explicitly. In the theorem, we moved X with respect to the action defined by the weights(0, 1, , 1) in An+1 . We can move the resulting cycle with respect to the action defined bythe weights (1, 0, 1, ,1) in An+1 . Repeating this for each new cycle, we get a cycle that isinvariant under the maximal torus of GL(An+1 ) acting on Pn. This means that the componentsof the cycle are defined by ideals of the form (xi,, xin _ k ). This gives an easy way to finddegT(X, Pn) and mu/tTy. (x, X), and,Chapter 5. Multiplicities on Varieties^ 51Proposition 5.1.9 degT(X, Pn ) is a polynomial over Z in the Ao, • • • , AnMUitT ,pn (X, X) is a polynomial over Z in Al — Ao,....An — Ao•Proof: if V is defined by the ideal (x;-1,• • •xin-k), then degT(V,Pn)^1-1.7 Ai, andmu/tT,pn (x, V) =^— A0). Since X is T-rationally equivalent to a cycle whose compo-nents are of this form, we have the result.^ ^We now consider how the equivariant multiplicities at the fixed points of Pn are related.Proposition 5.1.10 Suppose Pn has only isolated fixed points. If dim X > 0, thenni 0 ri (Ai  ^muitT,pn (pi , x) = 0.If dim X = 0 and X has only one component Po, thenmultT,pn (Po, X) = fl(A3 — Ao)m,where m is the geometric multiplicity of Po in X .(5.37)(5.38)Proof: We first consider Pn itself. The equivariant multiplicity of Pn at each fixed point is 1,and after taking common denominators in the sum, we find that the numerator is:/1 Ao A8 • • .^1 \1 Al Ai • • • Ar l- 1\1 An AF, • • • A;r4 1/If m < n, we inject Pm y Pn , where Pm = Proj(k[xo,... , xm]). Since multi Rn(Pi, X) = 0if Pi 0 Pm, by Corollary 5.1.6 and Proposition 5.1.3, we get,u,^Rio A — A^m)i-0^Ai) muitT,Pn (Pi , Pm) iE<In flj>mi((Ajj — A1) m^(Pi, pindet = 0.^ (5.39)=E j<nt^Ai)  multi pm (pi, pm)°= 0.(5.40)Chapter 5. Multiplicities on Varieties^ 52If dim X = 0, then the proposition results from Corolary 5.1.6, Proposition 5.1.3 part 6 andLemma 3 2.3.^ ^5.2 Equivariant Multiplicities on VarietiesWe define the equivariant multiplicity of a fixed point x of a T-subscheme X of a smooth T-variety Y. The definition we use is the the one used by Rossmann [17]. We show that in thecase we are concerned with, namely where there is an affine neighbourhood in Y containing xthat is T-isomorphic to a T-representation space, the equivariant multiplicity considered in thelast section can be extended to one for Y and it is the same as that considered by Rossmann.In particular, the equivariant multiplicity is an invariant of AT(X), and is defined even whenx X.Let X be an equidimensional T-subscheme of an ambient smooth T-variety Y. If x E X isa fixed point of T in X, we define the equivariant multiplicity muitT(x, X, Y) of x in X relativeto Y:Definition: Let NxY = Spec(ei>omix /n4+ 1 )^Spec(R), where mx is the maximal idealdefining x in Y. Since Y is nonsingular, mx is generated by weight vectors xl,^, xn . So,Nx (Y) Spec(k[xi,... ,x,]). Let Cx (X) be the cone to x in X. Oci (x) is a ring with T-actionand is a quotient ring of R. So, Ocz (x) is a R, T module. Let^multT(x, X, Y) = muitT(Oc(x),R) if x E X^(5.41)0^if x fl X.Note that Oc.(x) is homogeneous in the usual sense, so muitT(x, X, Y) is defined even whenAi = 0 for some i.Let U be an open affine T-subset of Y such that U is T-isomorphic to a representation spaceF. We considerX ju C U C P(F ED 1),^(5.42)Chapter 5. Multiplicities on Varieties^ 53where 1 has weight 0. For notational purposes, we assume the element in k[xo, x1 , . , xn ]corresponding to 1 is xo.Proposition 5.2.1 If x E Y is a T fixed point and there is an open affine T-subset of Ycontaining x that is T-isomorphic to a T-representation space, thenmultTx (F0i)(x , X) = multT(x, X ,Y).^ (5.43)Proof: Let x E X. Since U F Nx (Y)cte F and we can close Nz (Y) and Cx (X) in P(FED 1).muUT (x, X, Y) = muUT (Ocx (x) (U), ON. (Y)(U))^(5.44)= muitT,P(Fed)(x, Cx(X))^ (5.45)= muitT,P(Fen(x, X). (5.46)0Since the right-hand sides have been defined when x^X and are 0, definingmuitT(x, X, Y) = 0 in this case is consistent.For completeness we list all the properties of muUT (x, X, Y).Proposition 5.2.2 Suppose Y is a smooth T-variety, x E YT and there is an affine open T-subset U of Y with x E U such that U is T -isomorphic to a T representation space with weightsAi .1. If X is an equidimensional T-scheme with components V of geometric multiplicity mi,thenmultT(x, X,Y) = multT(x,[X],Y) = E mimuuT(x, Vi,Y).^(5.47)2. If I lu is a T-subvariety defined by a single function of weight —Am u and H intersectsX in codimension 1, thenmultT (x, X fl H, Y) = AH i u multT(x,X,Y).^ (5.48)Chapter 5. Multiplicities on Varieties^ 54In particular, if a is T -rationally equivalent to zero, then multT(x, a, Y) = 0 and multT(x, a, Y)is an invariant of a E^(X).3. With notation as in .1 and 2, we have the following Bezout theorem:A HmultT(x , X ,Y) = multT(x , X n H,Y).^(5.49)muuT(x,x,y). -N1 1 E aA AN ,^ (5.50)AEA(T)where N = codim(X, Y), as E Z, all but a finite number of the as are zero, aA 0 implies thatA is a sum of the Ai, and multT(x, X, Y) is a polynomial with coefficients in Z in the Ai.5. Let co :r -4 T be a morphism of tori, Y be a T and a T' invariant variety. SupposeTt X Y^cox 1T X Y\1^(5.51)commutes. If X is a T-subvariety of Y and the weights of T' in the open set containing x arethen(fl Ai (3 dco) muitr(x, X, Y) = (11^multT(x, X, Y) o dso.^(5.52)6. Let E be a T -vector bundle over Y, x E YT , let E be trivial over U, and let E haveweights pi, , ilk. If p : E^Y is the projection, and i is the zero section embedding of Y inE thenmultr (i(x), p* (X), E) = multT(x, X ,Y).^ (5.53)7. With notation as in 6,multT (i(x), i(X), E) = (II Pi)mult(x , X ,Y).^(5.54)Chapter 5. Multiplicities on Varieties^ 558. If X is 0-dimensional with x as its only component, thenmultT(x, X ,Y) = (11 As)m,^ (5.55)where m is the geometric multiplicity of x in X .Proof: These are all consequences of the results of the last section.^ ^Chapter 6Intersections of T-line bundlesIn order to do intersection theory, we want to be able to define the Chem classes of a T-linebundle. We do this by associating a T-Cartier divisor to the pair consisting of a T-line bundle Land an equivariant meromorphic section of L. To this T-Cartier divisor, we associate a T-Weildivisor. This allows us to define the intersection of a T-line bundle with a T-subvariety V ofX provided an equivariant section exists on V. As in chapter 3, our results are by and largethe equivariant versions of Fulton's [6]. The main problem in generalizing Fulton's work tothe equivariant case is, in fact, that equivariant sections need not necessarily exist. We alsoconsider how equivariant multiplicities behave with respect to intersections.6.1 T-Cartier DivisorsDefinition: A T-Cartier divisor D on a T-variety X is a collection of pairs (Ua , fa) whereUa is an open T-subset of X and fa E R(Ua ) * is a weight vector of weight —A a such that onUa fl us, --1 . ifafp is nvertible in Ouanuo. The support of a T-Cartier divisor is the set of pointsin X where the local equations are not invertible. We label this by [DI. We will also label thecollection of components of PI by IDS. We say a T-Cartier divisor is effective if fa E Ou. forall a.As with usual Cartier divisors, we have a group structure:1. If D and D' are two T-Cartier divisors represented locally by (Ua , fa) and(Ua , f.'„), then D + D' is represented by (Ua , fa fc,' ), and has support IA U WI.2. The identity element is represented by (Ua , 1) for all a.3. If D is represented locally by (Ua , fa ), then —D is represented locally by (Ua , f ; 1 ).56Chapter 6. Intersections of T-line bundles^ 57Definition: If f E R(X)* is a weight vector we label the associated T-Cartier divisor by divf .We say two T-Cartier divisors D and D' are T-linearly equivalent if there is a weight vectorf E R(X)* of weight 0 such thatD = D' + divf .^ (6.1)So, locally, if D and D' are represented by (Ua , fa ) and (Ua , ft), then fa = Af . .A T-Cartier divisor defines a T-line bundle OD. Let Fa be the 1-dimensional representationspace generated by 1/ fa . Let L be the 1-dimensional Ou-sheaf defined over U, by OUP, 0 Fa .We setOD = Spec(Symm(Lv )).