aa r X i v : . [ m a t h . AG ] S e p Topology of Blow–ups and Enumerative Geometry
Haibao Duan, Banghe LiAcademy of Mathematics and Systems Sciences,Chinese Academy of Sciences
Abstract
Let f M be the blow–up of a manifold M along a submanifold X whose normal bundle has a complex structure. We obtain formulae forthe integral cohomology ring and the total Chern class of f M .As applications we determine the cohomology rings of the varietiesof complete conics and complete quadrices on the 3–space P , andjustify two enumerative results due to Schubert [23, § § Email addresses: [email protected]; [email protected]
In this paper M is a manifold in the real and smooth category, which isconnected and paracompact, but not necessarily orientable. The notion X ⊂ M stands for a closed, smoothly embedded submanifold in M . Thecohomologies are over the ring of integers, unless otherwise stated.In the complex or symplectic geometry, the blow–up construction hasbeen a basic and useful tool to formulate new and important manifolds f M out of embeddings X ⊂ M in the relevant category (e.g. Gromov [10],McDuff [20], Thurston [27]). The topological invariants of blow–ups f M , suchas the integral cohomology ring H ∗ ( f M ) and total Chern class C ( f M ), are theessential ingredients of intersection theory and enumerative geometry (e.g.Fulton [8], Griffith–Harris [11]). In the first part of this paper, by examiningthe geometry of a blow–up f M for an embedding X ⊂ M in the real andsmooth category , we obtain these invariants in its natural generality. Inparticular, our formula for H ∗ ( f M ) in Theorem 4.1 is not completely knowneven for the cases of the blow–ups in the complex and symplectic sittings,while our formula for C ( f M ) in Theorem 4.4 is applicable to blow–ups in thecategory of almost complex manifolds, see also Remarks 4.2 and 4.5.The second part is devoted to applications to enumerative geometry[14, 15], in which effective computation of the characteristic numbers is thefirst test [23, Chapter 6]. Granted with Theorems 4.1 and 4.3 we obtain1xplicit presentations of the integral cohomology rings of the varieties ofcomplete conics and quadrics on the projective 3–space P in Sections 5.1and 5.2, respectively. They are applied to evaluate all the characteristicnumbers of the parameter spaces by a single procedure in Section 5.3, andto justify the following two enumerative results of Schubert [23, § §
22] inSection 5.4:
Given quadrics in the space P in general position, there are , , conics tangent to all of them;Given quadrics in the space P in general position, there are , , quadrics tangent to all of them. For a survey on the earlier studies on these problems, we refer to thearticle [9] by Fulton–Kleiman–MacPherson. In particular, to our knowledgethe cohomologies of these parameter spaces have not yet been decided before.The authors would like to thank P. Aluffi, W. Fulton, S. Kleiman, J.Harris and D. Laksov for valuable communications concerning this work.
In this section we obtain a subtle partition on the tangent bundle τ ( f M ) ofa blow–up f M in Theorem 2.3. It implies that f M carries an almost complexstructure when the center i X : X → M is an embedding of almost complexmanifold (Theorem 2.4). It will also be applied to present τ ( f M ) as anelement in the K –theory of the blow–up f M in Theorem 4.3, which plays acrucial role in our approach to the Chern class C ( f M ) in Theorem 4.4.Let i X : X → M be an embedding whose the normal bundle γ X isequipped with a complex structure J , dim R γ X = 2 k . Furnish M with anRiemannian metric so that the induced metric on γ X is Hermitian in thesense of Milnor [21, p.156]. For an Euclidean vector bundle ξ denote by D ( ξ ) (resp. by S ( ξ )) the associated disk bundle (resp. sphere bundle). Let π : E = P ( γ X ) → X be the projective bundle associated to the complexvector bundle ( γ X , J ). Regarding the tautological line bundle λ E on E as asubbundle of the induced bundle π ∗ γ X one has 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 . Identify E ⊂ D ( λ E ) and X ⊂ D ( γ X ) with the zero sections of the bundles λ E and γ X , respectively. Lemma 2.1.
The map G restricts to a diffeomorphism D ( λ E ) r E → D ( γ X ) r X , and satisfies the relation G | E = π . roof. By the presentations of the spaces D ( λ E ) and D ( γ X ) 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 for a vector v ∈ D ( γ X ) r X the symbol h v i ∈ E denotes the line spanned by v . (cid:3) By Lemma 2.1 the map G restricts to a diffeomorphism g = G | S ( λ E ) : S ( λ E ) → S ( γ X ) by which one forms 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 the boundaries using g . Piecing to-gether 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 ).The manifold f M , together with the map f , is called the blow–up of M along the submanifold X with exceptional divisor E [20]. Obvious butuseful properties of the map f are: Lemma 2.2.
