A K-theory approach to the tangent invariants of blow-ups
aa r X i v : . [ m a t h . A T ] D ec A K–theory approach to the tangent invariants ofblow–ups
Haibao Duan ∗ Academy of Mathematics and Systems Sciences,Chinese Academy of Sciences, Beijing [email protected]
Abstract
We extend the formula for the Chern classes of blow-ups of alge-braic varieties due to Porteous and Lascu–Scott, and of symplectic andcomplex manifolds due to Geiges and Pasquotto, to the blow–ups ofalmost complex manifolds.Our approach is based on a concrete partition for the tangent bun-dle of a blow–up. The use of topological K–theory of vector bundlessimplifies the classical approaches.
Key words and phrases:
Blow ups; Almost complex manifolds; K–theory, Chern classes.
In this paper all manifolds under consideration are in the real and smoothcategory, which are connected but not necessarily compact and orientable.Let X ⊂ M be a smooth submanifold whose normal bundle has a com-plex structure, and let f M be the blow–up of M along X . We present apartition for the tangent bundle τ f M of f M which implies that, if X ⊂ M isan embedding in the category of almost complex manifolds , then the blow-up f M has a canonical almost complex structure, see Theorem 2.3 and Corollary2.4.The partition on τ f M is ready to apply to deduce a formula for the tangentbundle τ f M of the blow–up f M in the K –theory K ( f M ) of complex bundlesover f M , see Theorem 3.2 and Remark 3.3, which in turn yields a formulafor the total Chern class C ( f M ) of f M , see Theorem 4.6. ∗ The author’s research is supported by 973 Program 2011CB302400 and NSFC11131008.
For an Euclidean vector bundle ξ over a topological space Y write D ( ξ ) and S ( ξ ) for the unit disk bundle and sphere bundle of ξ , respectively. If A ⊂ Y is a subspace (resp. if f : W → Y is a map) write ξ | A (resp. f ∗ ξ ) for therestriction of ξ to A (resp. the induced bundle over W ).Assume throughout that i X : X → M is a smooth embedding thatembeds X as a closed subset of M , and whose normal bundle γ X has a fixedcomplex structure J . Furnish M with an Riemannian metric so that theinduced metric on γ X is Hermitian in the sense of Milnor [10, p.156].Let π : E = P ( γ X ) → X be the complex projective bundle associatedwith γ X . Since the tautological line bundle λ E on E is a subbundle of theinduced bundle π ∗ γ X we can formulate the composition G : D ( λ E ) ⊂ D ( π ∗ γ X ) b π → D ( γ X ),where b π is the obvious bundle map over π . Regard E ⊂ D ( λ E ) and X ⊂ D ( γ X ) as the zero sections of the corresponding disk bundles, respectively. Lemma 2.1.
The map G agrees with the projection π on E , and restrictsto a diffeomorphism D ( λ E ) r E → D ( γ X ) r X . Proof.
Explicitly we have D ( λ E ) = { ( l, v ) ∈ E × π ∗ γ X | v ∈ l ∈ E, k v k ≤ } , D ( γ X ) = { ( x, v ) ∈ X × γ X | v ∈ γ X | x, k v k ≤ } .The inverse of G on D ( γ X ) r X is ( x, v ) → ( h v i , v ), where v ∈ γ X | x with v = 0 and where h v i ∈ E denotes the complex line spanned by a non–zerovector v . (cid:3) It follows from Lemma 2.1 that the map G restricts to a diffeomorphism g = G | S ( λ E ) : S ( λ E ) = ∂D ( λ E ) → S ( γ X ) = ∂D ( γ X ).2ith respect to the metric on M identifying D ( γ X ) with a tubular neigh-borhood of X in M we can formulate the adjoint manifold(2.1) f M = ( M \ ◦ D ( γ X )) ∪ g D ( λ E )by gluing D ( λ E ) to ( M \ ◦ D ( γ X )) along S ( λ E ) using g . Moreover, piecingtogether the identity on M \ ◦ D ( γ X ) and the map G yields the smooth map(2.2) f : f M = ( M \ ◦ D ( γ X )) ∪ g D ( λ E ) → M = ( M \ ◦ D ( γ X )) ∪ id D ( γ X ),which is known as the blow–up of M along X with exceptional divisor E [9]. Obvious but useful properties of the map f are listed below. Lemma 2.2.