^ (6.2)For notational purposes, we write OD = Spec(Oua 0 k[xj. Note that if fa has weight -A,then OD has weight A„, and x, has weight -A0 .The transition functions go : Ou.nus xF -p Our,nus xF , are defined by the function go =falb E (9ta nus .There is a section SD defined locally over the U„ by the f„ E R(Ua ) * . Since x, has thesame weight as fa , this is an equivariant section.Note that if D and D' are T-linearly equivalent, then OD and OD' are T-isomorphic.Some of the group properties of T-Cartier divisors pass to the associated T-line bundles:1. If D and V are two T-Cartier divisors, then 0D+D , = OD 0 OD', 8D-FD' =SD 0 8D', and supp(sDi-D , ) = IDI U ID'I•2. If D is a T-Cartier divisor, then 0—D is generated locally by the fa , s-D = sir i ,and supp(s_D) = suPP(sD)•Of course, if f E R(X)* is a weight vector of weight 0, then the T-line bundle associatedto divf is the trivial one with trivial T-action, and the associated equivariant section is definedby f.Chapter 6. Intersections of T-line bundles^ 58We associate a T-Weil divisor to D in the usual fashion. Over U, we set[D] = E ordvf,„[V]^ (6.3)vin Zn 1 (Ua ) where the sum is over all codimension one subvarieties of Ua . Since fa is a weightvector, the V for which ordfa is non-zero are T-subvarieties.Since ordvf is well defined up to units, this extends to a global definition.With respect to equivariant multiplicity, the results of the previous chapters give:Proposition 6.1.1 Suppose that X is a T-subvariety of a smooth T -variety Y, D is a T -Cartier divisor on X, x E YT, and there is an open affine T-subset U C Y containing x that isT-isomorphic to a T representation space. If x E [J, and f, has weight —A, then,multT(x , [D],Y) = AamultT(x , X , Y). (6.4)Proof: The result is a local result. Let (U„, fa ) represent D, where x E U. To be able touse Proposition 5.2.2 part 2 we need the T-Cartier divisor defined on an open T-subset that isT-isomorphic to a T-representation space. Since Ua is not necessarily of this form, we considera T-Cartier divisor D' that is defined on U and that restricts to D on Ua .Let U' = U n Ua . The closure of U' in U n x is u n x, so R(U')* ,.-, R(U n X)* . Let f,be the weight vector in R(U fl X)* associated to fa . Since the ideal defining X in U is prime,there is a weight vector i E R(U) * that restricts to fa in R(U fl X)*. The numerator and thedenominator of I can be chosen to be weight vectors, and they act as a non-zero divisors onOx (U n x). So,mu/ty, (x, divI, Y) = AamuitT (x, X,Y).^ (6.5)On U fl X, [divf] — [divfa] lies in U fl X — U', which does not contain x. So, the equivariantmultiplicity of x in the two cycles is the same. So,muUT (x, [D],Y) = A,multT(x, X, Y).^ (6.6)0Chapter 6. Intersections of T -line bundles^ 59Remark: If D is T-linearly equivalent to D', thenmultT(x, [D], Y) = multT (x, [D'], Y).^ (6.7)So, the equivariant multiplicity of the T-Weil divisor associated to a T-Cartier divisor is aninvariant with respect to T-linear equivalence. If D = D' + [divf] where f has weight -A, thenmultT (x , [D] , Y ) = A multT ( x, [DI , I' ) •^ (6.8)6.2 T-pseudo DivisorsDefinition: A T-pseudo divisor D is a triple (L, Z, s) where Z is a closed T-subset of X, L isa T-line bundle and s is an equivariant section of L invertible over X - Z. We will sometimeswrite Z as IDI. A T-pseudo divisor (L', Z', s') is T-equivalent to (L, Z, s) if Z = Z', and thereis a T-isomorphism of T-line bundles yo : L L' such that w(s) = s .We have some of the group properties we'd expect:If D = (L, Z, s), and D' = (L', Z', s') are two T-pseudo divisors on X, thenI. D-I-TY=(LOV,ZUV,s0.9')2. -D = (L-1 , Z, s -1 ).A T-Cartier divisor D determines a T-pseudo divisor (OD, ID( , s D). We say D T-represents(L, Z, s) if IDI C Z and there is a T-isomorphism of T-line bundles yo : OD ---+ L such thatco(sD) = s on X - Z. Note that if Z = X, two T-Cartier divisors T-represent the sameT-pseudo divisor if and only if they are T-linearly equivalent.Note that if (L, Z, s) and (L', Z', s') are T-represented by D and D', then1. (L ® L', Z U Z', s 0 s') is T-represented by D + D'2. (L-1 , Z, 8 -1 ) is T-represented by -D.Theorem 6.2.1 Every T -pseudo divisor is T -represented by a T -Cartier divisor that isChapter 6. Intersections of T-line bundles^ 601. unique if Z 0 X2. unique up to T-linear equivalence if Z = X .Proof: The proof is essentially contained in Fulton [6] Lemma 2.2. For an open cover {Ua } suchthat Lluc, is trivial, choose an ao and set fa = gaa0 for all a. gam) E R(Ua rlUa0 )* = R(Ua )* isa weight vector, so fa is a weight vector in R(Ua )*. Also, faL3-1 = gas is invertible on Ua fl U.This shows the existence of a representing (but not T-representing) T-Cartier divisor D forZ = X.To get T-representation, we assume for the moment that Z = supp(s). Since sa = goso,sa l fa = 8,31 fp, and there is a r E R(X)* such that r = sa l fa E Ox(U,) * for all a. We setD' = D + divr. We then have sly = s. Since sly is an equivariant section, D' T-represents(L, Z, s).For uniqueness, if D and D' both T-represent (L, Z, s) and have local equations fa and LC,then fa/ fa = f E R(Ua ) * . Also, fax = fol& So, f E R(X)*, and f is a weight vector ofweight 0. If Z = X, D and D' are T-linearly equivalent. If Z 0 X, then SD = SD, off Z, andwe have fa = fa on Ua - znuQ . so, D = D'. ^Definition: If D is a T-pseudo divisor on an n-dimensional T-variety X, we define its T-Weilclass [D] E AT_ 1 (IDD as the Weil class of a T-representing T-Cartier divisor of D.If f : X' --, X is a morphism, f (X') rt supps, and D is a T-pseudo divisor on X, then thepullback f*(D) = (f *(L), f -1 (Z), f*(s)) is a T-pseudo divisor on X'.The pullback property illustrates the main problem with T bundles. We can only guaranteethe existence of an equivariant section on X' if f (X') 0 supps.Definition: Let D = (L, Z, s) be a T-pseudo divisor on X and let V C X be a T-subvarietyof X. We say D is V-admissible if L has an equivariant section over V. If a E 4:(X), we saythat D is a-admissible if for every component V of a L is V-admissible. If L has an equivariantsection over every T-subvariety of dimension k of X, we will say that D is k-admissible.Chapter 6. Intersections of T-line bundles^ 61Proposition 6.2.2 Suppose that X C Y is a T-subscheme of a smooth T-variety Y, D =(L, Z, s) is a T -pseudo divisor on X, x E YT, and there is a open affine T-subset U C Ycontaining x which is T-isomorphic to a representation space. Then, if L has weight Ax over x,multT(x, [D], Y) = AzmultT(x, X, Y).Proof: This is a consequence of Proposition 6.1.1.Remark: If D and D' are T-equivalent T-pseudo divisors, thenmultT(x, [D],Y) = multT(x,[g], Y).If D = D' + divf for f of weight —A, thenmuUT (x, [D], Y) = multT(x, [DIY) + )tmultT(x, X ,Y).(6.9)0(6.10)(6.11)Example: We show that the pullback of a T-pseudo divisor is not necessarily defined.Let X = A3 and let T = Gm x Gm , be the action with weights1 0 1 )0 1 2)Let D = (L, X, s) be the T-pseudo divisor where L is the trivial line bundle with weight (1,3)and s is the section defined by the function x2x3 E R(X) * . The equivariant sections of L aredefined by the functions axi4 — bx2x3 for a, b E k. So, on the subvariety W defined by theideal (x2), L has no equivariant section and if i : W X is the injection, then i*(L, X, s) doesnot exist. Note, however, that since(1 1) (ni ) (1 )(6.13)0 2) n2) = 3 )has solution (-1/2, 3/2), L® 2 does have an equivariant section, and i*(.02 , X, 802 ) does exist.(6.12)Chapter 6. Intersections of T-line bundles^ 626.3 Intersecting T-pseudo DivisorsDefinition: Let D be a T-pseudo divisor on X and V a k-dimensional T-subvariety of X suchthat D is V-admissible. If j : V -4 X is the inclusion of V in X, we defineD • [V] = [j* D]^ (6.14)in Ail i (IDI n V).Note that if f : X' -4 X is a T-morphism of T-schemes, V is a T-subvariety of X' andD is f (V)-admissible, then OD has an equivariant section defined on f(V) and so f*D isV-admissible.We extend the definition linearly to cycles:Definition: If a = E nv [v] E 4(X) and D is an a-admissible T-pseudo divisor on X, thenD • a = E nvD • [V]^ (6.15)in AT__ 1 (IDI n (al).Proposition 6.3.1 I. Let a, a' E Zr (X). If D is an a and an a' -admissible T -pseudo divisor,thenD•(a+a')=D•a-I-D•a'^ (6.16)in 4_ 1 (1DI 11 (la( U Ica).2. If a E 4 (X) and D, D' are two a-admissible T -pseudo divisors, then(D-1-D')•a=D•a+D'•a^ (6.17)in 4_ 1 ((ig U WI) fl lap.3. Let f: X' -- X be a proper T-morphism, a E ZIT (r), D a f,.(a)-admissible T-pseudodivisor on X and g : f -1 (IDI) n ial -- ID) n f Ial the induced morphism. Theng.((f*D) • a) = D • f,,,a^ (6.18)Chapter 6. Intersections of T-line bundles^ 63in 4_ 1 (D n nal)).4. Let f : X' -4 X be a flat T-morphism of relative dimension n, a E Zr (X), D ana-admissible T-pseudo divisor on X and g : If*DinIf*(a)1 -4 IDI n la! the induced morphism.Then(f* D) • f* (a) = g* (D • a)^ (6.19)in A4,,,_ 1 (f-1 (1D1) n lal)•Proof: The proof is contained in Fulton [6]. For 3, note that since D is f.a-admissible, ODhas an equivariant section over every component of f.a, and f*D is a-admissible.For 4, recall that locally over U, any T-rational function is a ratio of elements of Ou. So,any T-Cartier divisor is locally a difference of effective T-Cartier divisors.[1 -1 (Z)] = f*[Z]^(6.20)then gives the result.