Let i E : E → f M (resp. i X : X → M ) be the embedding givenby the zero section of D ( λ E ) (resp. of D ( γ X ) ) in view of (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) f M Let p E : λ E → E (resp. p X : γ X → X ) be the vector bundle projection. Tosafe notation we reserve same notion for its restrictions to the subspaces S ( λ E ) ⊂ D ( λ E ) ⊂ λ E (resp. S ( γ X ) ⊂ D ( γ X ) ⊂ γ X ).Let τ ( N ) denote the tangent bundle of a manifold N . Then 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 ))), and where R ( α i ) is the trivial real line bundle spanned by the field α i . Moreover, letting τ g be the tangent map of the diffeomorphism g , then (2.1) implies that32.3) τ ( f M ) = τ ( M \ ◦ D ( γ X )) ∪ h τ ( D ( λ E )),where the gluing diffeomorphism h is the bundle map over g with h ( u, tα ) = ( τ g ( u ) , tα ), u ∈ τ S ( λ E ) , t ∈ R .Indeed, the restricted bundles τ ( D ( λ E )) | S ( λ E ) and τ ( M \ ◦ D ( γ X )) | S ( γ X ),as well as the map h in (2.3), possess remarkable and useful properties. Tosee this we 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 , 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 [21, Lemma 3.1]. With respect to the Hermitian metric on γ X one hasthe orthogonal decomposition π ∗ γ X = λ E ⊕ λ ⊥ E with λ ⊥ E the orthogonalcomplement of the subbundle λ E ⊂ π ∗ γ X . In addition, for a complex vectorbundle ξ write ξ r for its real reduction. Theorem 2.3.
The tangent bundle of 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 order of the three direct summands in i) andii), the bundle isomorphism κ is given, respectively, bya) κ | ( π ◦ p E ) ∗ τ ( X ) = id ; b) κ | ( p E ∗ λ E ) r = id ;c) κ ( b ) = b ( α ) ∈ p E ∗ ( λ ⊥ E ) r for b ∈ Hom ( p E ∗ λ E , p E ∗ λ ⊥ E ) r . Proof.
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 42.4) 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(2.5) f ∗ τ ( D ( γ X )) = ( p E ∗ λ E ) r ⊕ ( π ◦ p E ) ∗ τ ( X ) ⊕ ( p E ∗ λ ⊥ E ) r .The relations i) and ii) are obtained by restricting the decompositions (2.4)and (2.5) to the subspace S ( λ E ) = ∂D ( λ E ) = ∂D ( γ X ), respectively. Prop-erties a), b), c) are transparent in view of the relation h = b g ◦ κ , togetherwith the description of g indicated in the proof of Lemma 2.1. (cid:3) A manifold M is called almost complex if its tangent bundle τ ( M ) is fur-nished with a complex structure J M . Given two almost complex manifolds( X, J X ) and ( M, J M ) an embedding i X : X → M is called almost complex if τ ( X ) is a complex subbundle of the restriction τ ( M ) | X . In this situationthe normal bundle γ X of X has the induced complex structure J and there-fore, the blow–up f M of M along X is defined. In view of the decomposition(2.1) on f M we notify thati) J M restricts to an almost complex structure on M \ ◦ D ( γ X );ii) the neighborhood D ( λ E ) of E in f M has the almost complexstructure so that as a complex bundle (compare with (2.4)): τ ( D ( λ E )) = ( π ◦ p E ) ∗ τ ( X ) ⊕ p E ∗ λ E ⊕ Hom ( p E ∗ λ E , p E ∗ λ ⊥ E )iii) with respect to i) and ii) the clutching map h in (2.3) is C –linear by properties a), b), c) of Theorem 2.3.These imply that Theorem 2.4. If i X : X → M is an embedding of almost complex manifold,then the blow–up f M has an almost complex structure that is compatiblewith the ones on M \ ◦ D ( γ X ) and on D ( λ E ). (cid:3) Remark 2.5.
In the case where i X : X → M is an embedding of symplecticsubmanifold, it has been shown by Gromov [10], McDuff [20], Geiges andPasquotto [13] that the blow–up f M has a symplectic structure. (cid:3) Preliminaries in cohomology theories
In this section we develop preliminary constructions and results in cohomol-ogy and topological K –theory, requested by the latter calculation with blow–ups. For a topological space Y let 1 C (resp. 1 R ) be the 1–dimensional trivialbundle Y × C (resp. Y × R ) on Y . For a ring A and a finite set { t , · · · , t n } denote by A · { t , · · · , t n } the free A –module with basis { t , · · · , t n } . Given an oriented k –dimensional real Euclidean bundle ξ over a space Y the identification space T ( ξ ) := D ( ξ ) /S ( ξ ) is called the Thom space of thebundle ξ . Since the corresponding quotient map q ξ : ( D ( ξ ) , S ( ξ )) → ( T ( ξ ) , ∗ )(with ∗ ∈ T ( ξ ) the preferred base point) is a relative homeomorphism, itinduces an isomorphism on cohomologies. Therefore, the classical Thomisomorphism [21, p.206] implies that the cohomology H ∗ ( T ( ξ ) , ∗ ) is a moduleover the ring H ∗ ( Y ), and that there is a distinguished class u ξ ∈ H k ( T ( ξ ))so that its image under q ∗ ξ is the Thom class of the oriented bundle ξ . Let e ( ξ ) ∈ H k ( Y ) be the Euler class of ξ . In the following result we determinethe ring structure on H ∗ ( T ( ξ )). Lemma 3.1.
As a H ∗ ( Y ) module the ring H ∗ ( T ( ξ )) has the presentation (3.1) H ∗ ( T ( ξ )) = Z ⊕ H ∗ ( Y ) · { u ξ } with u ξ + e ( ξ ) · u ξ = 0. Proof.