Let i E : E → f M (resp. i X : X → M ) be the zero section of D ( λ E ) (resp. of D ( γ X ) ) in view of the decomposition (2.1). Theni) the normal bundle of E in f M is λ E ;ii) f − ( X ) = E with f ◦ i E = i X ◦ π ;iii) f restricts to a diffeomorphism: f M \ E → M \ X . (cid:3) For a manifold N write τ N for its tangent bundle. One has then theobvious bundle decompositions τ D ( λ E ) | S ( λ E ) = τ S ( λ E ) ⊕ R ( α ); τ M \ ◦ D ( γ X ) | S ( γ X ) = τ S ( γ X ) ⊕ R ( α ),where α (resp. α ) is the outward (resp. inward) unit normal field alongthe boundary S ( λ E ) = ∂D ( λ E ) (resp. S ( γ X ) = ∂ ( M \ ◦ D ( γ X ))) with R ( α i )the trivial real line bundle spanned by the field α i . Moreover, if we let τ g be the tangent map of the diffeomorphism g then the decomposition (2.1)of f M indicates the partition(2.3) τ f M = τ M \ ◦ D ( γ X ) [ h τ D ( λ E ) ,where the gluing diffeomorphism h : τ D ( λ E ) | S ( λ E ) → τ M \ ◦ D ( γ X ) | S ( γ X )is the bundle map over g with h ( tα , u ) = ( tα , τ g ( u )), u ∈ τ S ( λ E ) , t ∈ R .3ndeed, the two bundles τ D ( λ E ) | S ( λ E ) and τ M \ ◦ D ( γ X ) | S ( γ X ) admit moresubtle decompositions with respect to them the gluing map h in (2.3) admitsa useful presentation.Let p E : λ E → E and p X : γ X → X be the obvious projections. Thesame notions will be reserved for their restrictions to the subspaces S ( λ E ) ⊂ D ( λ E ) ⊂ λ E and S ( γ X ) ⊂ D ( γ X ) ⊂ γ X , respectively.For a topological space Y write 1 C (resp. 1 R ) for the trivial complexline bundle Y × C (resp. the trivial real line bundle Y × R ) over Y . Fora complex vector bundle ξ write ξ r for its real reduction. As example thetrivialization over S ( λ E )(2.4) ( p E ∗ λ E | S ( λ E )) r = R ( α ) ⊕ R ( J ( α ))indicates that p E ∗ λ E | S ( λ E ) = 1 C , where J ( α ) is a unit tangent vectorfield on S ( λ E ).Let b g : g ∗ ( τ M \ ◦ D ( γ X ) | S ( γ X )) → τ M \ ◦ D ( γ X ) | S ( γ X )be the induced bundle of g over S ( λ E ), and let κ : τ D ( λ E ) | S ( λ E ) → g ∗ ( τ M \ ◦ D ( γ X ) | S ( γ X ))be the bundle isomorphism over the identity of S ( λ E ) so that h = b g ◦ κ [10,Lemma 3.1]. With respect to the Hermitian metric induced from γ X onehas the orthogonal decomposition π ∗ γ X = λ E ⊕ λ ⊥ E in which λ ⊥ E denotes theorthogonal complement of λ E in π ∗ γ X . Theorem 2.3.