^ ^Proposition 6.3.2 Suppose that X C Y is a T-subscheme of a smooth T-variety Y, a EZr (X), D = (L, Z, s) is an a-admissible T-pseudo divisor on X, x E YT and there is an openaffine T-subset U of Y containing x that is T-isomorphic to a representation space. Supposethat L has weight A over x. ThenmultT(x, D • a, Y) = Arnulty , (x, a, Y).^ (6.21)Proof: This is a consequence of Proposition 6.2.2.^ ^Example: For flat pullback the admissibility conditions on X are necessary.Let X = A2 , let T = Gm x Gm and let X have the T-action defined by the weights ( 01 7)and let D = (L, X, s) be the T-pseudo divisor where L is the trivial T-line bundle with weight(0,1) and s is defined by x2 E 0A2 . Let X' .--- A3 with weights ( 01 77). Let f : A3 --+ A2 be theprojection onto the first and second coordinates. L has no equivariant section on the varietyChapter 6. Intersections of T-line bundles^ 64V defined by (x2) in A2 , but f*L has an equivariant section defined on f*V. So flat pullbackdoes require the admissibility condition on X.Example: Additivity of T-pseudo divisors requires the admissibility conditions.Let X = A3 , let T = Gm x Gm and let X have the T-action defined by the weights G 13 0,let D = (L, X, s) be the T-pseudo divisor where L is the trivial T-line bundle with weight (1,0)and s is defined by the function x3, and let D' = (L', X, s') be the T-pseudo divisor where L'is the trivial T-line bundle with weight (0,2) and s' is defined by the function xi/x3. Let V bethe T-subvariety of X defined by the ideal (x3). Then neither D nor D' are V-admissible, butD + D' is V-admissible.6.4 CommutativityTheorem 6.4.1 Let D and D' be two T -Cartier divisors on an n-dimensional T -variety X .Suppose D is PI-admissible, and D' is IDI-admissible. ThenD • [DI = D' • [I)]. (6.22)Proof: We proceed by cases. If D and D' are effective and intersect properly, the theorem islocal, and is purely algebraic and is contained in Fulton [6], Theorem 2.4, case 1. The proof isa rewriting of Fulton [6], Theorem 2.4.For the other cases we consider the blow-up X X along certain subschemes. We pull backD and D' to X and intersect there. Since we have problems with the existence of equivariantsections, the only difference between what we do and what Fulton does is that we have to checkfor the existence of equivariant sections on the subvarieties of X that map to the codimension1 subvarieties of X that we are interested in. Checking for the existence of equivariant sectionsand getting around their non-existence makes up the bulk of this section.If D and D' are effective, lete(D,D') = maxfordy (D)ordv (D') : codim(V, X) = 1}^(6.23)Chapter 6. Intersections of T-line bundles^ 65where the max is over all codimension 1 T-subvarieties of X. Note that E(D , D') = 0 preciselywhen D and D' intersect properly.Let D fl D' be the intersection scheme of D and D'. If the local equations for D and D'are a and a', then D fl D' is defined locally over U by the ideal I = (a, as). For convenience,we also use I = (s, t). Let 7r : X -+ X be the blow-up of X along D fl D'. Locally, X =Proj(631 7 Lo/n) = Proj(Ox e (®°° 1(s, On)).Let E = 7r-1 (D fl D') be the exceptional divisor. Let U, and Ut be the open T-subsets of7r-1 U obtained by inverting s and t. Then, on Ut , r*D is defined by a = fa', 7r*D' is definedby a', and E is defined by a'. On Us , ir*D is defined by a, 7r*D' is defined by a' = !a, and Eis defined by a. So, if on Ut we define C by f and C' by 1, and on U3 we define C by 1 and C'by 1, then r*D=E+C, 7r*D' = E+ C', and 7r*D, 7r*D', E, C and C' are effective T-Cartierdivisors on X.Lemma 6.4.2 With notation as above,1.ici n ic'l = 02. If e(D, D s) > 0, then e(C,E), e(C s ,E) are strictly less than e(D,D').Proof: These claims are local in nature, so we assume that X = Spec(A). Our explicitdescription of C and C' shows that ICI fl IC'I = 0. For part 2, consider the mapAES,71 -4 Cf_oinS1-4 s^ (6.24)T i-twhere S has weight As and T has weight A t . This map is equivariant, defines an injection ccandChapter 6. Intersections of T-line bundles^ 66X -4. X X P iN IP^(6.25)Xcommutes.We can also consider s and t as T-sections. Consider the pull back of 0(1) from X x P 1 toX. Let s and t be the sections on X defined by the pullbacks of the sections associated to Sand T. We represent the associated functions by s and t as well. Then ICI is the zero-schemeZ(s) of s, and ICI is Z(t).Since Z(S) and Z(T) are mapped isomorphically to subschemes of X by p, Z(s) and Z(t) arealso mapped isomorphically to subschemes of X by 7r. So, if V is a codimension 1 T-subvarietyof X contained in ICI or in ICI, then V = 71 - (17) is of codimension 1 in X. Also, [D] = 7.[E+C],so that,ordv(D) > ordi,-(E) + ork(C).^ (6.26)Repeating this for D' and E + C', we getordv (D') > ork(E) + ordi-i(C').^ (6.27)Now, suppose 0 < e(D,D'), and V is such that ordi-j-(E)ord i-j-(C) = c(C, E). Then,ordv (D)ordv (D') > (ordp- (E) + ordi; (C))(ordi; (E) + ordi; (C'))^(6.28)> ordy(E) 2 + e(C,E).^ (6.29)Since ordv—(E)2 > 0, e(C, E) < E(D, D'). 0For the final technique we (temporarily) redefine equivariant intersection. If D is V-admissible, then the intersection is defined as before. If D is not V-admissible, we set D -V = 0.Chapter 6. Intersections of T-line bundles^ 67Because of this redefinition, we have to show proper pushforward still works for our V and thatequivariant sections exist on the appropriate varieties.Lemma 6.4.3 Let D and D' be T -Cartier divisors on a T -scheme X, D be PI-admissible,it : X -+ X be a proper T-morphism, V C Ir*D'I be a T-subvariety of X' of codimension 1,and let V = r(i-7). Then,R-*(7r*D • [V]) = D • [V]^ (6.30)in lqi_2(X).Proof: If V is of codimension 1, then V C 1./Y1 and OD has an equivariant section on V. So,7r* (7r*D • [V]) = D • [V].^ (6.31)in .47,:_2(X).If V is of codimension greater than 1, thenr* (7r*D • [q) = 0 = D • [V],^ (6.32)in AnT_2 (X), independent of whether a section exists or not.^ 0Now, we need to know whether OE, Oc, Oc' have equivariant sections over the appropriatevarieties.Lemma 6.4.4 Suppose V. is a T -subvariety of X of codimension 1 and V is of codimension 1in X. If V C leg, then OE and Oc, are is7 -admissible. If V C lir*VI, then OE and Oc are17 -admissible.Proof: We check the result using our explicit descriptions of OE, Oc and Oc,.Suppose V C ICI — Ann. Locally 0c , has section 1 over V and OE has section a' overV which is invertible over V. Similarly, if V C IC'I — IElnIcl, then Oc has section 1 over 1/'and OE has section a over V which is also invertible on V .Chapter 6. Intersections of T-line bundles^ 68If V C 1C1n1E1, then V is of codimension 1 in X and is contained in 1D1n1D'1. So OD andOD, have equivariant sections on V, Oc, has section 1 on V, and sinceir*D' = E + C', (6.33)OE has an equivariant section over V. Similarly, since 7r*D = E + C, Oc has an equivariantsection on V. For V C ICI n 1E1, we have Oc with section 1 on V, and OE and Oc, haveequivariant sections defined on V as well.Finally, let V 1E1 — ( 1C1U ICI) n 1E1 and let V be of codimension 1 in X. OD has anequivariant section on V , and so Or.D has an equivariant section on V. Oc has an invertibleT-section over V and so OE has an equivariant section on V. Similarly, Oc , has an invertibleequivariant section over V. ^case 2Suppose that D and D' are both effective. We show that for the redefined intersectionD • [D'] = D' • [D] by induction on OD ,D'). For c(D, D') = 0 we have already shown the result.Suppose the result is true for all effective T-Cartier divisors B and B' with c(B , B') < f(D, D').For it : X —> X a proper bi-rational T-morphism, let r*D = E ± C, and ir*D' = E ± C' whereE, C and C' are effective T-Cartier divisors. Then,D • [DI= ir* (7r*D • [71-* D']) (6.34)=77-* ((E± C) • [E ± C']) (6.35)== r* (E - [E] ± E • [C] ± C • [E] ± C • [C1) (6.36)= 7r* (E • [E] ± C' • [E] ± E • [C] ± C • [CT (6.37)= D' • [D] (6.38)in A,T_ 2 (X). Note that we use bi-rationality to get ir*[7r*D] = [D].Since D is 1/4-admissible and D' is ID1-admissible, the usual equivariant intersection andthe redefined intersection agree. So, for the usual intersection,D • [D'] = D' • [D]^ (6.39)Chapter 6. Intersections of T-line bundles^ 69in A,7,1_ 2 (X).case 3.Suppose one of D or D' is effective, say D'. Let D be T-represented on U by d = alb andD' by a' for a, b, a E Ou = A. Let J be the sheaf of denominators of D. So, on U,J = fc E A : cd E Al.^ (6.40)Let I be the sheaf of numerators, i.e.I = dJ.^ (6.41)I and J are both T-sheaves. To show that J is a T-sheaf, note that if c E J,Xd(t)(t • c)d = t • (cd) E A.