Algebraic cobordism of bundles on varieties

ALGEBRAI C COBORDISM OF BUNDLES ON V ARIETI E S Y.-P . LEE AND R. P ANDHARIP AN DE Abstract. The double p oint relation defines a natur a l theory of algebraic cob ordism for bundles on v ar ieties. W e construct a simple basis (ov er Q ) of the cor resp onding cob ordis m groups ov er Sp ec( C ) for all dimensio ns of v a r ieties and ranks of bundles. The basis con- sists of split bundles ov er pro ducts of pro jective s paces. Mor eov er, we prov e the full theory for bundles on v a rieties is an ex tens io n of scalars of standard algebr a ic co b or dism. Contents In tro duction 1 1. Chern c lasses 8 2. Lists of line bundles 12 3. Higher rank 19 4. Results o ve r X 21 References 24 Intr oduction 0.1. Algebraic cobordism. A success ful theory of algebraic cob or- dism ha s b een construc ted in [6] from Quillen’s axiomat ic p ers p ectiv e. The result Ω ∗ is the univ ersal orien ted Borel-Mo o r e homolog y theory of sc he mes, yielding the univ ersal oriented Bor el- Mo ore cohomology theory Ω ∗ for the sub catego ry of smo oth sche mes. Let k b e a field o f c haracteristic 0. Let Sch k b e the catego r y of sepa- rated sc hemes of finite ty p e o v e r k , a nd let Sm k b e the full sub category of smo oth quasi-pro jectiv e k -sc hemes. A geometric presen ta tion of a l- gebraic cob ordism in c haracteristic 0 via double p oint relations is given in [7]. Date : F ebruary 20 10. 1 2 Y.-P . LEE AND R. P ANDHA RIP AND E 0.2. Double p oin t degenerations. Let Y ∈ Sm k b e of pure dimen- sion. A morphism π : Y → P 1 is a double p oint de g ener ation ov er 0 ∈ P 1 if π − 1 (0) can b e written as π − 1 (0) = A ∪ B where A and B are smoo th co dimension one closed subsc heme s of Y , in tersecting transv ersely . The intersec tion D = A ∩ B is the double p oint lo cus of π o v er 0 ∈ P 1 . W e do not require A , B , or D to b e connected. Moreo v e r, A , B , and D are allo w ed to b e empt y . Let N A/D and N B /D denote the normal bundles of D in A and B resp ectiv e ly . Since O D ( A + B ) is trivial, N A/D ⊗ N B /D ∼ = O D . Since O D ⊕ N A/D ∼ = N A/D ⊗ ( O D ⊕ N B /D ), the pro jectiv e bundles (0.1) P ( O D ⊕ N A/D ) → D and P ( O D ⊕ N B /D ) → D are isomorphic. Let P ( π ) → D denote either of (0.1). 0.3. M ( X ) + . F or X ∈ Sc h k , let M ( X ) denote the set of isomorphism classes o v er X of pro jectiv e mor phisms (0.2) f : Y → X with Y ∈ Sm k . The set M ( X ) is a monoid under disjoint union of domains and is graded b y the dimension of Y o v er k . Let M ∗ ( X ) + denote the g raded group completion of M ( X ). Alternativ ely , M n ( X ) + is the free ab elian group generated b y mor- phisms (0.2) where Y is irr educible and of dimension n o v er k . Let [ f : Y → X ] ∈ M ∗ ( X ) + denote the elemen t determined by the morphism. ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 3 0.4. Double p oin t relations. Let X ∈ Sch k , and let p 1 and p 2 denote the pro jections to the first and second factors of X × P 1 resp ectiv e ly . Let Y ∈ Sm k b e of pure dimension. Let g : Y → X × P 1 b e a pro jectiv e morphism for whic h t he comp osition (0.3) π = p 2 ◦ g : Y → P 1 is a double p oin t degeneration o v er 0 ∈ P 1 . Let [ A → X ] , [ B → X ] , [ P ( π ) → X ] ∈ M ( X ) + b e obtained fr o m the fib er π − 1 (0) and the morphism p 1 ◦ g . Definition 1. Let ζ ∈ P 1 ( k ) b e a regular v alue o f π . W e call the map g a double p oint c ob or dism with degenerate fib er o v er 0 and smo oth fib er ov er ζ . The asso ciated d o uble p oint r elation ov er X is (0.4) [ Y ζ → X ] − [ A → X ] − [ B → X ] + [ P ( π ) → X ] where Y ζ = π − 1 ( ζ ). The relation (0.4) dep ends not o nly on the mo r phism g and the p oint ζ , but also on t he c hoice of decomp osition o f the fib er π − 1 (0) = A ∪ B . W e view (0.4) as an analog of the classical relation of r ational equiv a - lence of a lgebraic cycles. Let R ∗ ( X ) ⊂ M ∗ ( X ) + b e the subgroup generated b y al l double p oin t relatio ns o v er X . Since (0.4) is a homogeneous elemen t o f M ∗ ( X ) + , R ∗ ( X ) is a graded subgroup of M ∗ ( X ) + . Definition 2. F or X ∈ Sc h k , do uble p oint c ob or dism ω ∗ ( X ) is defined b y the quotien t (0.5) ω ∗ ( X ) = M ∗ ( X ) + / R ∗ ( X ) . A cen tral result of [7] is the isomorphism (0.6) Ω ∗ ∼ = ω ∗ whic h provid es a geometric presen tation of algebraic cob ordism. Since resolution of singularit ies and Bertini’s results are used, t he isomor- phism is established only when k has c haracteristic 0 . 4 Y.-P . LEE AND R. P ANDHA RIP AND E 0.5. Over a p oin t. W e write Ω ∗ ( k ) a nd ω ∗ ( k ) for Ω ∗ (Sp ec ( k ) ) and ω ∗ (Sp ec ( k )) resp ectiv e ly . Let L ∗ b e t he Lazard ring [4 ]. The canonical map L ∗ → Ω ∗ ( k ) classifying the formal gr oup law fo r Ω ∗ is prov en t o b e an isomorphism in [6, Theorem 4 .3.7]. By Quillen’s result for complex cob or dism (in top ology), L n ∼ = M U − 2 n (pt) , and the we ll-kno wn generators of M U ∗ (pt) Q [11, Chapter VI I], w e see Ω ∗ ( k ) ⊗ Z Q is generated as a Q -algebra b y the classes of pro jectiv e spaces. The f o llo wing result is then a conseque nce of (0.6), (0.7) ω ∗ ( k ) ⊗ Z Q = M λ Q [ P λ 1 × ... × P λ ℓ ( λ ) ] , where the sum is ov er all partitions λ . The partition λ = ∅ corresp onds to [ P 0 ] in grade 0. 