The tangent bundle of the blow up f M has the partition τ f M = τ M \ ◦ D ( γ X ) [ b g ◦ κ τ D ( λ E ) , in whichi) τ D ( λ E ) | S ( λ E ) = ( π ◦ p E ) ∗ τ X ⊕ ( p E ∗ λ E ) r ⊕ p E ∗ Hom ( λ E , λ ⊥ E ) r ;ii) g ∗ ( τ M \ ◦ D ( γ X ) | S ( γ X )) = ( π ◦ p E ) ∗ τ X ⊕ ( p E ∗ λ E ) r ⊕ p E ∗ ( λ ⊥ E ) r .Moreover, with respect to the decompositions i) and ii) the bundle isomor-phism κ is given bya) κ | ( π ◦ p E ) ∗ τ X = id ; b) κ | ( p E ∗ λ E ) r = id ;c) κ ( b ) = b ( α ) for b ∈ Hom ( p E ∗ λ E , p E ∗ λ ⊥ E ) r . roof. It follows from the standard decompositions τ E = π ∗ τ X ⊕ Hom ( λ E , λ ⊥ E ) r , τ D ( λ E ) = ( p E ∗ λ E ) r ⊕ p E ∗ τ E that(2.5) τ D ( λ E ) = ( p E ∗ λ E ) r ⊕ ( π ◦ p E ) ∗ τ X ⊕ p E ∗ Hom ( λ E , λ ⊥ E ) r .Similarly, it comes from τ D ( γ X ) = p X ∗ τ X ⊕ p X ∗ γ X , π ∗ γ X = λ E ⊕ λ ⊥ E ,as well as the definition of f that(2.6) f ∗ τ D ( γ X ) = ( p E ∗ λ E ) r ⊕ ( π ◦ p E ) ∗ τ X ⊕ ( p E ∗ λ ⊥ E ) r .One obtains the relations i) and ii) by restricting the decomposition (2.5)and (2.6) to the subspace S ( λ E ) ⊂ D ( λ E ), respectively.Finally, properties a), b), c) are transparent in view of the relation h = b g ◦ κ , together with the description of g indicated in the proof of Lemma2.1. (cid:3) A manifold M is called almost complex if its tangent bundle is furnishedwith a complex structure J M [10, p.151]. Given two almost complex man-ifolds ( X, J X ) and ( M, J M ) an embedding i X : X → M is called almostcomplex if τ X is a complex subbundle of the restricted bundle τ M | X . Inthis situation the normal bundle γ X of X has the canonical complex struc-ture J induced from that on τ M | X and that on τ X , hence the blow–up f M of M along X is defined. Moreover, in view of the decomposition (2.1) wenote thati) J M restricts to an almost complex structure on M \ ◦ D ( γ X );ii) the tubular neighborhood D ( λ E ) of E in f M has the canonicalalmost complex structure so that as a complex vector bundle τ D ( λ E ) = ( π ◦ p E ) ∗ τ X ⊕ p E ∗ λ E ⊕ Hom ( p E ∗ λ E , p E ∗ λ ⊥ E )(compare this with (2.5)). Since with respect to the induced complex struc-tures on τ D ( λ E ) | S ( λ E ) and on τ M \ ◦ D ( γ X ) | S ( γ X ) the clutching map h in(2.3) is C –linear by Theorem 2.3, one obtains Corollary 2.4. If i X : X → M is an embedding in the category of almostcomplex manifolds, then the blow–up f M has a canonical almost complexstructure that is compatible with that on M \ ◦ D ( γ X ) and that on D ( λ E ). (cid:3) Remark 2.5.
The analogue of Corollary 2.4 in the symplectic setting isdue to McDuff [9, Section 3], which concludes that if i X : X → M is an em-bedding of symplectic manifolds, then the blow–up f M admits a symplecticform which coincides with the one on M \ X off the exceptional divisor E . (cid:3) The tangent bundle of a blow–up
For a topological space Y let K ( Y ) (resp. e K ( Y )) be the K –theory (re-duced K –theory) of complex vector bundles over Y . If i X : X → M is anembedding in the category of almost complex manifolds, then the blow–up f M has a canonical almost complex structure by Corollary 2.4. In partic-ular, the difference τ f M − f ∗ τ M can be regarded as an element of the ring e K ( f M ). In Theorem 3.2 below we obtain a formula expressing the element τ f M − f ∗ τ M ∈ e K ( f M ) in term of the decomposition p ∗ E γ X = p ∗ E λ E ⊕ p ∗ E λ ⊥ E .For a relative CW–complex ( Y, A ) the inclusion j : ( Y, ∅ ) → ( Y ; A )induces a homomorphism(2.1) j ∗ : K ( Y ; A ) → e K ( Y ),where K ( Y ; A ) is the relative K –group of the pair ( Y ; A ) defined by(3.2) K ( Y ; A ) =: e K ( Y /A ).In addition, the group K ( Y ; A ) admits another description useful in thesubsequent calculation. Lemma 3.1 ([2, Theorem 2.6.1]).