^ (6.42)So, t • c E J. Since d is a weight vector, I = dJ is a T-sheaf as well.Let K be the sheaf generated by both I and J. We blow-up X along the subschemedefined by K. On U this ideal is generated by (a, b). As before we label this by (s, t). LetU = Proj(E13,7_0Kn)) Proj(A ® (ED,',°_ 1 (s, t))). On Ut we have a = (s/t)b, and on U,,b = (t/s)a.We consider the map r :^X x P 1 induced by the local map on functions,k[S,T1 — ED,',°_0 (S,S - s^ (6.43)T t.As before, we can consider s and t as T-sections. We consider the pullback of 0(1) from P Ito I. Let s and t be the T-rational functions associated to the sections on X defined by thepullbacks from X x P 1 of the sections associated to S and T. We label these T-sections by sand t as well. Then C = Z(s), and C' = Z(t). These are mapped isomorphically by it to X, soif V is a codimension 1 T-subvariety of X contained in ICI or in ICI, then V is of codimensionChapter 6. Intersections of T-line bundles^ 701 in X. Now, on Ut, C is defined by i and C' is defined by 1. On Us , C is defined by 1 and C'is defined by t . So, on Ut, ir*D is defined by (slt)b • b-1 and on U, by a((tIs)a) -1 . We haveC n c' = 0, andir*D = C - C'.^ (6.44)We again check for the existence of equivariant sections.Lemma 6.4.5 Let D' be any T-Cartier divisor on X . If D is a ID'i-admissible T-Cartierdivisor on X and V C 17r*D'I is of codimension 1, then Ott.D, Oc and Oc , arel7 -admissible.If D' is IDI-admissible,17 C Ilr*DI and V is of codimension 1 in X, then (9,. DI is V -admissible.Proof: Suppose V C 17r*D'I. If V C 17r*D'i - ICI U ICI, then Oc and Oc, have invertibleequivariant sections defined on V.If V C lir*D'I n ICI, then V is of codimension 1 in X and Oir ►D has an equivariant sectionon V. Oc, has section 1 on V, so Cc has an equivariant section on V.Similarly, if V C 17r*D'1 n ICI, then V is of codimension 1 in X,^D has an equivariantsection defined on V, Oc has section 1 and Oc, has an equivariant section on V.Suppose that V C^If V is of codimension 1 in X, then OD' has an equivariant sectionover V and O►D, has an equivariant section over V.^ ^Note that if E is the canonical divisor on X then kr*D1 contains E. However, since 7r*D =C- C'+E - E, for the redefined equivariant intersection whether E has an equivariant sectionover any V in X is immaterial.Applying case 2, and in particular equations 6.34 to 6.38, since C,^and D are effectiveand 7r : X -4 X is bi-rational, for the redefined intersection,D • [DI = lr*(lr *D • [r * D']) = 1r* (7r*D' • [7*g) = D' • [D]^(6.45)in A,T_ 2 (X).Again, since D is ID'I-admissible and D' is IDI-admissible, the usual equivariant intersectionand the redefined intersection agree, so with the usual intersection,D•[D'} D' • [IA^ (6.46)Chapter 6. Intersections of T-line bundles^ 71in AT_2 (X).For the final case where D and D' are arbitrary, we blow-up along the T-subscheme definedby the sheaf of denominators and numerators of D as above to getir*D = C — C'^ (6.47)on X where C and C' effective and ICI fl^= 0. Applying case 3 then givesD • [D'] = ir.((C + C') • [7*^= ir.(r*D' • [C + C1) = D' • [D]^(6.48)for the redefined intersection. Again, as above, this implies commutativity for the usual equiv-ariant intersection.^ ^Corollary 6.4.6 Suppose D is a k and k + 1-admisssible T-pseudo divisor on X and supposethat for all k and k + 1-dimensional T-subvarieties V of X there is a weight vector f E R(V)*of weight 0. If a E 4(X) is T-rationally equivalent to 0, then D • a = 0 in Ak 1 (X).Proof: Let V be a k + 1-dimensional T-subvariety, and let f E R(V) * be a weight vector ofweight 0. Then,D • [divf] = divf • [D] = 0.^ (6.49)0Example: The condition on V is necessary.We use a previous example. Let X = A3 , with torus Gm x^and weights(1 0 1(6.50)0 1 2)Let L be the T-line bundle with weight (1,3). The equivariant sections of L are defined byfunctions ax2x3 — bxix3 for a, b E k. The T-rational functions of weight 0 are of the forma(xix3/x3)n for a E k and n E Z. Over the subvariety defined by (x3) L has an equivariantChapter 6. Intersections of T-line bundles^ 72section, but over the cycle defined by (xix3) which is T-rationally equivalent to it, no equivariantsection exists. So, we do need the second condition in the corollary.Because of the problems with rationality, we define:Definition: X is k-nice if for every k-dimensional T-subvariety V of X there is a weight vectorf E R(X)* of weight 0. Equivalently, the trivial T-line bundle of weight 0 has an equivariantsection on every k-dimensional T-subvariety of X.Remark: Niceness is not as strong a condition as admissibilty. Any T-variety has an opensubset U over which algebraic quotient U IIT exists. If k > dim(X — U), since OullT '-=-' Ox (U)T ,X is k-nice.6.5 Intersection with T-line bundlesDefinition: Let L be a T-line bundle on a T-scheme X. We will say L is k-admissible if L hasan equivariant section over every k-dimensional T-subvariety of X.Definition: Let L be a T-line bundle on X. If V is a k-dimensional T-subvariety of X and Lis k-admissible, then L is T-represented by a T-pseudo divisor D and we defineci(L)n v = D • [V]^ (6.51)in Ak I (V). We extend this definition linearly to a E 4 (X).Proposition 6.5.1 1. Let X be k and k + 1-nice, a E ZIT (X), and let L be a k and a k + 1-admisssible T-line bundle on X . If a is T -rationally equivalent to 0, thenco) n a = 0^(6.52)in 4 1 (X).L. Let X be k and k + 1-nice, a E AZ' (X), and let L and L' be k and k + 1-admisssibleT -line bundles on X . Then,co ® L') fl a = ci(L) fl a + ci(V) fl a^ (6.53)Chapter 6. Intersections of T-line bundles^ 73in Ak i (X)•3. Let f : X' -- X be a proper T-morphism, X and X' be k and k + 1-nice, and let L be ak and k + 1-admisssible T-line bundle on Y. If a E AT (X'), then10(ci(f *L) n a) = c1(L) n Ma)^ (6.54)in Ak 1 (x).4. Let f : X' --). X be a flat T-morphism of relative dimension n, X be k and k + 1-nice,X' be k + n and k + n + 1-nice and let L be a k and k + 1-admisssible T-line bundle on X . Ifa E AZ' (X), thenf * (ci(L) n a) = c i (f* L) n f *a^ (6.55)in AT" _ i (X I).5. Let X be k — 1, k and k + 1-nice, and let L and L' be k — 1, k and k + 1-admisssibleT-line bundles on X . If a E AT (X), thenci (L) n (ci (L') n a) = c1 (L') n (ci (L) 11 a),^(6.56)in Al 2 (X).Proof: We know the results hold if a E ZiT(X). Part 1 follows from Corollary 6.4.6. Theremainder follow from part 1, and Proposition 6.3.1.^ ^Proposition 6.5.2 Suppose that X C Y is a k and k + 1-nice T-subscheme of a smooth T-variety Y, a E AT (X), L is a k -admissible T -line bundle on X, x E YT and there is an openaffine T-subset of Y containing x that is T-isomorphic to a T-representation space. If L hasweight A over x, thenmu/tT(x, ci (L)11 a, Y) = Amu/ty , (x, a, Y).^ (6.57)Proof: This results from Proposition 6.3.2, and that mu/tT(x, a, Y) is an invariant of 4(X).aChapter 6. Intersections of T-line bundles^ 746.6 NotesWe could have considered different forms of equivariant intersection.If (L, Z, s) is a T-pseudo divisor, we could demand that IDI C Z, (p : OD -4 L be a weightisomorphism (but not necessarily a T-isomorphism), and w(sD) = s off Z. Note that since thetransition functions are equivariant, if 90 is a weight morphism of weight A on one U,„, then it isa weight morphism of weight A on all the U,„. This would lead to a different form of equivariantintersection for which a section always exists, but for which most of the equivariant propertiesare lost. In future, we will say such a T-Cartier divisor represents (but not T-represents) the T-pseudo divisor (L, Z, s). Note that for normal projective varieties, Theorem 4.7.3 shows that thestrongest equivalence relation we can use with this form of intersection is rational equivalence.If Z = X, D and D' represent the same T-pseudo divisor ifD = D' + divf^ (6.58)where f E R(X)* is a weight vector.For this form of intersection, all the results of Proposition 6.5.1 hold without the admis-sibility conditions, but instead of getting a cycle in AT (X), we get a cycle in Ak (X). Theequivariant multiplicity is also dependent on the choice of representing T-Cartier divisor. If(L, Z, .9) is represented by the T-Cartier divisor D, L has weight Az in the fibre over x E XTand D has weight A + A over x, thenmultT(x, D • a, Y) = (Az + A)multT(x, a, Y).