0.6. Bundles. F or X ∈ Sc h k , let M n,r ( X ) denote t he set of isomor- phism classes ov e r X of pairs [ f : Y → X , E ] with Y ∈ Sm k of dimension n , f pro jectiv e, and E a rank r v ec tor bundle on Y . The set M n,r ( X ) is a monoid under disjoint union of domains. Let M n,r ( X ) + denote the g roup completion of M n,r ( X ). Double p oin t relations are easily defined in the setting of pairs fo l- lo wing [7, Section 13]. Let Y ∈ Sm k b e of pure dimension n + 1. Let g : Y → X × P 1 b e a pro jectiv e morphism for whic h t he comp osition π = p 2 ◦ g : Y → P 1 is a double p oint degeneration o v er 0 ∈ P 1 . Let E b e a rank r v ector bundle on Y . Let [ A → X , E A ] , [ B → X , E B ] , [ P ( π ) → X , E P ( π ) ] ∈ M n,r ( X ) + b e obtained from the fib er π − 1 (0) a nd the morphism p 1 ◦ g . Here, E A and E B denote the restrictions of E to A and B r esp ectiv ely . The restriction E P ( π ) is defined b y pull-back from Y via P ( π ) → D ⊂ Y . ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 5 Definition 3. Let ζ ∈ P 1 ( k ) b e a regular v alue of π . The associated double p o i n t r ela tion o v er X is [ Y ζ → X , E Y ζ ] − [ A → X , E A ] − [ B → X , E B ] + [ P ( π ) → X , E P ( π ) ] where Y ζ = π − 1 ( ζ ). F or X ∈ Sc h k , let R n,r ( X ) ⊂ M n,r ( X ) + b e the subgroup gener- ated b y a ll double p o in t relations. Double p o int cob o r dism theory fo r bundles on v arieties is defined b y ω n,r ( X ) = M n,r ( X ) + / R n,r ( X ) . The sum ω ∗ ,r ( X ) = ∞ M n =0 ω n,r ( X ) is alw a y s a ω ∗ ( k )-mo dule via pro duct (and pull-bac k). If X ∈ Sm k , then ω ∗ ,r ( X ) is also a mo dule o v er t he ring ω ∗ ( X ). 0.7. Basis. The main result o f t he pap er is the construction of a basis of ω n,r ( k ) a nalogous to the fundamen tal presen tation (0.7). Our basis is indexed by pairs of partitions. A p artition p air of size n and t yp e r is a pair ( λ, µ ) where (i) λ is a par t it ion of n , (ii) µ is a sub-par t ition of λ of length ℓ ( µ ) ≤ r . The sub-partition condition means µ is obtained by deleting parts o f λ . The partition µ ma y b e empty and ma y equal λ if ℓ ( λ ) ≤ r . Sub- partitions µ, µ ′ ⊂ λ a r e equiv alent if they differ b y p erm uting equal parts of λ . Let P n,r b e the set of all partition pairs of size n and type r . F or example, P 3 , 2 =    (3 , ∅ ) , ( 3 , 3) , (21 , ∅ ) , (21 , 2 ) , (21 , 1 ) , (21 , 2 1 ) , (111 , ∅ ) , (111 , 1) , (111 , 1 1)    . T o eac h ( λ, µ ) ∈ P n,r , w e asso ciate an elemen t φ ( λ, µ ) ∈ ω n,r ( k ) b y the following construction. Let P λ = P λ 1 × . . . × P λ ℓ ( λ ) . T o eac h par t m of µ , let L m → P λ b e the line bundle obtained b y pulling- bac k O P m (1) via the pro jection to the fa ctor P λ → P m 6 Y.-P . LEE AND R. P ANDHA RIP AND E corresp onding to the part m . Since µ ⊂ λ , m is part of λ . W e define φ ( λ, µ ) = [ P λ , O r − ℓ ( µ ) ⊕ M m ∈ µ L m ] . The bundle o n P λ has a trivial factor o f rank r − ℓ ( µ ). Theorem 1. F or n, r ≥ 0 , we have ω n,r ( k ) ⊗ Z Q = M ( λ,µ ) ∈P n,r Q · φ ( λ, µ ) . In other words, the elemen ts φ ( λ, µ ) determine a basis of ω n,r ( k ) ⊗ Z Q . If r = 0, Theorem 1 sp ecializes to (0.7). In case ( n, r ) = (3 , 2), the basis of Theorem 1 is giv en b y [ P 3 , O 2 ] , [ P 3 , O ⊕ O (1)] , [ P 2 × P 1 , O 2 ] , [ P 2 × P 1 , O ⊕ O (1 , 0)] , [ P 2 × P 1 , O ⊕ O (0 , 1)] , [ P 2 × P 1 , O (1 , 0) ⊕ O (0 , 1 )] , [ P 1 × P 1 × P 1 , O 2 ] , [ P 1 × P 1 × P 1 , O ⊕ O (1 , 0 , 0)] , [ P 1 × P 1 × P 1 , O (1 , 0 , 0) ⊕ O (0 , 1 , 0 )] . Theorem 1 is pro v en in Section 3. The a rgumen t requires study- ing an alg ebraic cob o rdism theory for lists o f line bundles on v a rieties dev eloped in Section 2. The structure of ω ∗ ,r ( k ) ov e r Z is determined by the following result pro v en in Section 3.3. Theorem 2. F or r ≥ 0 , ω ∗ ,r ( k ) i s a fr e e ω ∗ ( k ) -mo dule with b asis ω ∗ ,r ( k ) = M λ ω ∗ ( k ) · φ ( λ, λ ) wher e the sum is over al l p artitions λ o f length at m ost r . 0.8. Over X . In fact, ω ∗ ,r ( k ) determines ω ∗ ,r ( X ) for all X ∈ Sc h k . There is a natural map γ X : ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) → ω ∗ ,r ( X ) of ω ∗ ( k )-mo dules defined by γ X  [ Y f → X ] ⊗ φ ( λ, λ )  = [ Y × P λ f ◦ p Y − → X , O r − ℓ ( λ ) ⊕ M m ∈ λ p ∗ P λ ( L m ) ] . ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 7 Here, λ has length at most r , and p Y and p P λ are the pro jections of Y × P λ to Y and P λ resp ectiv e ly . Theorem 3. F or r ≥ 0 and X ∈ Sch k , the map γ X is an isomorphism of ω ∗ ( k ) -mo dules. By Theorem 3, the algebraic cob ordism theory ω ∗ ,r of bundles on v arieties is simply an extension of scalars of the original theory ω ∗ . 