Any element in the group K ( Y ; A ) canbe represented by a triple [ ξ, η ; α ] in which ξ and η are vector bundles over Y and α : ξ | A → η | A is a bundle isomorphism.Moreover, with respect to this representation of the group K ( Y ; A ) onehas i) the triple [ ξ, ξ ; id ] represents the zero for any bundle ξ over Y ;ii) [ ξ, η ; α ] + [ ξ , η ; α ] = [ ξ ⊕ ξ , η ⊕ ξ ; α ⊕ α ] ;iii) [ ξ, η ; α ] ⊗ γ = [ ξ ⊗ γ, η ⊗ γ ; α ⊗ id ] ;iv) j ∗ [ ξ, η ; α ] = ξ − η ,where ⊕ means direct sum of vector bundles (homomorphisms), and where ⊗ denotes the action K ( X ; A ) ⊗ K ( X ) → K ( X ; A ) defined by the tensorproduct of vector bundles . (cid:3) For an embedding i X : X → M of almost complex manifolds let f : f M → M be the blow–up of M along X with exceptional divisor E . Consider thecomposition j E : K ( D ( λ E ) , S ( λ E )) → ∼ = K ( f M , f M r ◦ D ( λ E )) j ∗ → e K ( f M )6n which the first map is the excision isomorphism . By Lemma 3.1 thetrivialization ε : λ E | S ( λ E ) → C indicated by (2.4) defines an element[ p ∗ E λ E , C ; ε ] ∈ K ( D ( λ E ) , S ( λ E )). Theorem 3.2.
In the ring e K ( f M ) one has (3.3) τ f M − f ∗ τ M = j E ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ ⊥ E ). Proof.
The partition (2.1) of the blow up f M implies the relation τ f M | M \ ◦ D ( γ X ) = f ∗ τ M | M \ ◦ D ( γ X )which gives rise to an element [ τ f M , f ∗ τ M ; id ] ∈ K ( f M ; f M r ◦ D ( λ E )) byLemma 3.1. Moreover, with respect to the excision isomorphism K ( f M ; f M r ◦ D ( λ E )) ∼ = → K ( D ( λ E ); S ( λ E ))we have [ τ f M , f ∗ τ M ; id ] = [ τ D ( λ E ) , f ∗ τ D ( γ X ) ; κ ], where κ : τ D ( λ E ) | S ( λ E ) → g ∗ ( τ M \ ◦ D ( γ X ) | S ( γ X ))is the bundle isomorphism specified in Theorem 2.3. Granted with thedecompositions of the bundles τ D ( λ E ) and f ∗ τ D ( γ X ) in (2.4) and (2.5), aswell as the decomposition of bundle map κ in Theorem 2.3, we calculate[ τ D ( λ E ) , f ∗ τ D ( γ X ) ; κ ]= [( π ◦ p E ) ∗ τ X , ( π ◦ p E ) ∗ τ X ; id ] + [ p E ∗ λ E , p E ∗ λ E ; id ]+[ p E ∗ Hom ( λ E , λ ⊥ E ) , p E ∗ λ ⊥ E ; κ ′ ] (by ii) of Lemma 3.1)= [ p E ∗ Hom ( λ E , λ ⊥ E ) , p E ∗ λ ⊥ E ; κ ′ ] (by i) of Lemma 3.1)= [ p E ∗ ( λ E ⊗ λ ⊥ E ) , p E ∗ λ ⊥ E ; κ ′ ] (since Hom ( λ E , λ ⊥ E ) = λ E ⊗ λ ⊥ E )= [ p E ∗ λ E , C ; ε ] ⊗ p E ∗ λ ⊥ E (by iii) of Lemma 3.1)where κ ′ is the restriction of κ to the direct summand p E ∗ Hom ( λ E , λ ⊥ E ) of τ D ( λ E ) (see c) of Theorem 2.3). Summarizing, in the group K ( D ( λ E ) , S ( λ E ))we we have the relation[ τ f M , f ∗ τ M ; id ] = [ p E ∗ λ E , C ; ε ] ⊗ p E ∗ λ ⊥ E .Finally, applying the map j E to both sides yields the formula (3.3) by iv) ofLemma 3.1. (cid:3) Remark 3.3.