^(6.59)There is another form of equivariant intersection, which we call * intersection:Definition: For D a T-pseudo divisor on X, V a k-dimensional T-subvariety of X, we set1-(nD) • V if L° 11 is V-admissible^D * a = { n^(6.60)^0 if no n exists such that L®n is V-admissibleChapter 6. Intersections of T-line bundles^ 75in g_ i (X) 0 Q.If L is a T-line bundle, we can replace our usual equivariant intersection with * intersection.We write this form of intersection as ci (L) * a. This form of intersection is well-defined.With respect to * intersection, the only property of Theorem 6.5.1 to hold without theadmissibility condition is proper push forward. If p : X' -+ X is a proper T-morphism, V C X'is a k-dimensional T-subvariety and f E R(U) * is a weight vector where U is an open affine T-subset of V, then p* ([divp*(f)]) = d[divNR((x;) (f)], where d = [R(X1) : R(X)]. Since N 17((x) ) (f )has weight dAf, if (p*L)Øn has an equivariant section over V for some n, then L®d" has anequivariant section over p(V). So, pushforward holds for * intersection without the admissibilitycondition.The main reason for considering * intersection is that if T = Gm , a section is always definedfor some tensor power of L. This is just an application of the Chinese Remainder Theorem.For * intersection, if X is k and k -}- 1-nice thenAmultT(x, [V],Y) if L (8)n has an equivariant section overV for some nmuitT (x, ci (L)*[11,Y) =^ (6.61)0 if L®n has no equivariant section overV for any n.Chapter 7Intersections with T-Vector BundlesWe define the intersection of T-vector bundles with T-cycles in a T-variety X. In doing this,we follow Fulton [6]. First we define the T-Segre classes and then we define the T-Chern classesin terms of the T-Segre classes. As in the previous chapter, we have admissibility and nicenessconditions. For T-vector bundles these are a lot more restrictive than they were for T-linebundles.7 .1 T-Segre classesDefinition: Suppose that E is a T-vector bundle over an n-dimensional T-scheme X of ranke + 1. We say that E is a k+-admissible T-vector bundle if 0(1) is a k to (n + e)-admissibleT-line bundle on P(E). We will say that P(E) is k+-nice if P(E) is k to (n + e)-nice.In future, we shall call the admissibilty and niceness conditions just admissibility conditions.Definition: Let E be a (k — i + 1)+-admissible T-vector bundle, P(E) be (k — i + 1)+-niceand let p : P(E) -- X be the projection. For a E AT (X) we define the T-Segre class assi(E) fl a = p.(ci(0(1)) e+i n p*a)^ (7.1)in AT (X).We start with a basic fact:76Chapter 7. Intersections with T-Vector Bundles^ 77Lemma 7.1.1 Suppose f :^X is a flat T-morphism of relative dimension n and L is a(k — i +1) 1- -admissible T-line bundle on X . Then f*L is a (k + n — i +1)+ -admissible T-linebundle on X'.Proof: Note that if V E Zr(X t) for 1 > (k + n — i + 1), then dim f (V) > 1 — i + 1. So L hasan equivariant section over f (V) and f *L has an equivariant section over V.^^Corollary 7.1.2 Suppose that f : X' X is a flat T-morphism of relative dimension n, E isa (k — i + 1)+ -admissible T -vector bundle over X and P(E) is (k — i +1)+ -nice . Then f*E is(k + n — i +1) 4- -admissible on X' and P(f*E) is (k + n — i + 1)+-nice.Proof: Note that Op( f*E)(1) = f * (00.)). The lemma applied to 0E(1) and the trivial T-line bundle on P(E) with trivial T-action then show the admissibility and niceness conditionsrespectively. ^Proposition 7.1.3 1. Let E be (k — i +1) -1- -admissible, let P(E) be (k — i +1)+ -nice and leta E Ak (X). Then,a. si(E) n a= 0 if i< 0b. so(E) n a = a.2. Let F be a rank f +1 T-vector bundle on X, E and F be (k — i — j +1) 1- -admissibleT-vector bundles, P(E) and P(F) be (k — i — j + 1)+ -nice and a E^(X). Thensi(E) n [si(F) n a] = si(F) n [si(E) n (7.2)in 4 i_i (x).3. Let f :^-4 X be a proper T-morphism, E be a (k — i +1) 1- -admissible T -vector bundleover X . Suppose that f *E is a (k — i +1) -1- -admissible T-vector bundle over X' and P(E) andP(f*E) are both be (k — i +0+ -nice. For a E AT (X'),f * (si(f* E) n a) = si(E) n f * (a)^ (7.3)Chapter 7. Intersections with T-Vector Bundles^ 78in Ak i (X)•4. Let f : X' -4 X be a flat T-morphism of relative dimension n, E be a (k — i +1)+ -admissible T-vector bundle over X, and P(E) be (k — i +1)+ -nice. For a E AT(X'),f*(si(E) fl a) = si(f*E) fl f*a^ (7.4)in fiLn_ i (X).5. If a E Ak (X'), E is a k and (k + 1)-admissible T-line bundle on X, and P(E) is k and(k + 1)-nice, thensi(E) fl a =^(E) fl a^ (7.5)in Ak i (X)•Proof: The proof is contained in Fulton [6]. The only part that is different is part 1.b. For 3,we need the admissibility conditions on X' since if V is a j-dimensional T-subvariety of X' suchthat dimp(V) < k — i +1 the admissibility conditions on X do not guarantee an equivariantsection on V.To prove 1, by 3 we can restrict to the case where a = [V] and V = X. We have,so(E) n^= P*(C1(0E( 1 )) e n ply]) = m[v]^(7.6)in AT (V) for some m E Z. We need to show that m = 1. Since this is a local result we canassume that E is trivial over V.Since the map j : AT (V) -4 Ak(V) sending a T-cycle in AT(V) to its rational equivalenceclass in Ak(V) is a surjection and since A T(V) = Ak(V) = Z are both generated by [V], wehave j[V] = ±[V] in Ak(V). Since the only difference between equivariant intersection andusual intersection is that we demand that the sections be equivariant, if a E Ai (V), we havej(si(E) fl a) = si(E) fl a in Ai_i(V) provided the left-hand side is defined. In [6] Proposition3.1 Fulton shows that so(E) fl [V] = [V] in Ak(X). So, so(E) fl [V] = [V] in AT (V), and m = 1.0The proof of 1.b has the following corollary:Chapter 7. Intersections with T-Vector Bundles^ 79Corollary 7.1.4 If E is (k +1)+ -admissible and (k +1)+ -nice, thenp* : 4(X) -4 4+,(P(E))^ (7.7)is a split monomorphism.Proof: The inverse map is given by p,(ci(0(1))e n -).^ ^Proposition 7.1.5 Suppose that X is a T-subscheme of a smooth n-dimensional T-variety Y,x E YT and there is an affine open T-subset U' of Y containing x that is T-isomorphic to aT-representation space. Suppose that E is a (k — i + 1)+ -admissible T-vector bundle on X suchthat the weights po, , p e of E in the fibre over x are distinct and P(E) is (k — i + 1)+ -nice.If a E 4(X), then,e^(pj)e-1-iMUUT(X, Si(E) n a,Y) = E ,muitT(x, a,Y).j=0 ii/OW/ k/i)(7.8)Proof: As in Proposition 6.1.1, we need the open T-subset containing x over which E is trivialto be isomorphic to a T-representation space. Since this is not necessarily the case, we considera different T-vector bundle that is of this form. Let U be a T-subset of U' containing x overwhich E is trivial. We extend E to U' to get a trivial T-vector bundle E' over U' that restrictsto E on U. Let p' : P(E') = U' x P(F) U'. We need to show thatmultT(x, si (E) n a, Y) multT(x, si(g) n a, Y). (7.9)We first show thatmuitT(x,p.(c1(0E(1)) n a), Y) = muitT(x,//*(c1(0E4 1 )) n cei), (7.10)where a E Zk (P(E)) and a' E ZiT(P(E1)) is the cycle related to a. We can restrict to thecase where a is a T-subvariety of X . First, suppose that V C P(E)1u is a k-dimensional T-subvariety. We restrict V to P(E)lu and close it in P(E') to get a k-dimensional T-subvarietyV' C P(E') that restricts to V in P(E)Iu . Note that p* [V] = p',,,[111]. A T-section s of OE(1)Chapter 7. Intersections with T-Vector Bundles^ 80determines a weight vector in R(V)*. Since R(V*) = R(V')* , this in turn determines a T-section of 0E1(1) over V' which restricts to s on V. Let D and D' be the associated T-pseudodivisors on V and V'. D' • [V'] and the closure of D • [V] in P(E') agree, except possibly onp'-1 (U - U'). However, subvarieties of p'-1 (U - U') map to subvarieties of U - U'. Sincex U - U', these subvarieties have equivariant multiplicity 0. So,multT(x,p,(ci(OE(1)) n^= multT(x,1*(c1(0 E' (1)) n ), Y).^(7.11)If a E 4(X), this shows thatmuitT(x,p.(ci (0E(1))e+i n P*a), Y) = muUT (x, P'*(cl(C3E' ( 1 ) )e+i n P'*a), Y)^(7.12)andmultT(x, si(E) n a,Y) = multT(x, si(E')n a, Y).^(7.13)We now need to show thatmuitT(x,/*(11, 37)^1ri^ \mu/4(x x^P (e)).j=0 1.1.4j4/1(7.14)for [V'} E ZIT (P(E')).Any T-subvariety of P(E') is T-rationally equivalent to a cycle whose components are ofthe form W x where W is a (k - q)-dimensional T-subvariety of X and of the formW x Pi where W is a k-dimensional T-subvariety of X. In the first case,multT(x x Pi,W x^= multT(x, W, Y)mu/tT(Pi, Fo i ..i0P(F)).^(7.15)Proposition 5.1.10 shows that1(7.16)jE=0 nwi _ to muitT (pi ,^= 0.Since p'„(W x^= 0,multT(x,p',,(W x 1= E^ multT(x Itimespi,W^= 0.j=0(7.17)Chapter 7. Intersections with T-Vector Bundles^ 81In the second case, if j i, then multT(x x Pi, W x Pi, P(E')) = 0. Since (W x Pi) = W,Proposition 5.2.2 part 7 shows thatejE=0^j(µi µ') multT(x X Pj, W x Pi, P(E')) muitT(x,Pfk(W x Pi), Y).So, for any a E AT (P(E')),e 1InUitT(X,p s,a, y) E µ^,)  multT (x x pi , a, P(E')).j=0 nw tpi — p)Now,(7.18)(7.19)multT(x, si(E) fl a,Y) = multT(x, si(g) fl a,Y)=multT(x,p*(ci(OE , (1)) n p*a), )7)iti ,MttitT(X x Pj,C1(0E1(1))"+i n a,P(E'))— E t^hoicui — ) (7.20)e^e+i^pi ,)MUitT(X X Pi, P*a, P(E')))— EJ=c, e^e+imultT (x, a, Y)=E Fri=0 --/Ojutit —Remark: Let X be a T-subscheme of a smooth T-variety Y. If we consider the quotientgroup AT(X)/ ti where a 0 if multT(x, a, Y) --= 0 for all x E XT, the proposition holdsindependent of niceness conditions. The reason is that if mu/tT(x, a , Y) --= 0 for all x E XT ,then multT(x, c1(L) fl a, Y) = 0 for any a-admissible T-line bundle L.The following is the T-Segre class analogue of a characteristic number formula that has beenproved under various conditions by Iversen and Nielsen [9] and Brion [3]. While all of theserequire some use of the Riemann-Roch theorem, our result is purely combinatorial.Proposition 7.1.6 Suppose X C Pn is a T-subscheme where P" has isolated fixed points andweights Ao,.. , An. If x = Pi E XT , let A = Ai. Let E be a T-vector bundle on X such thatChapter 7. Intersections with T-Vector Bundles^ 82the weights pzo,... , Axe of E in the fibre over x E XT are distinct. Let a E Ak (X). If P is anisobaric polynonial of degree k in xo,... , x e where xi has degree i, then1 m^E n^\ P(TO (X )7 • • • Te(X))77TUUT(X, Pn ),xEXT nyExT (Ay — ^x/yOx(7.21)where e^e+iTi(x) Ej=0 moi cti./ —and m is the geometric multiplicity of P(si(E),... ,se (E))n a.