0.9. Chern inv arian ts. let Y be a nonsingular pro jectiv e v ariet y of dimension n , and let E b e a rank r v ector bundle on Y . The Chern in v arian ts of the pair [ Y , E ] are Z Y Θ  c 1 ( T Y ) , . . . , c n ( T Y ) , c 1 ( E ) , . . . , c r ( E )  where Θ is an y gra ded degree n p olynomial (with Q -co efficien ts) of the Chern classes o f the tangent bundle T Y and E . Let C n,r b e the finite dimensional Q -v ector space o f graded degree n p olynomials in the Chern classes. Theorem 4. The C hern invariants r esp e ct algebr aic c ob or dism. The r esulting map ω n,r ( k ) ⊗ Z Q → C ∗ n,r is an isom orphism. A simple counting argumen t (give n in Section 1) show s the dimension of C n,r equals the cardinalit y of P n,r . In case ( n, r ) = ( 3 , 2), there are 9 basic Chern in v ariants of [ Y , E ], c 3 ( T Y ) , c 2 ( T Y ) c 1 ( T Y ) , c 1 ( T Y ) 3 , c 2 ( T Y ) c 1 ( E ) , c 1 ( T Y ) 2 c 1 ( E ) , c 1 ( T Y ) , c 2 ( E ) , c 1 ( T Y ) c 1 ( E ) 2 , c 2 ( E ) c 1 ( E ) , c 1 ( E ) 3 . Theorem 4 is pro v en jointly with Theorem 1 in Section 3. 0.10. A pplications. F or studying a theory asso ciated to pair s [ Y , E ] whic h admits a m ultiplicativ e double p oin t degeneration formu la, al- gebraic cob ordism ω n,r ( C ) is a useful to ol. The full theory can b e calculated from the toric basis elemen ts sp ecified b y Theorem 1 . The determinations of ω 3 ( C ) and ω 2 , 1 ( C ) ha v e b een used in [7] to pro v e the conjectures of [1 , 9, 10] gov e rning the degrees of virtual classes on the Hilb ert sche mes of p oints of 3-fo lds. Rece n tly , Y. Tzeng [13] has used the 4-dimensional ba sis of ω 2 , 1 ( C ) in a b eautiful pro of of G¨ ottsc he’s conjecture [3] gov erning no dal curv e coun ting (in terpreted as degrees o f cycles in the Hilb ert sc hemes of p oin ts of surfaces). The basis of ω n,r ( C ) will b e used in [5] for the study o f flop inv ariance o f quan tum cohomology . 8 Y.-P . LEE AND R. P ANDHA RIP AND E 0.11. Sp eculations. Consider the algebraic group GL r o v er C . W e view ω ∗ ,r ( C ) as an algebraic mo del for M U ∗ ( B GL r ). Theorem 4 may b e in terpreted as sa ying ω ∗ ,r ( C ) is dual to M U ∗ ( B GL r ) = M U ∗ (pt)[[ c 1 , . . . , c r ]] . D. Maulik suggests defining an algebraic cob ordism theory ω ∗ , G for principal G -bundles on algebraic v arieties b y the double p oint relation of D efinition 3. P erhaps the resulting theory ov e r a p oint for classical groups G is dual to M U ∗ ( B G )? An algebraic appro a c h to M U ∗ ( B G ) for linear algebraic groups has b een prop osed in [2] by limits of Ω ∗ o v er algebraic appro ximations to B G . The construction is similar to T o t aro’s definition [12] o f the Cho w ring of B G , but requires also the coniv eau filtra t ion (see [8] for an alternative limit definition). F o r many examples, including B GL r , the isomorphism Ω ∗ ( B G ) ∼ = M U ∗ ( B G ) is obtained [2]. Suc h isomorphisms we re predicted in [14]. Another a ppro ac h to our pap er is p erhaps p ossible via a limit defi- nition of Ω ∗ ( B GL r ). There should b e a map Ω ∗ ( B GL r ) → ω ∗ ,r ( C ) whic h is injectiv e by Chern in v ariants and surjectiv e by Prop osition 11. 0.12. A c kno wled gmen ts. W e thank D . Abra movic h, J. Li, D. Maulik, B. T otaro, and Y. Tzeng fo r discussions ab out algebraic cob ordism and double p o in t degenerations. The basis of Theorem 1 w as guessed while writing [7]. Conv ersations with M. Levine play ed an essen tial role. He suggested the p ossibilit y of the extension of scalars result established in Theorem 3. Y.-P . L. w as supp orted b y NSF grant DMS-090109 8. R. P . w as supp orted by NSF grant DMS-05 00187. 1. Chern classes 1.1. Cob or dism in v ariance. Let n, r ≥ 0. There is canonical bilin- ear map ρ : M n,r ( k ) + ⊗ Z Q × C n,r → Q defined by integration, ρ ([ Y , E ] , Θ) = Z Y Θ  c 1 ( T Y ) , . . . , c n ( T Y ) , c 1 ( E ) , . . . , c r ( E )  . Prop osition 5. The p airing ρ annihilates R n,r ( k ) . ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 9 Pr o of. In case r = 0, the in v ariance of the Chern n um b ers of the t a n- gen t bundle is a well-kno wn prop ert y of algebraic cob ordism ov er a Sp ec ( C ), see [6, 11]. Let Y ∈ Sm k b e of pure dimension n + 1. Let (1.1) π : Y → P 1 b e a pro jectiv e morphism whic h is a double p oint degeneration ov er 0 ∈ P 1 . Let L b e a line bundle on Y . Supp ose L is ve ry ample o n Y . Cutting Y with s g eneric sections of L yields an nonsingular subv ariet y of co dimension s , D s ⊂ Y π → P 1 . The comp osition D s → P 1 is a double p oin t degeneration o v er 0 ∈ P 1 . Let Y ζ , A , B , and P ( π ) b e the four spaces whic h o ccur in the double p oin t relation f or (1.1) in Definition 3. Let Y ζ ∩ D s , A ∩ D s , B ∩ D s , P ( π ) ∩ D s b e the f our spaces whic h o ccur in the r elation for D s → P 1 . Since the tangen t bundle of Z ∩ D s satisfies 0 → T Z ∩ D s → T Z | Z ∩ D s → s M i =1 L → 0 in eac h of the four cases, w e hav e c ( T Z ∩ D s ) = c ( T Z ) (1 + c 1 ( L )) s , c i ( T Z ∩ D s ) = c i ( T Z ) − s · c i − 1 ( T Z ) c 1 ( L ) + · · · , where w e ha v e suppressed the restrictions. The a pplication of the r = 0 case of the Prop osition to the degenerations D s → P 1 for a ll s implies (b y descending induction) the r = 1 case for double p oin t relations where L is ample. Similarly if L 1 , . . . , L m are v ery ample line bundles on Y , w e can consider D s 1 ,...,s m ⊂ Y π → P 1 obtained b y cutting with s 1 sections of L 1 , s 2 sections of L 2 , . . . , and s m sections of L m . The application of the r = 0 case of the Prop o sition to the degeneration D s 1 ,...,s 2 → P 1 for all s 1 , . . . , s m implies inv ari- ance under the double p oint relation o f graded degree n p olynomials in the Chern classes of the tangen t bundle a nd the Chern classes of L 1 , . . . , L m . The r = 1 case of the Prop osition follo ws since ev ery line bundle L ma y b e written as the difference of tw o very ample line bundles. 