In the group K ( f M ) formula (3.3) has the concise expression7 f M = f ∗ τ M + i E ! ( λ ⊥ E )where i E ! is the Gysin map in K –theory i E ! : K ( E ) ψ E → ∼ = K ( D ( λ E ) , S ( λ E )) j E → e K ( f M ),and where ψ E is the Thom isomorphism x → ψ E ( x ) = U E ⊗ p ∗ E x , x ∈ K ( E ),with U E ∈ K ( D ( λ E ) , S ( λ E )) the Thom class of λ E . Indeed, from the generalconstruction [2, p.98–99] of Thom classes from the exterior algebras of vectorbundles one finds that U E = [ p ∗ E λ E , ε ]. (cid:3) Based on the formula (3.3) we deduce a formula for the total Chern class C ( f M ) of a blow up f M in the category of almost complex manifolds. In thissection the coefficients for cohomologies will be the ring Z of integers. Let BU be the classifying space of stable equivalent classes of complex vectorbundles, and let c r ∈ H r ( BU ) be the r th Chern class of the universalcomplex vector bundle over BU . Then H ∗ ( BU ) = Z [ c , c , · · · ].For a topological space Y let [ Y, BU ] be the set of homotopy classes ofmaps from Y to BU . In view of the standard identification e K ( Y ) = [ Y, BU ]([13, p.210]) we can introduce the total Chern class for elements in e K ( Y ) asthe co–functor C : e K ( Y ) → H ∗ ( Y ) defined C ( β ) = 1 + f ∗ c + f ∗ c + · · · , β ∈ e K ( Y ) , where f : Y → BU is the classifying map of the element β . Clearly one has Lemma 4.1.
The transformation C satisfies the next two properties.i) If ξ i , i = 1 , , are two complex vector bundles over Y with equaldimension and with (the usual) total Chern classes C ( ξ i ) , then C ( ξ − ξ ) = C ( ξ ) C ( ξ ) − . ii) For a closed subspace A ⊂ Y let j A : ( Y, ∅ ) → ( Y, A ) and q A : Y → Y /A be the inclusion and quotient maps, respectively. Then the nextdiagram commutes: ( Y ; A ) = e K ( Y /A ) j ∗ A → e K ( Y ) C ↓ ↓ CH ∗ ( Y /A ) q ∗ A → H ∗ ( Y ) . (cid:3) For two complex vector bundles λ and ξ over a space Y with dim λ = 1,dim ξ = m , and with the total Chern classes C ( λ ) = 1 + t ; C ( ξ ) = 1 + c ( ξ ) + c ( ξ ) + · · · + c m ( ξ ),respectively, assume that the Chern roots of ξ is s , · · · , s m . That is C ( ξ ) = Q ≤ i ≤ m (1 + s i )with c r ( ξ ) = e r ( s , · · · , s m ) the r th elementary symmetric function in theroots s , · · · , s m . The calculation C ( λ ⊗ ξ ) = Q ≤ i ≤ m (1 + t + s i ) = (1 + t ) m Q ≤ i ≤ m (1 + s i t )= (1 + t ) m [1 + c ( ξ )(1+ t ) + c ( ξ )(1+ t ) + · · · + c m ( ξ )(1+ t ) m ]shows that(4.1) C ( λ ⊗ ξ ) = X ≤ r ≤ m (1 + t ) m − r c r ( ξ ).For an Euclidean complex line bundle λ over Y with associated diskbundle p λ : D ( λ ) → Y the Thom space T ( λ ) of λ is the quotient space D ( λ ) /S ( λ ). In term of Lemma 3.1 the trivialization ε : p ∗ λ ( λ ) | S ( λ ) → C indicated by (2.4) defines the element[ p ∗ λ ( λ ) , C ; ε ] ∈ K ( D ( λ ) , S ( λ )) = e K ( T ( λ )).Given a ring A and a set { u , · · · , u k } of k elements let A { u , · · · , u k } bethe free A module with basis { u , · · · , u k } .In addtion to Theorem 3.2, the next result will play a key role in estab-lishing the formula for Chern class. Lemma 4.2.