(7.22)Proof: The result is purely combinatorial. Let V be a k-dimensional T-subvariety of P(E).Let the fixed points of Pe be Pi and the open T-subset of Pe defined by inverting xi be Ui. Thefixed points of P(E) over x E XT are then x x Pi. We prove the proposition by first summingover the x x Pj where x is fixed and then summing over the fixed points of Pn .As in the previous proposition, let E be trivial over U. We can extend E to Ui to obtain atrivial T-vector bundle Ei overLet [V] E (P(E)). Since 0E(1) is not necessarily k+-admissible, we can only guaranteethe existence of a weight section s of weight v, say. As in the previous proposition, we canextend s to a weight section si of OE; (1). Let D be the representing T-Cartier divisor definedby s and Di, that defined by si. As in the previous proposition, if we identify the variety inP(Ei) associated to V with V,eTTj=0 II/0j1multT(x x Pi, D • [V],P(E)) =— Ax;)e^10 nio joixi fixi) multT(x x P;, D; • [V], P(Ei)) =^(7.23)^multT(x x Pi,[V],P(Ei)).If dim V > 0, either p.[V] = 0 or dimp.[V] > 0. In either case,E n 1^, x , TriuuT(x,p4vb pn) = 0.^(7.24)ArExT lAyEXT lAy — )yxChapter 7. Intersections with T-Vector BundlesSincewe have,83e 1multT(x,p,,[11,Pn) = E ,^muitT(x x Pj , [V], P(Ei)),i=o MA/U./xi — i ) (7.25).ExT1yexT (Ay — Ax ) pxj)MuitT(X X Pi, [11,P (Ei)) =y$s^r11:( zttix / + —v : Hio (pfixxii_ px j) MUitT(X X Pj, [11,P (Ei))•ypx1TEXT il yexT (Ay — A ) E0 ^j(7.26)Recalling that for a E AT(X),multT(x x Pj, p* a,P (Ei)) = multT(x, a, Pn )^(7.27)and using the above we find,1n^ ,x, muitT (x, si(E) fl a, Pn) =xEXT 11YEXT (Ay — A )yx1 enioi(pzi _ iaxi) muitT (x,„e-Fi^ (7.28)P'xj zEXT 11 YEXT (Ay — Az ) jo_yOsSo,E^/^As^ multT(x, P(so(E), , (E)) n a, X) =xExT 11 yEX (Ay — A )1^ (7.29)sEXTE 11 YEXT (-y —n^Ax)muitT(x,P(To(x), —,Te(x)), Pn).y$xRecalling that if a E 4(X), 1 M E^ x, multT(x, a, X),xEX II yExT kAy — Ayx(7.30)we get the result.0While the sums in the proposition may look formidable, note that Fly0x (Ay — Ax ) is just theChapter 7. Intersections with T-Vector Bundlese^„e-FiH TT^P' xj=0 iik$j(itxk Axj)is the determinant of the matrix/ 1 Ax0VDM(pxj, i) det1 itx i‘ 1 Axe84e+iauxo(7.31)Px1(7.32)e-1,e-1Van Dermonde determinant and that the numerator of the fraction/40/4 114e^AX;1 AXV7.2 T-Chern ClassesWe define the T-Chern classes in terms of the T-Segre classes and we show their basic properties.The bulk of this section involves showing that the splitting construction works in the equivariantsetting.We imitate the construction of the T-Chern classes in Fulton. If E is (k — j+1)+-admissibleand P(E) is (k — j + 1)+-nice, we define the polynomialst(E) E si(E)t i .^ (7.33)i=oWe define the T-Chern series to be,00ct (E)^ci(E)ti,i=0where ct (E) = st (E) -1 . The first j T-Chern classes are the first j terms of the series.Explicitly,(7.34)co (E) =1^(7.35)cl (E) =^(E) (7.36)c2(E) — s1(E)c1 (E) — s2 (E)^ (7.37)ci(E)= —si(E)ci_i(E) — 82(E)ci_2(E) — • • • — si(E).^(7.38)Chapter 7. Intersections with T-Vector Bundles^ 85Theorem 7.2.1 1. Suppose that E is a (k -i+1)+ -admissible T-vector bundle on a T-schemeX such that P(E) is (k - i +1)+ -nice. If i > rankE, thenci(E) = 0.^ (7.39)2. Let F be a rank f +1 T-vector bundle on X, E and F be (k - i - j +1)+ -admissibleT-vector bundles, P(E) and P(F) be (k - i - j +0+ -nice and a E 4(X). Thenci(E) n [ci (F) n a] = ci(F) n [ci(E) n^ (7.40)in A iL i_i (X).3. Let f : X' -4 X be a proper T-morphism, E a (k - i +1) 1- -admissible T-vector bundleon X, f*E be a (k - i +1)+ -admissible T-vector bundle on X' and let P(E) and P(f*E) be(k - i +1)± -nice. If a E AT (X), thenf.(ci(f* E) n a) = ci(E) n fka^ (7.41)in 4_ i (X).(. Let f : X' -> X be a flat T-morphism, E a (k - i +1)+ -admissible T-vector bundle onX and P(E) and be (k - i +1)+ -nice. If a E AT (X), thenci(f* n^= f * (ci(E) n a)^ (7.42)in 4+n_ i (X).5. Let a E AT (X). If E is of rank r = e +1,0 -> E' -4 E E" -4 0^ (7.43)is an exact sequence of (k - r)+-admissible T-vector bundles and P(E'), P(E) and P(E") are(k - r +1)+ -nice, thenct(E) = ct(g)ct(e).^ (7.44)6. If E is a [X]-admissible T-line bundle on X, D a T-pseudo divisor on X with OD E,thencl (E) n X = [OD].^ (7.45)Chapter 7. Intersections with T-Vector Bundles^ 86Proof: 2, 3, 4 and 6 follow from Proposition 7.1.3. We show 1 and 5 by using the splittingconstruction.Splitting ConstructionGiven a (k - r +1)+-admissible T-vector bundle E of rank r = e + 1 on X such that P(E) is(k -r +1)+-nice, we construct a space f : X' X such that f is a flat T-morphism of relativedimension n, say, f* : AT (X) -4 4+n (r) is injective, f *E is (k +n - r +1)+-admissible, andf *E has a filtration,0 = Mo C^C C Mr = f*E^ (7.46)where the quotient bundlesMi/Mi-i = Li^ (7.47)are (k + n - r + 1)+-admissible T-line bundles.The construction is the usual splitting construction. Considerp5E^EI^IP(E) X.Po(7.48)p'6E has a T-subbundle OE(-1). Let E 1 be the quotient T-vector bundle p'd E/0(-1) on P(E).E 1 is of rank r - 1. If we set X 1 = P(E), we can repeat the construction to get X 2 = P(E 1 ),and E2 = 14E 1 /0Ei(-1). Continuing this, we arrive at X' = X'. Let f be the composition ofall the pi. Since each pi is flat, f is flat, and since pi : Ak (X i ) -+ 4+e_ i (P(E i )) injective byCorollary 7.1.4, f* : AT (X') 2417:(X) injective. This gives us the diagram:Chapter 7. Intersections with T-Vector Bundles^ 87Er(4 E2 --• • . -+ E2qi El ____4^______ pl'El ______* El----+ PIA E ----*q5E _____+^ p6E ______* EI^I^i^IX '^ • -4 P(E 1) ----) P (E) --, X,(7.49)Pr^ P1^ Powhere qt = pr 0 pr-1 0 • - • o pi.Let Mi be the kernel of the map q5E -+ q7E i . We have Mi_.1 C Mi and0 =-  Mo C Mi. C • • • C Mr =1*E.^ (7.50)Finally, since the E i are T-vector bundles of rank r - i, the M2 are T-vector bundles of rank i.We still need to show admissibility.Lemma 7.2.2 If E is a (k - i +1)+ -admissible T -vector bundle of rank e and P (E) is j+ -nicefor any j > 0, then any rank e -1 T-subbundle E' and any quotient T -line bundle L of E are(k - i + 1)+ -admissible and j+ -nice.Chapter 7. Intersections with T-Vector Bundles^ 88Proof: we have the commuting diagramOE, (1) --+ OE( 1 )P (E')^P (E)^(7.51)X i -4 X/.0E1(1) is the restriction of OE(1) to P(E'), so, since E is (k — i + 1)+-admissible, and anyT-subvariety of P(E') is a T-subvariety of P(E), E' is (k — i + 1)+-admissible.Suppose that OL is locally generated over Ou by x. Since E L, there is a morphismon the structure sheaves cio : Ou k[x] -4 OE. Let yo(1 (8) x) = y. This induces a morphismP(E) --4 P(L), and we also get a morphism 0E(-1) OL (-1) .cs2 L. Since 0E(-1) has anequivariant section over any T-subvariety of X of dimension greater than (k — i + 1), composingwith the morphism above gives a T-section of L. So, L is (k — i + 1)+-admissible.To show the niceness conditions, note that if P(E) is j+-nice, then the trivial T-line bundlewith trivial T-action over any T-subvariety of P(E) of dimension greater than j has a T-section.If E' is a T-subbundle of E, then any T-subvariety of P(E') is contained in P(E), so any T-subvariety of P(E') of dimension greater than j has a T-section of the trivial T-line bundlewith trivial T-action. Similarly, if L is a quotient T-line bundle, then P(L) = X. Since X canbe embedded is P(E), any T-subvariety of X of dimension greater than j has a T-section ofthe trivial T-line bundle with trivial T-action defined on it. ^For future reference, we would like to know the weights of the L. We find them inductively.Let Fuo... ji ) be the T-representation space with basisX^0 , We let Uji be the opensubset of P(Fuo ,... Ji )) obtained by inverting xi i . Locally, P(E i ) is of the form Uxqx•-•xChapter 7. Intersections with T-Vector Bundles^ 89and locally E' has weights {A, :1 0 jt, for all i}. Pulling this back to X', and taking kernels,we find Mi has weights Ajc,, , Aii over Ux U,0 x • • • x . So, the quotient bundle Mi/Mi_i = Lihas weight Ai over U x x • • • xSuppose i > r. We want to show thatct (E) = H(1 + (4)0.