10 Y.-P . LEE AND R. P ANDHA RIP AND E T o prov e t he r > 1 case of the Prop osition, we use a splitting argu- men t. Let π b e a double p o in t degeneration as ab o v e (1 .1). Let E b e a rank r bundle on Y . Let F ( E ) → Y π → P 1 b y the complete flag v ar iety ov er Y obtained from E . The comp o si- tion F ( E ) → P 1 is a double p oint degeneration with tautological line bundles L 1 , . . . , L r whic h sum in K -theory to the pull-back of E . The established line bundle results then yield the r > 1 case.  As a consequenc e of Prop o sition 5, the pairing ρ descends, (1.2) ρ : ω n,r ( k ) ⊗ Z Q × C n,r → Q . Our first go al is to b ound the rank of the pairing from b elow. 1.2. Indep endence. 1.2.1. Mono m ials of C n,r . F or notatio na l conv enience, w e write ele- men ts Θ ∈ C n,r as p olynomials Θ( u 1 , . . . , u n , v 1 , . . . , v r ) where u i = c i ( T Y ) a nd v i = c i ( E ). Both u i and v i ha v e degree i . A canonical basis o f C n,r is obtained b y monomials of graded degree n . Let Q n,r b e the set of partit io n pairs ( ν, µ ) where (i) µ is a pa r tition of size | µ | ≤ n with larg est part at most r , (ii) ν is a partition of n − | µ | . The corresp ondence (1.3) n Y i =1 u l i i r Y j = 1 v m j j ↔ (1 l 1 · · · n l n , 1 m 1 · · · r m r ) yields a bijection b et w e en the monomial basis of C n,r and the set Q n,r . Let C ( ν, µ ) denote the monomial asso ciated to ( ν , µ ) ∈ Q n,r . Lemma 6. Ther e is a natur al b i j e ction ǫ : Q n,r → P n,r . Pr o of. Giv en ( ν, µ ) ∈ Q n,r , define ǫ ( ν, µ ) = ( ν ∪ µ t , µ t ) ∈ P n,r . Here, µ t is the pa rtition obtained b y tr a nsp o sing the Y oung diagram asso ciated to µ . Hence, µ t has length at most r .  ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 11 1.2.2. O r dering. The v -degree o f a monomial in C n,r is the vec tor deg v n Y i =1 u l i i r Y j = 1 v m j j ! = ( m 1 , m 2 , . . . , m r ) ∈ Z r ≥ 0 . W e define a to t al ordering on Z r ≥ 0 b y the f ollo wing rule: ( m 1 , . . . , m r ) > ( m ′ 1 , . . . , m ′ r ) if either m r > m ′ r or if m j = m ′ j for all j > i and m i > m ′ i . T he resulting partial order on the monomials on C n,r (indexed b y Q n,r ) is sensitiv e only the v ariables v i . 1.2.3. B iline ar p airing. Let M b e the matrix with row s and columns indexed b y Q n,r and elemen ts M n,r [( ν, µ ) , ( ν ′ , µ ′ )] = ρ  φ  ǫ ( ν, µ )  , C ( ν ′ , µ ′ )  for ( ν, µ ) , ( ν ′ , µ ′ ) ∈ Q n,r . Recall, t he map φ : P n,r → ω n,r ( k ) w as defined in Section 0.7. The ro ws and columns of M n,r are ordered b y the partial ordering on Q n,r defined in Section 1.2.2. Lemma 7. If ( ν, µ ) < ( ν ′ , µ ′ ) in the p artial or der of Q n,r , then M n,r [( ν, µ ) , ( ν ′ , µ ′ )] = 0 . Pr o of. Let µ = 1 m 1 · · · r m r and µ ′ = 1 m ′ 1 · · · r m ′ r . If ( ν , µ ) < ( ν ′ , µ ′ ), then, in the highest index i where a difference o ccurs, m i < m ′ i . Supp ose the difference o ccurs in the index i = r . Then, m r is the minimal part of µ t . F or the pair [ Y , E ] = φ ( ν, µ ), the bundle E is a direct sum of r line bundles pulled-back f r om the O (1) factors of a pro duct of r pro jective spaces (with minimal dimension m r ). Since m r < m ′ r , the class c m ′ r r ( E ) v anis hes on Y b y dimension considerations. If the hig hest difference o ccurs in an index i < r , the argument is the same (following a gain from elemen tary dimension considerations).  Prop osition 8. M n,r is a nonsingular ma trix. Pr o of. By Lemma 7, the mat r ix M n,r is blo c k lo w er triangular with resp ect to the partial ordering o n Q n,r . The blo c ks are determined b y all ( ν, µ ) ∈ Q n,r with the same µ . Let µ = 1 m 1 . . . r m r . Consider the bundle E = M m ∈ µ t L m − → P µ t , 12 Y.-P . LEE AND R. P ANDHA RIP AND E follo wing the notatio n of Section 0.7. Since (1.4) Z P µ t c 1 ( E ) m 1 c 2 ( E ) m 2 . . . c r ( E ) m r = 1 , the blo c k in M n,r corresp onding to µ is the mat r ix M n −| µ | , 0 . The latter is nonsingular by well-kno wn results a b out the usual r = 0 theory of algebraic cob ordism [6, 11].  As a consequenc e of Prop o sition 8, the generators prop osed in The- orem 1 span a subspace of ω n,r ( k ) ⊗ Z Q of rank at least |P n,r | . In particular, dim ( ω n,r ( k ) ⊗ Z Q ) ≥ |P n,r | . Moreo v e r, the pairing (1.2) has rank at least |P n,r | . T o complete the pro ofs of Theorem 1 and 4, w e will pro v e the reve rse inequalit y dim ( ω n,r ( k ) ⊗ Z Q ) ≤ |P n,r | in Section 3. 