The formula for H ∗ ( T ( ξ )) comes from the Thom isomorphism,by which taking product with u ξ yields an additive isomorphism H r ( Y ) ∼ = H r + k ( T ( λ )) of degree k [21, p.206]. It remains to show the relation u ξ + e ( ξ ) · u ξ = 0 in (3.1) that characterizes the cohomology H ∗ ( T ( ξ )) as a ring.Let p : S ( ξ ⊕ R ) → Y be the sphere bundle of the Euclidean bundle ξ ⊕ R , and set D +( − ) ( λ ) = { ( u, t ) ∈ S ( λ ⊕ R ); t ≥ t ≤ } .Then(3.2) S ( ξ ⊕ R ) = D − ( ξ ) ∪ D + ( ξ ) with S ( ξ ) = D − ( ξ ) ∩ D + ( ξ ),(3.3) both D ± ( ξ ) can be identified with the disk bundle D ( ξ ) of ξ .In view of (3.2) one can form the map onto the Thom space h ξ : S ( ξ ⊕ R ) → T ( ξ ) = S ( ξ ⊕ R ) /D − ( ξ ).6et u = h ∗ ξ ( u ξ ). Then by [4, Lemma 4](3.4) H ∗ ( S ( ξ ⊕ R )) = H ∗ ( Y ) { , u } with u + e ( ξ ) · u = 0.Since the map h ∗ ξ is monomorphic onto the direct summand Z ⊕ H ∗ ( Y ) { u } of the ring H ∗ ( S ( ξ ⊕ R )), one gets the relation in (3.1) from (3.4). (cid:3) For an oriented subbundle η ⊂ ξ of an Euclidean bundle ξ let γ beits orthogonal complement. The inclusion j : ( D ( η ) , S ( η )) ⊂ ( D ( ξ ) , S ( ξ ))induces the map T ( j ) : ( T ( η ) , ∗ ) → ( T ( ξ ) , ∗ )between Thom spaces. In view of the homeomorphism( D ( ξ ) , S ( ξ )) ∼ = ( D ( η ) , S ( η )) × ( D ( γ ) , S ( γ ))of topological pairs one can show that Lemma 3.2.
With respect to the presentations of the rings H ∗ ( T ( ξ )) and H ∗ ( T ( η )) in (3.1), the induced map T ( j ) ∗ is given by (3.5) T ( j ) ∗ ( x · u ξ ) = ( x ∪ e ( γ )) · u η , x ∈ H ∗ ( Y ). (cid:3) Let i Y : Y → N be a smooth embedding of a closed manifold Y intoa Riemannian manifold N with oriented normal bundle ξ . With respect tothe induced metric on ξ identify D ( ξ ) with a tubular neighborhood of Y in N , and set ◦ D ( ξ ) = D ( ξ ) \ S ( ξ ). The quotient map onto the Thom space j Y : N → N/ ( N \ ◦ D ( ξ )) ∼ = T ( ξ )will be called the normal map of the embedding Y ⊂ N . In the cohomologyexact sequence of the pair ( N, N \ ◦ D ( ξ )) using the isomorphisms H ∗ ( T ( ξ ) , ∗ ) q ∗ ξ → ∼ = H ∗ ( D ( ξ ) , S ( ξ )) ∼ = H ∗ ( N, N \ ◦ D ( ξ )), H ∗ ( N \ ◦ D ( ξ )) ∼ = H ∗ ( N \ Y )to substitute the group H ∗ ( T ( ξ ) , ∗ ) in place of H ∗ ( N, N \ ◦ D ( ξ )), and to re-place H ∗ ( N \ ◦ D ( ξ )) by H ∗ ( N \ Y ), one obtains the exact sequence(3.6) · · · δ → H ∗ ( T ( ξ ) , ∗ ) j ∗ Y → H ∗ ( N ) → H ∗ ( N \ Y ) δ → · · · .7t can be shown that (see [21, Theorem 11.3]) Lemma 3.3.
With respect to the presentation (3.1) of the ring H ∗ ( T ( ξ )) ,the map j ∗ Y in (3.6) has the following properties:i) i ∗ Y ◦ j ∗ Y ( x · u ξ ) = e ( ξ ) ∪ x , x ∈ H ∗ ( Y ); ii) j ∗ Y ( x · u ξ ) ∪ y = j ∗ Y (( x ∪ i ∗ Y y ) · u ξ ), x ∈ H ∗ ( Y ), y ∈ H ∗ ( N ). In addition, if N is closed and oriented, then the class j ∗ Y ( u ξ ) ∈ H k ( N ) is the Poincar`e dual of the oriented cycle class i Y ∗ [ Y ] ∈ H ∗ ( N ) . (cid:3) Suppose that we are given an exact ladder of abelian groups · · · → A → A → A → A → · · · i ↓∼ = i ↓ i ↓ i ↓∼ = · · · → B → B β → B → B → · · · .in which the vertical maps i and i are isomorphic. We shall need thefollowing result from homological algebra. Lemma 3.4.