^ (7.52)j=1Lemma 7.2.3 Suppose E is filtered as above, E is (k — i+1)+ -admissible for i > r and P(E)is (k — i + 1)+ -nice. Let s be an equivariant section of E with support Z. Then, for a E (X),there exists a cycle Q E 14 such thatH ci (Lj) n a =^ (7.53)j=iIn particular, if s is trivial, i.e. Z = 0, then IT ci (4) n a = 0.Proof: The proof is as in Fulton [6]. We proceed by induction. If r = 1, since E is ak+-admissible T-line bundle, E is T-represented by the T-pseudo divisor (E, Z, s) where Z =supp(s). So, ci (E) n a C Z. Suppose that we have the result for T-vector bundles of rank r —1.Consider the exact sequence,0 Mr_i^E Lr^O. (7.54)Let g = cp(s). Then, s is a (possibly 0) T-section of Lr . Since E is (k — i + 1)+-admissible, Lis as well, and we set{(Lr, Z ,."§) if g 0 0Dr =(Lr , Z, si) if g = 0,where s' is an equivariant section of Lr over X.Ifj:Z-4Xis the inclusion,(7.55)ci (Li.) fl a = j.(D,. • a).^ (7.56)Since Mr_i is a rank r — 1 T-subbundle of Mr, Mr-1 has a T-section induced by s with supportZ.r-1H ci (4) n a = ( 1-1 cO.0) n ./.(Dr • a) = 0J=0 (7.57)Chapter 7. Intersections with T-Vector Bundles^ 90for E 14. 0To complete the splitting, let E be a (k - i + 1)+-admissible T-vector bundle for i > r overX which is filtered as above. Consider p : P(E) -4 X. 0E(-1) is a T-subbundle of p*E and0(-1) 0 0(1) is a trivial T-line subbundle of weight 0 of p*E 0 0(1). Since 0(-1) 0 0(1) is(k-i+1)+-admissible it has an equivariant section over any T-subvariety of P (E) of dimensiongreater than k - i +1. So we have a trivial T-section of p* E 0 0(1). p*E 00(1) has a filtrationby T-subbundles with T-line bundle quotients p*Lj 0 0(1). So,ci(P*Li 0 0(1)) = 0.^ (7.58)Let E = ci(0 ( 1 )), ak (resp. &k) be the k th symmetric function in the c1 (Li) (resp. ci(P*Li))•Since p*Lj, and 0(1) are (k - i + 1)+-admissible,4.lici(p*Lj 0 0(1))^7*^ 0.Multiplying by e-1 for 1 < 1 < i +1, and recalling that r e + 1,e Fl 5^. 1. e+/-1^+ Err Cl-1^0 .So,(7.59)(7.60)P*(ci(0 ( 1 )) e+i np*a) +p*(5-ici(0(1))e+1-1 -n + . +p*Carci (0(1)) 1-1 p*a) = 0 (7.61)si(E) fl a +^(E) n a + + arsi,_1(E) 11 a = 0.^(7.62)So,(1 + alt +^+ artr )st (E ) = 1,^ (7.63)andct(E) = H(1 + ci(Li)t).^ (7.64)i=iChapter 7. Intersections with T-Vector Bundles^ 91We now show 1 a and 5.Injectivity of f*, andci(f* E) fl f *a = f * (ci(E) fl a)^ (7.65)now imply la. For 5, we find f : X' X such that f*E' and f*E" split, with (k — r + 1)+-admissible T-line bundle quotients and L. f *E then has an induced filtration with quotientsand LZ. So,ct(E) = ci (e)ci (E'').^ (7.66)0The splitting construction also yields,Proposition 7.2.4 Suppose that X C Y is a T-subscheme of a smooth T-variety Y, x E YTand there is an open affine T-subset U of Y containing x which is T-isomorphic to a T-representation space. Let E be a (k — i + 1)+ -admissible T-vector bundle on X with weightsPo, • • • Pe, and let P(E) is (k — i+1)+-nice. If a E AT (X), thenmu/tT(x,ci(E) fl a, Y) = Cli (itO, • • • Pe)MUltT (X, a, X).^(7.67)Proof: We note that in the spliting construction, X' is locally of the form U x F where U isan open T-subset of X and F is a T-representation space. If x x 0 E U x F, thenmuitT (x, q (E) n a, Y) = multT(x x 0, ci(f*(E)) n f *a, U x F).^(7.68)Since ci(f* E) n f* a = m=i (ci (Li) fl f *a), and Li has weight Ai over x,multT(x x 0, ci(f *E) n f* a, X') = vi(/p0, • • • , pe)multT(x x 0, f* a , X')^(7.69)multT(x, ci(E) fl a, X).^(7.70)0Chapter 7. Intersections with T-Vector Bundles^ 92Corollary 7.2.5 If X, Y and E satisfy the conditions of the proposition,^multT (x , si(E) fl a, X) = ( p 0 ,^, fie )mu/t(x, a, X)^(7.71)where ri(po,^, pe ) is the sum of all monomials of degree i in the pj 's.Proof: This results from the relations between the T-Segre and T-Chern classes. These rela-tions are precisely the ones between the elementary symmetric functions and the Ti.^^We also get the following seemingly purely algebraic fact:Corollary 7.2.6^= Ti (110 , • • • ite)11j0k(iij Pk) •Oddly enough, the determinant VDM(//ii, i) does not seem to have been calculated before.If we omit the admissibility conditions, since we can not determine the equivariant multiplic-ity of the intersection except when the final cycle is of dimension 0, we only get the characteristicnumber formula mentioned before:Proposition 7.2.7 Let X C Pin be an n-dimensional T -variety. Suppose that P rn has iso-lated fixed points and weights A0,..., Am . Let E be a T-vector bundle over Pm with weights{No, • • • /lie} over Pi E Pm. Let a E (X). If P(xo, , x e ) is an isobaric polynomial ofdegree n, where xi has degree i, thenP (ci (E),^1, ce (E) n a) = E^P(aii,• • • ,aie)mutt,(p,,a,Pn),^(7.72)i=o Ai)where a ik are the k th symmetric functions in the tiiiProof: This is merely a consequence of the result concerning T-Segre classes. If we replace theci by its expression in terms of T-Segre classes, we get an isobaric polynomial as above, but inthe T-Segre classes. Proposition 7.1.6 and the expression of the T-Segre classes in tems of theT-Chern classes then yield the result. ^Chapter 7. Intersections with T-Vector Bundles^ 937.3 NotesIf we consider * intersection (i.e. we have ci(O(1))e+i *^in the definition of intersection)then all the properties of Proposition 7.1.3 and Proposition 7.2.1 hold, and the only ones to holdwithout admissibility conditions are Proposition 7.1.3 part la and the push forward propertiesfor both the Segre and Chern classes. The various properties concerning multiplicities hold aswell, provided we do not omit the admissibility conditions.Chapter 8Applications and RelationsWe consider how AT(X) is related to the other equivariant objects. The objects we are in-terested in are A"(X//T) where X//T is the algebraic quotient, vector bundles on X//T andKT(X). The main result we show is that if X is a complex variety with free T-action anddim T = r, then AT (X) 0 Q = Ak+ ,(XIIT) 0 Q. In the third section we consider 4(X) forG a reductive algebraic group.8.1 A(X//T)Suppose dim T = r. Let X be a complex T-variety with free T-action, and suppose that Xcan be covered by open affine T-subsets. We show that A.(XIIT) 0 Q = AT(X) 0 Q. Iff : X -4 X//T is the quotient morphism, we show that if E is a bundle on X//T, then f*E isan r-admissible T-vector bundle. We show that in some cases, multiplicity is an invariant ofA.(XIIT).We start with a result of Vistoli [191.Proposition 8.1.1 Let f : X -+ X//T =Y be the quotient variety, Y' be the normalization ofY and let X' be the normalization of X . There exists a normal variety Y" such that if X" isthe normalization of Y" x y X, thenxn ---1--4 X' —2.-- Xf'l^i f^(8.1)______> yt _______+ yP 1^P94Chapter 8. Applications and Relations^ 95commutes. Furthermore, there are finite groups F' and F acting on Y" such that Y" F' =and r F = Y .Proof: The proof is in Vistoli [19], Lemma 4. In fact, the result he proves is more generalsince it concerns reductive algebraic groups.^ ^Definition: Let G be a group acting on X' and let p : X'^X = X'//G be the quotient.Let W be a subvariety of X and let ew be the order of the inertia group of a general point ofp-1 (W). We define p* : ZIT (X) -4 ZIT (X') by,= elvielf[P-1 (W)].^ (8.2)Lemma 8.1.2 Suppose G is finite. Then the set of cycles T -rationally equivalent to 0 inZIT (X') 0 Q are generated by cycles of the formE[div(g-1 • r)]^ (8.3)gEGwhere V is a k + 1-dimensional T-subvariety of X' and r E R(V)* is a weight vector of weight0. In particular, AT (X') Q = 4(x) Q.Proof: Suppose that a is G stable and T-rationally equivalent to 0. There exists a collectionof k+ 1-dimensional T-subvarieties VI, , Vn of X' and T-weight vectors ft E R(Vi) * of weight0 such that a = EriL i divfi. Averaging over G we have,n(E g) • a = E E [div(g -1 • fi)] = IGI a.^ (8.4)gEG^1=1 gEGSo, after tensoring with Q, cycles of this form generate the set of G stable cycles that areT-rationally equivalent to 0 in ZT(X') 0 Q.^ ^Theorem 8.1.3 1. f* passes to T -rational equivalence2. f* : Ak(X IIT) Q 4+r (X) 0 Q is an isomorphism.Chapter 8. Applications and Relations^ 96Proof: Vistoli shows [19] Lemmas 2 and 3, that rq'*, f"*p'* : Zk(110 Q -4 4 (X") Q aredefined and are equal. We know that fll* : Ak(Y")0Q A4(X") 0)Q from Proposition 4.8.2.Since IP and q'* are also isomorphisms on the AT  0 Q, f* passes to T-rational equivalenceand is an isomorphism as well.Vistoli [19] also shows in the proof of Theorem 1, that f*p., q.1* : Zk(r)are defined and are equal. As we have just seen, f* passes to T-rational equivalence and is anisomorphism after tensoring with Q. Since p* and q* are also isomorphisms after tensoring withQ, f* : Zk(Y) 0 Q 4+, (X) Q passes to T-rational equivalence and is an isomorphism onthe Chow groups..