2. Lists of line bundles 2.1. Lists. F o r X ∈ Sc h k , let M n, 1 r ( X ) denote the set of isomorphism classes o v er X of tuples [ f : Y → X , L 1 , . . . , L r ] with Y ∈ Sm k of dimension n , f pro jectiv e, and L 1 , . . . , L r an ordered list of line bundles on Y . The set M n, 1 r ( X ) is a monoid under disjoin t union of domains. Let M n, 1 r ( X ) + denote the group completion of M n, 1 r ( X ). Let Y ∈ Sm k b e of pure dimension n + 1, and let g : Y → X × P 1 b e a pro jectiv e morphism for whic h t he comp osition π = p 2 ◦ g : Y → P 1 is a double p oint degeneration ov e r 0 ∈ P 1 . Let L 1 , . . . , L r b e a list of line bundles o n Y . Let [ A → X , L 1 ,A , . . . L r,A ] , [ B → X , L 1 ,B . . . L r,B ] , [ P ( π ) → X , L 1 , P ( π ) , . . . , L 1 , P ( π ) ] ∈ M n, 1 r ( X ) + b e obtained fr o m the fib er π − 1 (0) and the morphism p 1 ◦ g . ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 13 Definition 4. Let ζ ∈ P 1 ( k ) b e a regular v alue of π . The associated double p o i n t r ela tion o v er X is [ Y ζ → X , { L i,Y ζ } ] − [ A → X , { L i,A } ] − [ B → X , { L i,B } ] + [ P ( π ) → X , { L i, P ( π ) } ] where Y ζ = π − 1 ( ζ ). F or X ∈ Sch k , let R n, 1 r ( X ) ⊂ M n, 1 r ( X ) + b e the subgroup generated b y all double p oint relations. Double p oin t cob ordism theory for lists of line bundles on v arieties is defined by ω n, 1 r ( X ) = M n, 1 r ( X ) + / R n, 1 r ( X ) . The sum ω ∗ , 1 r ( X ) = ∞ M n =0 ω n, 1 r ( X ) is alw a ys a ω ∗ ( k )-mo dule via pro duct. If X ∈ Sm k , then ω ∗ , 1 r ( X ) is also a mo dule ov er the ring ω ∗ ( X ). 2.2. Basis. A p artition list of size n and t yp e r is a tuple ( λ, ( m 1 , . . . , m r )) where (i) λ is a par t it ion of n , (ii) ( m 1 , . . . , m r ) is a list with m i ≥ 0 whose union of non - z e r o parts is a sub-partition µ ⊂ λ . Let P n, 1 r b e t he set of all partition lists of size n and type r . F or example, P 3 , 1 2 =        (3 , (0 , 0)) , (3 , (3 , 0 )) , (3 , (0 , 3)) , (21 , (0 , 0)) , ( 2 1 , (2 , 0)) , (2 1 , (1 , 0)) , (21 , (0 , 1)) , (21 , (0 , 2)) , (21 , (2 , 1 )) , (21 , ( 1 , 2)) , (111 , (0 , 0 ) ) , (111 , (1 , 0)) , (111 , (0 , 1)) , (111 , (1 , 1))        . T o eac h ( λ, ( m 1 , . . . , m r )) ∈ P n, 1 r , w e asso ciate an elemen t φ ( λ, ( m 1 , . . . , m r )) ∈ ω n,r ( k ) b y the following construction. Let P λ = P λ 1 × . . . × P λ ℓ ( λ ) . T o eac h non-zero part m i , let L m i → P λ b e t he line bundle obta ined b y pulling-bac k O P m i (1) via the pro jection to the fa ctor P λ → P m i 14 Y.-P . LEE AND R. P ANDHA RIP AND E corresp onding to the pa rt m i . If m i = 0, let L m i b e the trivial line bundle on P λ . W e define φ ( λ, ( m 1 , . . . , m r )) = [ P λ , ( L m 1 , . . . , L m r ) ] . Theorem 9. F or n, r ≥ 0 , we have ω n, 1 r ⊗ Z Q = M ( λ, ( m 1 ,...,m r )) ∈P n, 1 r Q · φ ( λ, ( m 1 , . . . , m r )) . Theorem 9 will b e prov en in Section 2.6 with a mix of tec hniques from [6, 7] and new metho ds fo r studying algebraic cob o rdism relations for line bundles on v arieties. 2.3. Chern in v arian ts. Let C n, 1 r b e the Q -ve ctor space of g raded de- gree n p olynomials in the Chern classes c 1 ( T Y ) , . . . , c n ( T Y ) , c 1 ( L 1 ) , . . . , c 1 ( L r ) . There is canonical bilinear map ρ : M n, 1 r ( k ) + ⊗ Z Q × C n, 1 r → Q defined by integration, ρ ([ Y , E ] , Θ) = Z Y Θ  c 1 ( T Y ) , . . . , c n ( T Y ) , c 1 ( L 1 ) , . . . , c 1 ( L r )  . The pro o f of Prop osition 5 implies the pairing ρ annihilat es R n, 1 r ( k ). Hence, ρ descends, ρ : ω n, 1 r ( k ) ⊗ Z Q × C n, 1 r → Q . The monomial basis of C n, 1 r is easily seen to hav e the same cardinality as the set P n, 1 r . A straigh tforw ard extension of the metho ds of Section 1.2.3 implies the elemen ts o f { φ ( λ, ( m 1 , . . . , m r )) | ( λ, ( m 1 , . . . , m r )) ∈ P n, 1 r } ⊂ ω n, 1 r ⊗ Z Q span a subspace of dimension |P n, 1 r | . In particular, dim( ω n, 1 r ⊗ Z Q ) ≥ |P n, 1 r | . 2.4. Globally generated line bundles . Let m ⊂ ω ∗ ( k ) b e the ideal generated b y all elemen ts of p ositiv e dimension, 0 → m → ω ∗ ( k ) → Z → 0 . Since ω ∗ , 1 r ( k ) is a ω ∗ ( k )-mo dule, w e can define the g r a ded quotien t e ω ∗ , 1 r ( k ) = ω ∗ , 1 r ( k ) m · ω ∗ , 1 r ( k ) , e ω ∗ , 1 r ( k ) = ∞ M n =0 e ω n, 1 r ( k ) . ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 15 F or ( λ, ( m 1 , . . . , m r )) ∈ P n, 1 r , let e φ ( λ, ( m 1 , . . . , m r )) ∈ e ω n, 1 r ( k ) denote the class of φ ( λ , ( m 1 , . . . , m r )) in the quotient. Prop osition 10. L e t Y ∈ Sm k b e a pr oje ctive variety of dim e nsion n with line bund les L 1 , . . . , L r al l gen er ate d by glo b al s e ctions. Then, [ Y , L 1 , . . . , L r ] ∈ e ω n, 1 r ( k ) lies in the Z -line ar sp a n of ( e φ ( λ, ( m 1 , . . . , m r ))    ( λ, ( m 1 , . . . , m r )) ∈ P n, 1 r , r X i =1 m i = n ) in e ω n, 1 r ( k ) . Pr o of. Since L 1 , . . . , L r are all generated by global sections on Y , there exists a pro jectiv e morphism f : Y → P d 1 × · · · × P d r , L i = f ∗ ( O P d i (1)) . W e view f as determining an elemen t of algebraic cob ordism, [ f : Y → P d 1 × · · · × P d r ] ∈ ω n ( P d 1 × · · · × P d r ) . A fundamen tal result of [6, Theorem 1.2.1 9] is the isomorphism (2.1) A ∗ ( X ) ∼ = e ω ∗ ( X ) = ω ∗ ( X ) m · ω ∗ ( X ) , where A ∗ ( X ) is t he Cho w theory of X (with Z co efficien ts). The Cho w group A n ( P d 1 × · · · × P d r ) is generated by linear subv arieties ι m 1 ,...,m r : P m 1 × · · · × P m r ֒ → P d 1 × · · · × P d r where P r i =1 m i = n . W e conclude [ f ] is a Z -linear com bination of the elemen t s [ ι m 1 ,...,m r ] ∈ e ω n ( P d 1 × · · · × P d r ) . Relations in ω n ( P d 1 × · · · × P d r ) lift canonically to ω n, 1 r ( P d 1 × · · · × P d r ) b y pulling-bac k the list (2.2) O P d 1 (1) , . . . , O P d r (1) ev ery where. Since all double p oint relations in ω n ( P d 1 × · · · × P d r ) o ccur o v er P d 1 × · · · × P d r , the pull-bac k of the list (2.2) is w ell-defined a nd canonical. The pull-bac k of the list (2.2) via ι m 1 ,...,m r yields the elemen t φ ( λ, ( m 1 , . . . , m r )) ∈ ω n, 1 r ( k ) , 16 Y.-P . LEE AND R. P ANDHA RIP AND E where P r i =1 m i = n . Here, λ is obtained simply by removing the 0 parts m i . Hence, after pushing-forw ard fro m P d 1 × · · · × P d r to Sp ec ( k ), the argumen t is complete.  2.5. Pr o jective bundles. W e will need auxiliary results on pro jectiv e bundles to remov e the global g eneration h ypot hesis of Prop osition 1 0. Let Z ∈ Sm k b e a pro jectiv e v ariet y equipp ed with a list of line bundles L 1 , . . . , L r and a split rank 2 v ector bundle B = O Z ⊕ N . W e are in terested in the classes [ P ( B ) , L 1 , . . . , L r ] , [ P ( B ) , L 1 ( ± 1) , . . . , L r ( ± 1)] ∈ ω ∗ , 1 r ( k ) . Here, P ( B ) denotes the pr o jectivization b y sub-lines, and L i ( ± 1) stands for L i ⊗ O P ( B ) ( ± 1). Let s b e t he section Z → P ( B ) determined b y the f actor N ⊂ B . The divisor s is an elemen t of the linear series asso ciated to O P ( B ) (1). The degeneration to the norma l cone of s yields a double p oint relation in ω ∗ ( Z ). After pulling-ba c k the list L 1 , . . . , L r , we o btain a double p oin t relation in ω ∗ , 1 r ( Z ). Twisting t he list by the exceptional divisor of the degeneration yields the relation [ P ( B ) , L 1 , . . . , L r ] − [ P ( B ) , L 1 (1) , . . . , L r (1)] − [ P ( O Z ⊕ N ∗ ) , L 1 ( − 1) , . . . L r ( − 1)] +[ P ( B ) , L 1 ⊗ N ∗ , . . . , L r ⊗ N ∗ ] = 0 ∈ ω ∗ , 1 r ( Z ) . Since ( O Z ⊕ N ∗ ) ⊗ N ∼ = B , w e may r ewrite the ab ov e relatio n in the follo wing form: [ P ( B ) , L 1 , . . . , L r ] − [ P ( B ) , L 1 (1) , . . . , L r (1)] − [ P ( B ) , L 1 ⊗ N ∗ ( − 1) , . . . L r ⊗ N ∗ ( − 1)] +[ P ( B ) , L 1 ⊗ N ∗ , . . . , L r ⊗ N ∗ ] = 0 ∈ ω ∗ , 1 r ( Z ) . After replacing L i with L i ⊗ N eve rywhere, w e obtain our main pro- jectiv e bundle r elation in ω ∗ , 1 r ( Z ): [ P ( B ) , L 1 ( − 1) , . . . L r ( − 1)] = [ P ( B ) , L 1 ⊗ N , . . . , L r ⊗ N ] − [ P ( B ) , L 1 ⊗ N (1) , . . . , L r ⊗ N (1)] +[ P ( B ) , L 1 , . . . , L r ] . ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 17 Prop osition 11. L e t Y ∈ Sm k b e a pr oje ctive variety of dim e nsion n with arbitr ary line bund les L 1 , . . . , L r . T hen, [ Y , L 1 , . . . , L r ] ∈ e ω n, 1 r ( k ) lies in the Z -line ar sp a n of ( e φ ( λ, ( m 1 , . . . , m r ))    ( λ, ( m 1 , . . . , m r )) ∈ P n, 1 r , r X i =1 m i = n ) in e ω n, 1 r ( k ) . Pr o of. Let Z ⊂ Y b e a nonsingular divisor suc h that L 1 ( Z ) , . . . , L r ( Z ) are a ll g lobally generated. Consider the double p oin t relation in ω n, 1 r ( Y ) obtained from degenerating to the normal cone of Z , pulling-bac k the list L 1 , . . . , L r , and twis ting b y the exceptional divisor of the degener- ation: [ Y , L 1 , . . . , L r ] (2.3) − [ Y , L 1 ( Z ) , . . . , L r ( Z )] − [ P ( O Z ⊕ O Z ( Z )) , L 1 ( − 1) , . . . L r ( − 1)] +[ P ( O Z ⊕ O Z ( Z )) , L 1 ( Z ) , . . . , L r ( Z )] = 0 ∈ ω n, 1 r ( Y ) . Prop osition 10 applies to the second and fourth term of relatio n (2.3). The third term, ho w ev er, requires further analysis. Using our main pro jectiv e bundle relation in ω n, 1 r ( Z ), w e can trade the third term for − [ P ( O Z ⊕ O Z ( Z )) , L 1 ( Z ) , . . . , L r ( Z )] +[ P ( O Z ⊕ O Z ( Z )) , L 1 ( Z )(1) , . . . , L r ( Z )(1)] − [ P ( O Z ⊕ O Z ( Z )) , L 1 , . . . , L r ] . The last tw o terms are not co v ered by Prop osition 10. W e ha v e pro v en the Prop osition mo dulo elemen ts of the form [ P ( B ) , L ′ 1 , . . . , L ′ r ] , [ P ( B ) , L ′ 1 (1) , . . . , L ′ r (1)] ∈ ω n, 1 r ( Z ) where B = O Z ⊕ N is a split rank 2 bundle and L ′ i are arbitrary line bundles on Z . Let π : P ( B ) → Z b e the pro jection. Let Z ′ ⊂ Z b e a nonsingular divisor suc h tha t L ′ 1 ( Z ′ ) , . . . , L ′ r ( Z ′ ) , L ′ 1 ( Z ′ )(1) , . . . , L ′ r ( Z ′ )(1) are all g lobally generated on P ( B ). 18 Y.