If the map i is monomorphic, then i is monomorphic.In addtion, if the short exact sequence → A → B → B / Im i → is splitable, theni) the sequence → A → B → B / Im i → is splitable,ii) the map β induces an isomorphism B / Im i → B / Im i . (cid:3) For a CW –complex Y let K ( Y ) (resp. e K ( Y )) be the topological K –theory(resp. reduced K –theory) of complex vector bundles over Y . For a relative CW –pair ( Y, A ) the inclusion j : ( Y, ∅ ) → ( Y ; A ) induces a homomorphism(3.7) j ∗ : K ( Y ; A ) → e K ( Y ),where K ( Y ; A ) is the relative K –group defined by(3.8) K ( Y ; A ) =: e K ( Y /A ).Alternatively, the group K ( Y ; A ) admits the following characterization. Lemma 3.5 ([1, 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 one has ) 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 ] , γ ∈ K ( X ) ;iv) j ∗ [ ξ, η ; α ] = ξ − η ,where ⊕ (resp. ⊗ ) denotes direct sum (resp. tensor product) of vectorbundles (resp. homomorphisms). (cid:3) Let BU be the classifying space of the stable equivalent classes of complexvector bundles, and let c r ∈ H r ( BU ) be the r th Chern class of the universalbundle on BU . Then H ∗ ( BU ) = Z [ c , c , · · · ]. For a CW –complex Y let[ Y, BU ] be the set of homotopy classes of maps from Y to BU . In view ofthe canonical identification e K ( Y ) = [ Y, BU ] ([26, p.210]) the total Chernclass is seen to be the co–functor C : e K ( Y ) → H ∗ ( Y ) defined by C ( ξ ) = 1 + f ∗ c + f ∗ c + · · · , where f ∈ [ Y, BU ] is the classifying map of the element ξ ∈ e K ( Y ). Clearlyone has Lemma 3.6.
The transformation C satisfies the next two properties.i) If ξ i , i = 1 , , are two complex vector bundles over Y with equaldimension, then C ( ξ − ξ ) = C ( ξ ) C ( ξ ) − .ii) For a relative CW –pair ( Y, A ) let j A : ( Y, ∅ ) → ( Y, A ) and h A : Y → Y /A be the inclusion and quotient maps, respectively. Then the nextdiagram commutes: K ( Y ; A ) = e K ( Y /A ) j ∗ A → e K ( Y ) C ↓ ↓ CH ∗ ( Y /A ) h ∗ A → H ∗ ( Y ) . (cid:3) We conclude this section with some computational aspects of Chernclasses. For an m –dimensional complex vector bundle ξ over CW –complex Y with total Chern class C ( ξ ) = 1 + c + · · · + c m ∈ H ∗ ( Y )let π ξ : P ( ξ ) → Y be the associated projective bundle. The tautological linebundle on P ( ξ ) is denoted by λ ξ . We shall set t = e ( λ ξ ) ∈ H ( P ( ξ )) with λ ξ the complex conjugation of λ ξ [21, p.167]. Lemma 3.7. H ∗ ( P ( ξ )) = H ∗ ( Y ) · { , t, · · · , t m − } subject to the relation m + c · t m − + · · · + c m − · t + c m = 0. (cid:3) Assume from now on that λ and ξ are two complex Euclidean bundlesover Y with dim λ = 1, dim ξ = m , and with the total Chern classes C ( λ ) = 1 + t and C ( ξ ) = 1 + c ( ξ ) + · · · + c m ( ξ ),respectively. Lemma 3.8. C ( λ ⊗ ξ ) = X ≤ r ≤ m (1 + t ) m − r c r ( ξ ). Proof.
By the splitting principle we can assume that C ( ξ ) = Q ≤ i ≤ m (1 + s i )where s , · · · , s m are the Chern roots of ξ . The lemma is shown by thecalculation 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 ]. (cid:3) Let p λ : D ( λ ) → Y be the disk bundle of λ . Along the subspace S ( λ ) ⊂ D ( λ ) the induced bundle p ∗ λ ( λ ) has the trivialization ε : p ∗ λ ( λ ) | S ( λ ) = R ( α ) ⊕ R ( α ) → C ,where α is the unit tangent field along the fiber circles, and α is the conju-gation of the field α . By Lemma 3.5 it defines an element(3.9) [ p ∗ λ ( λ ) , C ; ε ] ∈ K ( D ( λ ) , S ( λ )) = e K ( T ( λ )) (by (3.8)).Recall by Lemma 3.1 that H ∗ ( T ( λ )) = Z ⊕ H ∗ ( Y ) { u λ } with u λ ∈ H ( T ( λ ))the Thom class. Lemma 3.9.
In the ring H ∗ ( T ( λ )) one has C ([ p ∗ λ ( λ ) , C ; ε ] ⊗ p ∗ λ ξ ) = ( X ≤ r ≤ m (1 − u λ ) m − r c r ) C ( ξ ) − . Proof.