Let e ∈ H ( Y ) and x ∈ H ( T ( λ )) be respectively the Eulerclass and the Thom class of the oriented bundle λ . Theni) the integral cohomology ring of T ( λ ) is determined by the additivepresentation (4.2) H ∗ ( T ( λ )) = Z ⊕ H ∗ ( Y ) { x } , ogether with the single relation x + xe = 0 ;ii) for a complex vector bundle ξ over Y with total Chern class C ( ξ ) =1+ c + · · · + c m ∈ H ∗ ( Y ) , the total Chern class of the element [ p ∗ λ ( λ ) , C ; ε ] ⊗ p ∗ λ ξ ∈ e K ( T ( λ )) is (4.3) C ([ p ∗ λ ( λ ) , C ; ε ] ⊗ p ∗ λ ξ ) = ( X ≤ r ≤ m (1 + x ) m − r c r ) C ( ξ ) − ∈ H ∗ ( T ( λ )) . Proof.
The presentation (4.2) comes immediately from the Thom isomor-phism theorem, which states that product with Thom class x yields anadditive isomorphism H r ( Y ) ∼ = H r +2 ( T ( λ )), y → y · x , y ∈ H r ( X )for all r ≥
0. For i) it remains to justify the relation x + xe = 0.Let p : S ( λ ⊕ R ) → Y be the sphere bundle of the Euclidean bundle λ ⊕ R and set D +( − ) ( λ ) = { ( u, t ) ∈ S ( λ ⊕ R ) | t ≥ t ≤ } .It is clear thata) S ( λ ⊕ R ) = D − ( λ ) ∪ D + ( λ ), S ( λ ) = D − ( λ ) ∩ D + ( λ ),b) both D ± ( λ ) can be identified with the disk bundle D ( λ ) of λ .In view of a) we have the canonical map onto the Thom space T ( λ ) q : S ( λ ⊕ R ) → T ( λ ) = S ( λ ⊕ R ) /D − ( λ ).Moreover, letting u = q ∗ x ∈ H ( S ( λ ⊕ R )) the ring H ∗ ( S ( λ ⊕ R )) has thepresentation (see [3, Lemma 4]) H ∗ ( S ( λ ⊕ R )) = H ∗ ( Y ) { , u } / (cid:10) u + ue (cid:11) .The relation x + xe = 0 on H ∗ ( T ( λ )) is verified by the relation u + ue = 0on H ∗ ( S ( λ ⊕ R )), together with the fact that the induced ring map q ∗ ismonomorphic onto the direct summand Z ⊕ H ∗ ( Y ) { u } of H ∗ ( S ( λ ⊕ R ).For ii) define over S ( λ ⊕ R ) the complex line bundle λ u by λ u = p ∗ λ | D + ( λ ) ∪ ε C | D − ( λ ) (in view of the partition a)).Then C ( λ u ) = 1 + u ∈ H ∗ ( S ( λ ⊕ R ). Moreover, for a vector bundle ξ over Y the element[ λ u , C ; ε ] ⊗ p ∗ ξ ∈ K ( S ( λ ⊕ R ) , D − ( λ ))10orresponds to the element [( p ∗ λ ( λ ) , C ; ε ) ⊗ p ∗ λ ξ ] ∈ K ( D ( λ ) , S ( λ )) under