^ ^The theorem allows us to calculate the Chow groups for quotients of affine spaces fairlyeasily.Corollary 8.1.4 Let V be a T-representation space, and let V" be the open set of V overwhich the T-action is free. Then,f* : Ak(V"IIT)0 Q 4-1-r(V ss) Q.^ (8.5)Note that this is essentially the same result that Ellingsrud and Stromme [5] get. They alsoshow that if V"//T is non-singular then AT (V") is a free group andf* : Ak(liss/a) 4+,(v") 0 Q.^ (8.6)Example: The morphism f* is not necessarily surjective (even if defined) if we do not tensorwith Q.Let X = A2 with the Gm -action with weights A x = 1 and Ay = 2. X//T L-f P l , butAr(X) = Z ED Z/(2, —1). f* : X//T -4 X sends the generator of Ao(X//T) to (0, 1) E AT (X).Since (0, 1) ti (2, 0), (1, 0) has no pre-image and f* is not surjective.Chapter 8. Applications and Relations^ 97Example: The map f. : ZT(X) -- Z(X//T) induced by f.[V] = [f (V)] does not induce a mapon the equivariant Chow groups.We use the previous example. Let P and Q be the points of X//T defined by the semi-invariant ideals (x) and (y). Since x 2/y is T-rationally equivalent to 0, if f. : AT (X) -+Ak_ r (XIIT) were a morphism, then f.(div(x 2/y)) = 2[P] - [Q] would be rationally equivalentto 0. However, since X IIT 'z'..131 , this is not the case. So, f* : 4 (X) -+ Ak_r (XIIT) is not amorphism.We relate the vector bundles on X//T to the T-vector bundles on X. We start with atheorem whose proof is due to F. Knop [11].Theorem 8.1.5 If the action of T on the complex variety X is free, then every T-vector bundleis isomorphic (not necessarily T -isomorphic) to the pull back of a vector bundle on X IIT . Ifthe weights of the T-vector bundle are all 0, then the isomorphism is a T-isomorphism.Proof: The proof of the first statement is in Kraft [11]. All we have to note is that the pullback bundles really are T-bundles. Let p: X -- X//T. If E is a vector bundle over X//T, thenlocally over some open U C X//T, E is trivial. p-1U is an open T-subset of X over which p*Eis trivial. For the second statement, note that E has the trivial T-action. So, the weights of Eover U are all 0, and those of p*E over p -1(U) are also 0. ^We consider equivariant multiplicities and quotient varieties.Let Y be a smooth T-variety such that for x E YT there exists an open T-subset U containingx that is T-isomorphic to a T-representation space. Let X be a T-subvariety of Y. Let A be thefree group generated by the multT(x, [V],Y) for all k-dimensional T-subvarieties V of X. LetB be the free group generated by all the multT(x, [V], Y) for all k-dimensional T-subvarietiesof X contained in X - X". Then multT(x, a, Y) is an invariant of a E Ak, (X") in A/Bwhere we identify multT(x, a, Y) with its class in A/B. This follows from the exact sequenceA IT(X - X") -- AT ^-> 4(X") --4 0.^ (8.7)Chapter 8. Applications and Relations^ 98Remark: We would like to understand the relationship between A(X//T) and AT (X) better.We would like to know if in Theorem 8.1.3 the morphism is an injection without tensoring withQ. In many examples, we do find that it is an injection.As far as intersections go, if E is a vector bundle on X//T, we would like to know howsi(E) fl — and ci(E) fl — behave when pulled back to X.8.2 KT (X)Since AT (X) involves T-cycles, we would expect some relationship between 400 and KT (X).This is in fact the case, but the relation is not quite the one we would expect.First of all, the obvious morphism cp : ZT(X) -4 KT (X) does exist and is an injection. IfM is a T, Ox module, then M has a composition series with quotients locally T-isomorphic toOx/Pi vi. Let Vi be the variety associated to R/ Pi. Then E i (-1)ZR/Pi is in the image of so,but E i (-1)jR/Pi • vi need not be. So, yo is not an isomorphism.Repeating S.G.A. VI, we do get a map that respects T-rational equivalence. So,A PX) —* KT (X)^mod KT (X ) k+ 1^(8.8)exists. On resolving, we get.14.71:(X) Q -+ KT(X) Q.^ (8.9)Remark: As in usual K-theory, we would expect the map to have an inverse at least onIm(itT(X) Q). The proposed inverse would be the Chern class. However, since we haveadmissibility problems, the Chern class does not necessarily exist, and the inverse need notexist either.We also have other problems. Since AT (X) does not necessarily have a product structure,the localization result we would expect from Nielsen's [16] result does not hold.Chapter 8. Applications and Relations^ 998.3 4(X)We consider a definition of 4(X) for G a reductive algebraic group and X a G-scheme. If Gis a reductive group with maximal torus T, we can define a G/T-action on 4(X) as follows:let g E G be a representative for g' E G/T. For [V] E Zr (X) we set g' • V = g • V. Since V isa T-variety, this action is independent of the choice of representative. This action extends toone onDefinition: If G is reductive with maximal torus T we say a E Zr(X )G/T = Zk(X) G isG-rationally equivalent to 0 if there is a collection pairs {(171, f 1) , • • • , f.)} where Vi is aT-variety, f E MVO* is a weight vector of weight 0 and the collection is G-stable. By this wemean that the pair (g • Vi,g- 1 • fi) is in the collection for every g E G and every 1 < i < n.Using Lemma 2 of Vistoli [19], if G/T is finite, we have4(X) 0 Q = AT  Q = Ak,(XIIT)G/T 0 Q = Ak,(X//G) 0 Q. (8.10)where r = dim T, and provided X//G exists.For this definition of the equivariant Chow groups most of the properties Fulton considersin chapter 1 hold. For the alternate definition of G-rational equivalence we require that thecollection of pairs {(Vi, fill be G-stable where V is a T-subvariety of X xPl where Pl has thetrivial G-action and f : V -+ P l is the T-morphism induced by the projection onto P 1 andis dominant As above, by G-stable we mean that the pair (g • o g- 1 ) is in the collectionfor every g E G and for every i. We also have the exact sequence of section 4.6, provided werequire that the subscheme X be G stable. We also have flat pullback. What we do not have,though, is the moving result of section 4.4, nor the change of groups results of section 4.7, northe affine bundle results of section 4.8. The reason for this is that the part involving G/T isnot always satisfied.This should give a fairly good idea of why we did not consider G-rational equivalence. Thedefinition is too hard to work with, and not all the properties we want necessarily exist.Chapter 8. Applications and Relations^ 100We consider another possible definition.Example: As with AT(X), we could demand that the subvarieties generating 4(X) be Ginvariant. This however, is too restrictive. Consider X = A2 with the G = Z/2Z-action givenby interchanging X1 and X2. For this definition of equivariant Chow groups, let B be the freegroup on the elements of the field k. Then 4(X) = B/2B and is generated by the cycles ofthe form (Xi — a, X2 - a) E A2 , where a E k.Bibliography[1] W. Borho, J.-L. Brylinski and R. Macpherson. Nilpotent Orbits, Primitive Ideals, andCharacteristic Classes, volume 78 of Progress in Mathematics. Birkhauser, 1989.[2] M. Brion. Groupes de Picard et nombres caracteristiques des varietes spheriques. DukeMath. J., 58, 1989, 397-424.[3] M. Brion. Variete spheriques et theorie de Mori. pre-print, 1992.[4] V. Danilov. The geometry of tonic varieties. Russ. Math. Surveys, 33, 1978, 97-154.[5] G. Ellingsrud and S. Stromme On the Chow ring of a geometric quotient. Ann. Math.,130, 1989, 159-187.[6] W. Fulton. Intersection Theory. Springer, 1984.[7] R. Godement. Cours d'Algebre. Hermann, 1963.[8] A. Grothendieck, J. Dieudonne. Elements de la Geometrie Algebrique I et II. IHES, 4, 8,1961.[9] B. Iversen and H. Nielsen. Chern numbers and diagonalizable groups. J. London Math.Soc., 11, 1975.[10] J. Jurkiewicz. Chow ring of projective non-singular torus embedding. Colloq. Math., 43,261-270.[11] H. Kraft. G-vector bundles and the linearization problem. In Group Actions and InvariantTheory, pages 111-124, 1989.[12] S. Lang. Algebra. Addison-Wesley, second edition, 1984.[13] D. Mumford. Geometric Invariant Theory. Springer, 1965.[14] D. Mumford. Algebraic Geometry 1: Complex Projective Varieties. Springer, 1976.[15] H. Matsumura. Commutative Algebra. Benjamin-Cummings, second edition, 1980.[16] H. Nielsen. Diagonalizably linearized coherent sheaves. Bull. Soc. Math, France, 102, 1974,85-97.[17] W. Rossmann. Equivariant multiplicities on complex varieties. Asterisque, 173-174, 1989,313-330.[18] P. Samuel. Methodes d'Algebre Abstraite en Geometrie Algebrique. Springer, 1955.[19] A. Vistoli. Chow groups of quotient varieties. J. Alg., 107, 1987, 410-424.101


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