By the partition S ( λ ⊕ R ) = D + ( λ ) ∪ D − ( λ ) in (3.2) and using ε as a clutching function, one defines the line bundle λ u on S ( λ ⊕ R ) by λ u = p ∗ λ | D + ( λ ) ∪ ε C | D − ( λ ).10hen C ( λ u ) = 1 − u with respect to the formula (3.4) of H ∗ ( S ( λ ⊕ R )).On the other hand, under the excision isomorphism K ( S ( λ ⊕ R ) , D − ( λ )) ∼ = K ( D ( λ ) , S ( λ )) (by (3.3))the element [ λ u , C ; ε ] ⊗ p ∗ λ ξ corresponds to [ p ∗ λ ( λ ) , C ; ε ] ⊗ p ∗ λ ξ , which is alsomapped to the element λ u ⊗ p ∗ ξ − p ∗ ξ ∈ e K ( S ( λ ⊕ R ))under the induced map of the map j : ( S ( λ ⊕ R ) , ∅ ) → ( S ( λ ⊕ R ) , D − ( λ ))by iv) of Lemma 3.5. It follows that C ◦ j ∗ ([ λ u , C ; ε ] ⊗ p ∗ ξ ) = C ( λ u ⊗ p ∗ ξ ) C ( p ∗ ξ ) − (by i) of Lemma 3.6)= ( X ≤ r ≤ m (1 − u ) m − r c r ) C ( ξ ) − (by Lemma 3.8).By ii) of Lemma 3.6 we get in H ∗ ( S ( λ ⊕ R )) the relation h ∗ λ C ([ p ∗ λ ( λ ) , C ; ε ] ⊗ p ∗ λ ξ ) = ( X ≤ r ≤ m (1 − u ) m − r c r ) C ( ξ ) − .The lemma is shown by h ∗ λ ( u λ ) = u and by the injectivity of h ∗ λ (see in theproof of Lemma 3.1). (cid:3) For an almost complex manifold M write C ( M ) for the total Chern classof its tangent bundle τ ( M ). Note that if the base space Y of ξ is an almostcomplex manifold, then P ( ξ ) is canonically an almost complex manifold with(3.10) τ ( P ( ξ )) = π ∗ ξ τ ( Y ) ⊕ Hom ( λ ξ ⊗ λ ⊥ ξ ) = π ∗ ξ τ ( Y ) ⊕ λ ξ ⊗ λ ⊥ ξ . Lemma 3.10. C ( P ( ξ )) = C ( Y ) · ( X ≤ r ≤ m (1 + t ) r c m − r ) . Proof.
Let 1 C be the 1–dimensional trivial bundle on P ( ξ ). The standardbundle isomorphisms λ ξ ⊗ λ ⊥ ξ ⊕ C = λ ξ ⊗ λ ⊥ ξ ⊕ λ ξ ⊗ λ ξ = λ ξ ⊗ π ∗ ξ ξ implies that C ( λ ξ ⊗ λ ⊥ ξ ) = C ( λ ξ ⊗ π ∗ ξ ξ ). The proof is done by Lemma 3.8and the relation by (3.10) C ( P ( ξ )) = C ( Y ) · C ( λ ξ ⊗ λ ⊥ ξ ). (cid:3) The topological invariants of a blow–up
Carrying on discussion of Section 2 we present in this section formulae forthe invariants H ∗ ( f M ) , τ ( f M ) and C ( f M ) of a blow–up f M . Assume thereforethat i X : X → M is an embedding whose normal bundle γ X has a complexstructure with total Chern class C ( γ X ) = 1 + c + · · · + c k ∈ H ∗ ( X ), k = dim C γ X .Let π : E = P ( γ X ) → X be the associated projective bundle of γ X . Let λ E be the tautological line bundle on E . Denote by λ ⊥ E the complement of thesubbundle λ E ⊂ π ∗ γ X .Let f : f M → M be the blow–up of M along X with exceptional divisor i E : E → f M . The normal maps of the embeddings X ⊂ M and E ⊂ f M willbe denoted, respectively, by j X : M → T ( γ X ) and j E : f M → T ( λ E ). ByLemma 3.1 (and Lemma 3.7) one has(4.1) H ∗ ( T ( γ X ) , ∗ ) = H ∗ ( X ) · { u X } H ∗ ( T ( λ E ) , ∗ ) = H ∗ ( E ) · { u E } (= H ∗ ( X ) · { u E , tu E , · · · , t k − u E } ),where u X ∈ H k ( T ( γ X )) and u E ∈ H ( T ( λ E )) are abbreviations of theThom classes u γ X and u λ E , respectively, and where t = e ( λ E ) ∈ H ( E ). H ∗ ( f M ) In view of the formula H ∗ ( E ) = H ∗ ( X ) · { , t, · · · , t k − } by Lemma 3.7 wecan form the quotient group H ∗ ( E ) = H ∗ ( E ) /H ∗ ( X ) · i ∗ E : H ∗ ( f M ) i ∗ E → H ∗ ( E ) → H ∗ ( E ) = H ∗ ( X ) · { t, · · · , t k − } .Set ω X := j ∗ X ( u X ) ∈ H ∗ ( M ), ω E := j ∗ E ( u E ) ∈ H ∗ ( f M ).We note by Lemma 3.3 that, if M is closed and oriented, then ω X (resp. ω E ) is the Poincar`e dual of the oriented cycle class i X ∗ [ X ] (resp. i E ∗ [ E ]). Theorem 4.1.
The maps f ∗ and i ∗ E fit into the short exact sequence (4.2) 0 → H ∗ ( M ) f ∗ → H ∗ ( f M ) i ∗ E → H ∗ ( E ) → that admits a split homomorphism j : H ∗ ( E ) → H ∗ ( f M ) given by j ( x · t r ) := − j ∗ E ( x · u E t r − ), x ∈ H ∗ ( X ), 1 ≤ r ≤ k − he ring H ∗ ( f M ) has the additive presentation (4.3) H ∗ ( f M ) = H ∗ ( M ) ⊕ H ∗ ( X ) · { t, · · · , t k − } , k = dim R γ X that is subject to the following two relations:i) ω X + Σ ≤ r ≤ k c k − r · t r = 0 ;ii) y ∪ t = i ∗ X ( y ) · t , y ∈ H ∗ ( M ) . Remark 4.2.