theexcision isomorphism K ( S ( λ ⊕ R ) , D − ( λ )) ∼ = K ( D ( λ ) , S ( λ )) indicated byb), which is also mapped to the element λ u ⊗ p ∗ ξ − p ∗ ξ ∈ e K ( S ( λ ⊕ R ))under the induced homomorphism j ∗ of the inclusion j : ( S ( λ ⊕ R ) , ∅ ) → ( S ( λ ⊕ R ) , D − ( λ )) by iv) of Lemma 3.1. It follows from i) of Lemma 4.1that C ( j ∗ ([( λ u , C ; ε ) ⊗ p ∗ ξ ])) = C ( λ u ⊗ p ∗ ξ ) C ( p ∗ ξ ) − = ( X ≤ r ≤ m (1 + u ) m − r c r ) C ( ξ ) − (by the formula (4.1)).The commutativity of the diagram in ii) of Lemma 4.1 implies that q ∗ C ([( λ u , C ; ε ) ⊗ p ∗ ξ ]) = ( X ≤ r ≤ m (1 + u ) m − r c r ) C ( ξ ) − .One obtains the formula (4.3) from q ∗ ( x ) = u and the injectivity of q ∗ . (cid:3) Let f : f M → M be the blow up of M along a submanifold i X : X → M whose normal bundle γ X has a complex structure and with total Chern class C ( γ X ) = 1+ c + · · · + c k ∈ H ∗ ( X ) , k = dim C γ X . Regard D ( λ E ) as a normaldisk bundle of the exceptional divisor E and consider the quotient map ontothe Thom space of λ E q : f M → f M / ( M \ ◦ D ( γ X ) = T ( λ E ).According to i) of Lemma 4.2 one has Lemma 4.3.
Let t ∈ H ( E ) and x ∈ H ( T ( λ E )) be the Euler classand Thom class of the oriented bundle λ E , respectively. Then the ring H ∗ ( T ( λ E )) is determined by the additive presentation (4.5) H ∗ ( T ( λ E )) = Z ⊕ H ∗ ( X )[ x, tx, · · · , t k − x ], together with the relationsi) x + tx = 0 ; ii) t k + c · t k − + · · · + c k − · t + c k = 0 . (cid:3) Consider the Gysin maps of the embeddings i X : X → M and i E : E → f M in cohomology 11 X ! : H ∗ ( X ) ψ X → ∼ = H ∗ ( D ( γ X ) , S ( γ X )) → ∼ = H ∗ ( M, M r ◦ D ( γ X )) j ∗ → H ∗ ( M ) i E ! : H ∗ ( E ) ψ E → ∼ = H ∗ ( D ( λ E ) , S ( λ E )) → ∼ = H ∗ ( f M , f M r ◦ D ( λ E )) j ∗ → H ∗ ( f M )where ψ X and ψ E are the Thom isomorphisms. We shall set ω E = i E ! (1) ∈ H ( f M ), ω X = i X ! (1) ∈ H k ( M ).Geometrically, if M is closed and oriented the class ω X (for instance) is thePoincare dual of the cycle class i X ∗ [ X ] ∈ H ∗ ( M ).The next result is shown in [4, Theorem 1]. Theorem 4.4.