-P . LEE AND R. P ANDHA RIP AND E Consider the double p oin t relation in ω n, 1 r ( Z ) obtained f rom degen- erating to the normal cone of π − 1 ( Z ′ ) ⊂ P ( B ), pulling- bac k the list L ′ 1 , . . . , L ′ r , a nd t wisting by the exceptional divisor of the degeneration: [ P ( B ) , L ′ 1 , . . . , L ′ r ] (2.4) − [ P ( B ) , L ′ 1 ( Z ′ ) , . . . , L ′ r ( Z ′ )] − [ P ( B Z ′ ) × Z ′ P ( O Z ′ ⊕ O Z ′ ( Z ′ )) , L ′ 1 (0 , − 1) , . . . L ′ r (0 , − 1)] +[ P ( B Z ′ ) × Z ′ P ( O Z ′ ⊕ O Z ′ ( Z ′ )) , L ′ 1 ( Z ′ ) , . . . , L ′ r ( Z ′ )] = 0 in ω n, 1 r ( Z ) . A similar relation holds fo r [ P ( B ) , L ′ 1 (1) , . . . , L ′ r (1)]. W e treat the third term of (2.4) in b oth cases with o ur main pro jectiv e bundle relation for the P ( O Z ′ ⊕ O Z ′ ( Z ′ )) pro jectivization. W e no w ha v e pro v en the Prop o sition mo dulo elemen ts of the form [ P ( B 1 ) × Z ′ P ( B 2 ) , L ′′ 1 , . . . , L ′′ r ] , [ P ( B 1 ) × Z ′ P ( B 2 ) , L ′′ 1 (1 , 0) , . . . , L ′′ r (1 , 0)] , [ P ( B 1 ) × Z ′ P ( B 2 ) , L ′′ 1 (0 , 1) , . . . , L ′′ r (0 , 1)] , [ P ( B 1 ) × Z ′ P ( B 2 ) , L ′′ 1 (1 , 1) , . . . , L ′′ r (1 , 1)] ∈ ω n, 1 r ( Z ′ ) where B i = O Z ′ ⊕ N i are split r a nk 2 bundles and L ′′ i are arbitra ry lines bundles on Z ′ . W e iterat e the pro cedure b y selecting a sufficien tly p ositive divisor Z ′′ ⊂ Z ′ . Since the dimensions of the divisors are dropping, the pro ce- dure terminates when dimension 0 is reached with the elemen ts [ P 1 × · · · × P 1 | {z } n , O ( l 1 , . . . , l n ) , . . . , O ( l 1 , . . . , l n ) | {z } r ] ∈ ω n, 1 r ( k ) with l i ∈ { 0 , 1 } . These elemen t s are co v ered b y Prop o sition 10.  2.6. Pr o of of Theorem 9 . W e pro v e the result b y induction on n . The n = 0 case is clear. W e assume the result f o r all n ′ < n . Using Theorem 9 for n ′ < n , w e conclude the g rade n part of m · ω ∗ , 1 r ⊗ Z Q is equal to the Q -linear span of ( e φ ( λ, ( m 1 , . . . , m r ))    ( λ, ( m 1 , . . . , m r )) ∈ P n, 1 r , r X i =1 m i < n ) in ω n, 1 r ( k ) ⊗ Z Q . By Prop osition 1 1, w e see dim( ω n, 1 r ( k ) ⊗ Z Q ) ≤ |P n, 1 r | . Since w e ha v e already established the rev erse inequalit y in Section 2.3, w e obtain dim( ω n, 1 r ( k ) ⊗ Z Q ) = | P n, 1 r | , ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 19 concluding the pro of of Theorem 9.  3. Higher rank 3.1. Splitting. As b efore, let m ⊂ ω ∗ ( k ) b e the ideal generated by all elemen t s of p ositiv e dimension. Since ω ∗ ,r ( k ) is a ω ∗ ( k )-mo dule, w e can define the gra ded quotien t e ω ∗ ,r ( k ) = ω ∗ ,r ( k ) m · ω ∗ ,r ( k ) , e ω ∗ ,r ( k ) = ∞ M n =0 e ω n,r ( k ) . F or ( λ, µ ) ∈ P n,r , let e φ ( λ, µ ) ∈ e ω n,r ( k ) denote the class of φ ( λ , µ ) in the quotient. Prop osition 12. L e t Y ∈ Sm k b e a pr oje ctive variety of dim e nsion n with r ank r ve ctor bund l e E . Then, [ Y , E ] ∈ e ω n,r ( k ) lies in the Z -line ar sp a n of n e φ ( λ, µ )    ( λ, µ ) ∈ P n,r , | µ | = n o in e ω n,r ( k ) . F or t he pro of of Prop osition 12, we will require the follo wing basic result. Lemma 13. Ther e exists a nons i n gular p r oje ctive variety b Y and a bir ationa l m orphism b Y → Y for which the p ul l-b ac k of E to b Y h a s a filtr at ion by sub-bund l e s 0 = E 0 ⊂ E 1 ⊂ E 2 ⊂ . . . ⊂ E r = E satisfying rank( E i /E i − 1 ) = 1 . Pr o of. Consider the complete flag v ariety ov er Y , π : F ( E ) → Y . There is a r a tional section s o f π . The v ariet y b Y is obtained from the resolution of singularit ies of the graph closure of s in Y × F ( E ).  T o pro v e Prop osition 12, let [ Y , E ] b e give n. Since [ b Y → Y ] = [ Y → Y ] ∈ e ω n ( Y ) b y (2.1), w e conclude [ b Y → Y , E ] = [ Y → Y , E ] ∈ e ω n,r ( Y ) 20 Y.-P . LEE AND R. P ANDHA RIP AND E as b efore. After pushing-forward to Sp ec ( k ), w e obta in [ b Y , E ] = [ Y , E ] ∈ e ω n,r ( k ) . On b Y , let L 1 , . . . , L r b e the list of line bundle obtained fr o m the sub quotien ts of the filtration of E . Sending the extension parameters to 0, we see [ b Y , E ] = [ b Y , L 1 ⊕ · · · ⊕ L r ] ∈ ω n,r ( k ) . Finally , Prop osition 11 applied to the list [ b Y , L 1 , . . . , L r ] concludes t he pro of of Prop o sition 12.  3.2. Pr o ofs of Theorems 1 and 4. W e prov e the r esult b y induction on n . The n = 0 case is clear. W e assume the result for all n ′ < n . Using Theorem 1 for n ′ < n , w e conclude the g rade n part of m · ω ∗ ,r ⊗ Z Q is equal to the Q -linear span of n e φ ( λ, µ )    ( λ, µ ) ∈ P n,r , | µ | < n o in ω n,r ( k ) ⊗ Z Q . By Prop osition 1 2, w e see dim( ω n,r ( k ) ⊗ Z Q ) ≤ |P n,r | . Since w e hav e already established the rev erse inequalit y in Section 1.2.3, w e obtain dim( ω n,r ( k ) ⊗ Z Q ) = | P n,r | , concluding the pro of of Theorems 1 and 4.  3.3. Pr o of of Theorem 2. Since Prop osition 12 holds ov er Z , w e see ω n,r ( k ) is generated ov e r Z b y n φ ( λ, µ )    ( λ, µ ) ∈ P n,r , | µ | = n o and the subgroups ω n ( k ) · ω 0 ,r ( k ) , ω n − 1 ( k ) · ω 1 ,r ( k ) , . . . , ω 1 ( k ) · ω n − 1 ,r ( k ) . W e no w prov e Theorem 2 b y induction o n n . Ce rtainly , ω i ( k ) is a free Z -mo dule of rank equal to the n um ber o f partitions of i . Using the induction hy p othesis, w e see ω n,r ( k ) ha s |P n,r | generators o v er Z . Since w e know dim( ω n,r ( k ) ⊗ Z Q ) = |P n,r | , no relations a mong these generators are p ossible.  ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 21 3.4. Pr o duct structure. There is a natural commutativ e ring struc- ture on ω ∗ , + ( k ) = ∞ M r =1 ω ∗ ,r ( k ) = ∞ M n =0 ∞ M r =1 ω n,r ( k ) giv en b y external pro duct [ Y 1 , E 1 ] · [ Y 2 , E 2 ] = [ Y 1 × Y 2 , p ∗ 1 ( E 1 ) ⊗ p ∗ 2 ( E 2 )] . Here, p 1 and p 2 are the pro jections of Y 1 × Y 2 on to the first and second factors resp ectiv ely . There is an inclusion of rings ω ∗ ( k ) ֒ → ω ∗ , + ( k ) , [ Y ] 7→ [ Y , O ] . By the basis result of Theorem 1, the pro duct o n ω ∗ , + ( k ) ⊗ Z Q is completely determined b y the sp ecial case [ P a , O (1)] · [ P b , O (1)] = [ P a × P b , O (1 , 1)] . Question. What is the de c omp osition of [ P a × P b , O (1 , 1)] in the b asis of ω a + b, 1 ( k ) ⊗ Z Q given in The or em 1 ? Of course, Theorem 4 pro vides a computational approach to the question for an y fixed a and b . Is there a closed form ula or an y structure in the answ er? 4. Resul ts over X 4.1. Surjectivity. F ollo wing the notatio n of Section 2.4, let e ω ∗ , 1 r ( X ) = ω ∗ , 1 r ( X ) m · ω ∗ , 1 r ( X ) , e ω ∗ , 1 r ( X ) = ∞ M n =0 e ω n, 1 r ( X ) . Consider the elemen t [ Y → X , L 1 , . . . , L r ] ∈ ω n, 1 r ( X ) . If all the L i are globally generated o n Y , then there exists a pro jectiv e morphism f : Y → X × P d 1 × · · · × P d r , L i = f ∗ ( O P d i (1)) . W e view f as determining an elemen t of algebraic cob ordism, [ f : Y → X × P d 1 × · · · × P d r ] ∈ ω n ( X × P d 1 × · · · × P d r ) . The Chow group A n ( X × P d 1 × · · · × P d r ) is generated ov er A ∗ ( X ) b y linear sub v arieties ι m 1 ,...,m r : P m 1 × · · · × P m r ֒ → P d 1 × · · · × P d r 22 Y.-P . LEE AND R. P ANDHA RIP AND E where P r i =1 m i ≤ n . Using (2.1) f or X × P d 1 × · · · × P d r , w e see [ f ] is a Z -linear com bination of elemen ts of the for m [ ι × ι m 1 ,...,m r ] ∈ e ω n ( X × P d 1 × · · · × P d r ) where ι : W → X is a resolution of singularities of an irreducible sub v ariet y of X and n = dim( W ) + r X i =1 m i . Concluding as in the pro of of Prop osition 10, w e find [ Y → X , L 1 , . . . , L r ] ∈ e ω n, 1 r ( X ) lies in the subspace spanned by pro ducts of elemen ts o f ω δ ( X ) with basis terms of ω n − δ, 1 r ( k ). The pro jectiv e bundle analysis in the pro of of Prop osition 11 o ccurs en tirely o v er Y a nd th us ov er X . Hence, w e can remov e the global generation hypothesis on the bundles L i just as b efore. Since the splitting of Lemma 13 also o ccurs o v er Y , w e conclude the comp osition ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) γ X − → ω ∗ ,r ( X ) − → e ω ∗ ,r ( X ) is surjectiv e. Prop osition 14. The natur al map γ X : ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) → ω ∗ ,r ( X ) is surje ctive. Pr o of. W e hav e already seen γ X surjects o nto ω ∗ ,r ( X ) / m · ω ∗ ,r ( X ). But then, m · ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) surjects via γ X on to m · ω ∗ ,r ( X ) m 2 · ω ∗ ,r ( X ) . The result follo ws b y iteration since T i ≥ 1 m i = 0.  4.2. Injectivity. Let c 1 , . . . , c r b e v ariables with c i of degree i . Let Ψ b e the space o f p olynomials in c 1 , . . . , c r with Z -co efficien ts. F or homogeneous ψ ∈ Ψ o f degree d , there a re natural Chern op erations C ψ : ω ∗ ,r ( X ) → ω ∗− d ( X ) ALGEBRAIC COBOR DISM OF BUNDLES ON V ARIETIES 23 defined by (4.1) C ψ ([ Y f − → X , E ]) = f ∗  ψ ( c 1 ( E ) , . . . , c r ( E )) ∩ [ Y → Y ]  ∈ ω ∗− d ( X ) where the a ction of ψ on the right is via the standar d Chern class op erations [6, Section 7.4] in algebraic cob ordism. T o sho w definition (4.1) resp ects the do uble p oin t relation in ω ∗ ,r ( X ), w e argue as follo ws. Supp ose g : Y → X × P 1 is a pro jectiv e morphism for whic h the comp osition π = p 2 ◦ g : Y → P 1 is a double p oint degeneration o v er 0 ∈ P 1 , a nd E is a rank r v e ctor bundle o n Y . The Chern op erat ion ψ ( c 1 ( E ) , . . . , c r ( E )) is w ell-defined on ω ∗ ( Y ), ψ : ω ∗ ( Y ) → ω ∗− d ( Y ) . Hence, for r egular v alues ζ ∈ P 1 ( k ) of π , ψ ∩  [ Y ζ → Y ] − [ A → Y ] − [ B → Y ] + [ P ( π ) → Y ]  = 0 ∈ ω ∗ ( Y ) . Pushing-forw ard to X a nd using the functoriality of the Chern class, w e obtain C ψ ([ Y ζ → X , E Y ζ ]) − C ψ ([ A → X , E A ]) − C ψ ([ B → X , E B ]) + C ψ ([ P ( π ) → X, E P ( π ) ]) = 0 ∈ ω ∗ ( X ) whic h is the required compatibility . By the characterization of ω ∗ ,r ( k ) in Theorem 2, w e ha v e (4.2) ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) = M λ ω ∗ ( X ) ⊗ φ ( λ, λ ) where the sum is o v er all partitions λ o f length at most r . Consider the pairing ρ X : ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) × Ψ → ω ∗ ( X ) defined by ρ X  ζ , ψ ) = C ψ ( γ X ( ζ )) . Using the ba sis (4.2), w e see the pairing ρ X is triang ula r with 1’s o n the diagonal by calculation (1.4). W e ha v e pro v en the following result. 24 Y.-P . LEE AND R. P ANDHA RIP AND E Prop osition 15. The natur al map γ X : ω ∗ ( X ) ⊗ ω ∗ ( k ) ω ∗ ,r ( k ) → ω ∗ ,r ( X ) is inje ct ive. Prop ositions 14 and 15 together complete the pro of of Theorem 3. 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