Combining the presentation (4.3) with the relations i) and ii)determines the ring structure on H ∗ ( f M ) completely. Indeed, granted withthe fact that H ∗ ( M ) ⊂ H ∗ ( f M ) is a subring via the map f ∗ , the relationsi) and ii) suffice to reduce, respectively, the products of elements in thesecond summand, and the products between elements in the first and secondsummands, as elements in the decomposition (4.3).For the blow–ups of complex manifolds Griffiths and Harris obtainedthe decomposition (4.3) in [11, p.605]. For cohomology with real coefficientsMcDuff obtained the decomposition (4.3) in [20, Proposition 2.4]. In ourcontext the manifold M is in the real and smooth category, which is notassumed to be compact or orientable, and may be of odd dimensional. (cid:3) Proof.
We organize the proof in view of the exact ladder:(4.4) → H r − ( M \ X ) → H r ( T ( γ X )) j ∗ X → H r ( M ) → H r ( M \ X ) →k T ( f ) ∗ ↓ f ∗ ↓ k→ H r − ( f M \ E ) → H r ( T ( λ E )) j ∗ E → H r ( f M ) → H r ( f M \ E ) → induced by the map of manifold pairs (see (3.6)) f : ( f M ; f M \ D ( λ E )) → ( M ; M \ D ( γ X ))where the vertical identification H ∗ ( M \ X ) = H ∗ ( f M \ E ) comes from prop-erty iii) of Lemma 2.2.By Lemma 3.2 and in term of (4.1) the map T ( f ) ∗ is given by(4.5) T ( f ) ∗ ( x · u X ) = ( x ∪ e ( λ ⊥ E )) · u E , x ∈ H ∗ ( X ).From π ∗ γ X = λ E ⊕ λ ⊥ E with C ( γ X ) = 1 + c + · · · + c k and C ( λ E ) = 1 − t one finds the following expression of the Euler class e ( λ ⊥ E ) ∈ H ∗ ( E )(4.6) e ( λ ⊥ E ) = c k − · c k − · t + · · · + 1 · t k − with e ( λ ⊥ E ) ∪ t = − c k .13n addition, by i) of Lemma 3.3 the composition j E ◦ i E : E → T ( λ E ) satisfies(4.7) i ∗ E ◦ j ∗ E ( x · t r u E ) = − x · t r +1 , x ∈ H ∗ ( X ) , r ≥ T ( f ) ∗ is monomorphic with cokernel H ∗ ( T ( λ E )) / Im T ( f ) ∗ = H ∗ ( X ) · { u E , u E t, · · · , u E t k − } .Consequently, the corresponding short exact sequence0 → H ∗ ( T ( γ X )) T ( f ) ∗ → H ∗ ( T ( λ E )) → H ∗ ( T ( λ E )) / Im T ( f ) ∗ → H ∗ ( T ( λ E )) = H ∗ ( T ( γ X )) ⊕ H ∗ ( X ) · { u E , u E t, · · · , u E t k − } .By Lemma 3.4 the map f ∗ is injective and induces the decomposition H ∗ ( f M ) = f ∗ H ∗ ( M ) ⊕ j ∗ E ( H ∗ ( X ) · { u E , u E t, · · · , u E t k − } ).Moreover, with respect to this presentation the additive map i ∗ E annihi-lates the first summand by ii) of Lemma 2.2, and carries the second sum-mand isomorphically onto H ∗ ( E ) by (4.7). This shows the first statementof Theorem 4.1, as well as the decomposition (4.3).For the relations i) and ii) we notice that the following equalitiesa) f ∗ ◦ j ∗ X ( x · u X ) = j ∗ E (( x ∪ e ( λ ⊥ E ) · u E ), x ∈ H ∗ ( X );b) f ∗ ( y ) ∪ j ∗ E ( e · u E ) = j ∗ E ( i ∗ X ( y ) ∪ e · u E ), y ∈ H ∗ ( M ); e ∈ H ∗ ( E );hold in the ring H ∗ ( f M ). Indeed, a) comes directly from the commutivity ofthe second diagram in (4.4), as well as the formula (4.5), while b) is verifiedby the following calculation: f ∗ ( y ) ∪ j ∗ E ( e · u E ) = j ∗ E ( i ∗ E ( f ∗ ( y )) ∪ e · u E ) (by ii) of Lemma 3.2)= j ∗ E ( i ∗ X ( y ) ∪ e · u E ) (since i ∗ E ◦ f ∗ = π ∗ ◦ i ∗ X by ii) of Lemma 2.2)Finally, taking x = 1 in a) (resp. e = 1 in b)) shows i) (resp. ii)). (cid:3) τ ( f M ) ∈ K ( f M ) According to Theorem 2.4, if i X : X → M is an embedding of almostcomplex manifold, then the blow–up f M has a canonical almost complexstructure. Moreover, since both M and f M admit smooth triangulations [21,p.240] one has τ ( f M ), f ∗ τ M ∈ K ( f M ). Our formula for τ ( f M ) shall make useof the element [ p ∗ E λ E , C ; ε ] ∈ K ( D ( λ E ) , S ( λ E )) specified by (3.9), as wellas the composition 14 E : K ( D ( λ E ) , S ( λ E )) e − → ∼ = K ( f M , f M r ◦ D ( λ E )) j ∗ → e K ( f M )in which e is the excision isomorphism in K –theory. Theorem 4.3.