The ring map f ∗ : H ∗ ( M ) → H ∗ ( f M ) embeds the ring H ∗ ( M ) as a direct summand of H ∗ ( f M ), and induces the decomposition (4.6) H ∗ ( f M ) = f ∗ ( H ∗ ( M )) ⊕ H ∗ ( X ) { ω E , · · · , ω k − E } , k = dim R γ X that is subject to the two relationsi) f ∗ ( ω X ) = P ≤ r ≤ k ( − r − c k − r · ω rE ; ii) f ∗ ( y ) · ω E = i ∗ X ( y ) · ω E , y ∈ H r ( M ). Moreover, with respect to the presentations of the rings H ∗ ( T ( λ E )) and H ∗ ( f M ) in (4.5) and (4.6), the induced ring map q ∗ is determined byiii) q ∗ ( t r x ) = ( − r ω r +1 E , r ≥ (cid:3) Remark 4.5.
In [6, p.605] Griffiths and Harris obtained the decomposition(4.6) for blow ups of complex manifolds, while the relations i) and ii) wereabsent. In the non–algebraic settings partial information on the ring H ∗ ( f M )was also obtained by McDuff in [9, Proposition 2.4].In comparison the structure of H ∗ ( f M ) as a ring is completely determinedby the additive decomposition (4.6), together with the relations i) and ii).Indeed, granted with the fact that f ∗ ( H ∗ ( M )) ⊂ H ∗ ( f M ) is a subring, therelations i) and ii) are sufficient to express, respectively, the products ofelements in the second summand, and the products between elements in thefirst and second summands, as elements in the decomposition (4.6). Thisidea has been applied in [4] to determine the integral cohomology rings ofthe varieties of complete conics and complete quadrics in the 3–space P ,and justify two enumerative problems due to Schubert. (cid:3) .3 The Chern class of a blow up Assume now that f : f M → M is a blow up in the category of almost complexmanifolds, and that the total Chern class of the normal bundle γ X is C ( γ X ) = 1 + c + · · · + c k , dim R γ X = 2 k .Applying the transformation C : e K ( f M ) → H ∗ ( f M ) to the equality (3.3) andnoting the obvious relation p ∗ E λ ⊥ E = p ∗ E γ X − p ∗ E λ E in the K –group K ( D ( λ E )) one obtains C ( f M ) · f ∗ C ( M ) − = C ( j E ([ p ∗ E λ E , C ; ε ] ⊗ ( p ∗ E γ X − p ∗ E λ E ))= C ( j E ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E γ X )) C ( j E ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ E )) (by i) of Lemma 4.1)= q ∗ C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E γ X ) C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ E ) (by ii) of Lemma 4.1).Furthermore, from C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E γ X ) = ( X ≤ r ≤ k (1 + x ) k − r c r ) C ( γ X ) − , C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ E ) = (1 + x + t )(1 + t ) − by (4.3) one gets C ( f M ) · f ∗ C ( M ) − = q ∗ ( X ≤ r ≤ k (1+ x ) k − r c r )(1+ t )(1+ x + t ) − C ( γ X ) − = ( X ≤ r ≤ k (1 + ω E ) k − r c r )(1 − ω E ) C ( γ X ) − .where the second equality comes from iii) of Theorem 4.4. It implies that C ( f M ) − f ∗ C ( M ) = f ∗ C ( M ) · g ( ω E )with g ( ω E ) = ( X ≤ r ≤ k (1 + ω E ) k − r c r )(1 − ω E ) C ( γ X ) − − ∈ H ∗ ( f M )a polynomial in ω E with coefficients in H ∗ ( X ).It is crucial for us to observe from the obvious relation g (0) = 0 that thepolynomial g ( ω E ) is divisible by ω E . Therefore, the relation ii) in Theorem4.4 is applicable to yield 13 ( f M ) − f ∗ C ( M ) = f ∗ C ( M ) · g ( ω E )= i ∗ X C ( M ) g ( ω E )= C ( X ) C ( γ X ) g ( ω E ) (since τ M | X = τ X ⊕ γ X ).Summarizing we get Theorem 4.6.
With respect to decomposition of the ring H ∗ ( f M ) in (4.6),the total Chern class of the blow up f M is C ( f M ) = f ∗ C ( M )+ C ( X )( X ≤ r ≤ k (1+ ω E ) k − r c r )(1 − ω E ) − X ≤ r ≤ k c r ). (cid:3) Examples 4.7.