In the ring e K ( f M ) one has (4.8) τ f M − f ∗ τ M = j E ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ ⊥ E ). Proof.
By Lemma 3.5 the partition (2.1) on f M gives rise to the element[ τ f M , f ∗ τ M ; id ] ∈ K ( f M ; f M r ◦ D ( λ E ))that satisfies also the relation e ([ τ f M , f ∗ τ M ; id ]) = [ τ D ( λ E ) , f ∗ τ D ( γ X ) ; κ ],where κ is the bundle isomorphism specified in Theorem 2.3. Granted withthe formulae (2.4) and (2.5) of the bundles τ D ( λ E ) and f ∗ τ D ( γ X ) , as well asTheorem 2.3, one computes[ 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 3.5)= [ p E ∗ Hom ( λ E , λ ⊥ E ) , p E ∗ λ ⊥ E ; κ ′ ] (by i) of Lemma 3.5)= [ p E ∗ ( λ E ⊗ λ ⊥ E ) , p E ∗ λ ⊥ E ; κ ′ ] (by Hom ( λ E , λ ⊥ E ) = λ E ⊗ λ ⊥ E )= [ p E ∗ λ E , C ; ε ] ⊗ p E ∗ λ ⊥ E (by iii) of Lemma 3.5)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, we get the relation e ([ τ f M , f ∗ τ M ; id ]) = [ p E ∗ λ E , C ; ε ] ⊗ p E ∗ λ ⊥ E ∈ K ( D ( λ E ) , S ( λ E )).Applying j E to both sides yields the formula (4.8) by iv) of Lemma 3.5. (cid:3) C ( f M ) ∈ H ∗ ( f M ) Assume that i X : X → M is an embedding of almost complex manifold.Combining formula (4.8) with the formulae by Lemma 3.9(4.9) C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E γ X ) = ( X ≤ r ≤ k (1 − u E ) k − r c r ) C ( γ X ) − C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ E ) = (1 − u E − t )(1 − t ) −
15e show that
Theorem 4.4.
With respect to decomposition of the ring H ∗ ( f M ) in (4.3),the total Chern class C ( f M ) of the blow up f M is (4.10) C ( f M ) = C ( M ) + C ( X )( X ≤ r ≤ k (1 + t ) k − r c r )(1 − t ) − X ≤ r ≤ k c r ). (cid:3) Proof.
By ii) of Lemma 3.6 the formula (4.8) implies that C ( f M ) ∪ f ∗ C ( M ) − = j ∗ E ( C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ ⊥ E )).Equivalently, one has in the reduced cohomology H ∗ ( f M , ∗ ) that C ( f M ) ∪ f ∗ C ( M ) − − j ∗ E ( g E − g E := C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ ⊥ E ) ∈ H ∗ ( T ( λ E )). It implies that C ( f M ) − f ∗ C ( M ) = f ∗ C ( M ) ∪ j ∗ E ( g E − j ∗ E ( i ∗ X C ( M ) ∪ ( g E − j ∗ E ( C ( X ) ∪ C ( γ X ) ∪ ( g E − τ ( M ) | X = τ ( X ) ⊕ γ X ).Therefore, with respect to the decomposition (4.3),(4.11) C ( f M ) = C ( M ) + i ∗ E ◦ j ∗ E ( C ( X ) ∪ C ( γ X ) ∪ ( g E − g E = C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ ⊥ E )= C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E γ X ) C ([ p ∗ E λ E , C ; ε ] ⊗ p ∗ E λ E ) (by p ∗ E λ ⊥ E = p ∗ E γ X − p ∗ E λ E )= X (1 − u E ) k − r c r ) C ( γ X ) − (1 − t )(1 − u E − t ) − (by (4.9))and since the composition i ∗ E ◦ j ∗ E : H ∗ ( T ( λ E )) → H ∗ ( E )is a H ∗ ( X ) module map with i ∗ E ◦ j ∗ E ( u E ) = − t . (cid:3) Remark 4.5.
Formula (4.10) expresses the Chern class C ( f M ) by the initialdata of the blow–up. As examples we have c ( f M ) = c ( M ) + ( k − t ; c ( f M ) = c ( M ) + ( k − tc ( X ) + ( (cid:0) k (cid:1) − k ) t +( k − tc ( γ X ); c ( f M ) = c ( M ) + ( k − tc ( X ) + (( (cid:0) k (cid:1) − k ) t +( k − tc ( γ X )) c ( X )+( (cid:0) k (cid:1) − (cid:0) k (cid:1) ) t +( (cid:0) k − (cid:1) − ( k − t c ( γ X ) + ( k − tc ( γ X ).16n algebraic geometry the formula for the Chern class of a blow–up ofa nonsingular variety was first conjectured by J. A. Todd and B. Segre[29, 24], confirmed by I. R. Porteous and Lascu–Scott [22, 18, 19]. It hasbeen generalized to the blow ups of possibly singular varieties along regularlyembedded centers by Aluffi [2]. Recently H. Geiges and F. Pasquotto [13]extended the formula to the blow–ups of symplectic and complex manifolds.Theorem 4.4 is applicable to the general situation of blow ups of almostcomplex manifolds. The 6–dimensional sphere S has a canonical almostcomplex structure with total Chern class 1 + 2 y , where y ∈ H ( S ) is anorientation class. The blow–up of S at a point X ∈ S is diffeomorphic tothe complex projective 3–space P , together with an induced almost complexstructure J by Theorem 2.4. By Theorem 4.4 we have C ( P , J ) = 1 + 2 t + 2 y − t = 1 + 2 t − t ,where t = e ( λ E ) ∈ H ( E ) with E ⊂ P the exceptional divisor, and wherethe second equality comes from the relation y = − t by i) of Theorem4.1. This computation shows that J is different with the standard complexstructure on P . (cid:3) Granted with Theorems 4.1 and 4.4, we compute the cohomologies of thevarieties of complete conics and quadrics on the 3–space P . They are appliedto justify the two enumerative results stated in Section 1.Let P n be the projective space of lines through the origin in C n +1 . Thecanonical Hopf complex line bundle on P n is denoted by λ n . P Let α be the orthogonal complement of the subbundle λ ⊂ P × C , and let Sym ( α ) ⊂ α ⊗ α be its symmetric product. Consider the projective bundle P ( Sym ( α )) → P associated to the complex vector bundle Sym ( α ) with dimension 6. Eachpoint in P ( Sym ( α )) consists of a pair ( l, v ) in which l ∈ P , and v is aquadratic form on the orthogonal complement l ⊥ of l ⊂ C , hence defines aconic on P lying on the plane l ⊥ . For this reason the variety M := P ( Sym ( α ))is known as the space of conics on P .The bundle map s : α → Sym ( α ) by v → v ⊗ v satisfies that s ( λv ) = λ s ( v ), λ ∈ C , hence induces a smooth embedding175.1) i : P ( α ) → M = P ( Sym ( α )).of the associated projective bundles. Its image consists of the degenerateconics of the double lines. For this reason the blow-up f M of M along P ( α )provide us with the variety of complete conics on P .To apply Theorem 4.1 to compute the cohomology of f M we need to knowthe cohomologies H ∗ ( M ), H ∗ ( P ( α )), the Chern classes of the normal bundle γ P ( α ) , the induced map i ∗ of the embedding (5.1) on cohomology, as well asthe Poincar`e dual ω P ( α ) of cycle class i ∗ [ P ( α )] ∈ H ∗ ( M ). These informationare summarized in the following result. Lemma 5.1.
Let x := e ( λ ) ∈ H ∗ ( P ) , y := e ( λ Sym ( α ) ) ∈ H ∗ ( M ) and ρ := e ( λ α ) ∈ H ∗ ( P ( α )). Then (5.2) H ∗ ( M ) = Z [ x, y ] / (cid:10) x , y + 4 xy + 10 x y + 20 x y (cid:11) ;(5.3) H ∗ ( P ( α )) = Z [ x, ρ ] / (cid:10) x , ρ + ρ x + ρx + x (cid:11) In addition, with respect to the presentations in (5.2) and (5.3) a) i ∗ ( x ) = x , i ∗ ( y ) = 2 ρ ;b) C ( γ P ( α ) ) = 1+ (3 x + 9 ρ )+ (30 ρ + 20 xρ + 6 x )+ (32 ρ + 32 xρ + 16 x ρ );c) ω P ( α ) = 4 y + 8 xy + 8 x y Proof.
By the definition of α we have in the ring H ∗ ( P ) = Z [ x ] / (cid:10) x (cid:11) that C ( α ) = 1 + x + x + x ; C ( Sym ( α )) = 1 + 4 x + 10 x + 20 x .These show the formulae (5.2) and (5.3) by Lemma 3.7. The relation a) istransparent since i is a bundle map over the identity on the base P whoserestriction on the fiber is the Veronese embedding of P on P [11].By Lemma 3.10 the total Chern class of M and P ( α ) are C ( M ) = (1 − x ) ((1 − y ) +4 x (1 − y ) +10 x (1 − y ) +20 x (1 − y ) ) ,C ( P ( α )) = (1 − x ) ((1 − ρ ) + (1 − ρ ) x + (1 − ρ ) x + x ).One obtains b) from C ( γ P ( α ) ) = i ∗ ( C ( M )) ∪ C ( P ( α )) − and a).In term of the basis { y , xy , x y, x } of H ( M ) by (5.2) assume that ω P ( α ) = ay + bxy + cx y + dx , a, b, c, d ∈ Z .Since the restriction of ω P ( α ) to the fiber P is equal to ω P = 4 ω ∈ H ( P )(the Poincar`e dual of the Veronese surface on P ) we have a = 4. It followsthen from i ∗ ( ω P ( α ) ) = c ( γ P ( α ) ) by (3.2) that18 ∗ ( ω P ( α ) ) = 32 ρ + 4 bxρ + 2 bx ρ + dx = (32 − d ) ρ + (4 b − d ) xρ + (2 c − d ) x ρ = 32 ρ + 32 xρ + 16 x ρ ,where the second equation comes from the relation x = − ( ρ + xρ + x ρ )on P ( α ) by (5.1). Coefficients comparison in the third equality tells that d = 0, b = c = 8. This shows the formula c). (cid:3) Let E ⊂ f M be the exceptional divisor and set z = e ( λ E ) ∈ H ( E ). Interm of Theorem 4.1 one formulates the ring H ∗ ( f M ) from Lemma 5.1. Theorem 5.2.
The cohomology of the variety of complete conics on P is (5.6) H ∗ ( f M ) = Z [ x,y ] h x ; y +4 xy +10 x y +20 x y i ⊕ Z [ x,ρ ] h x ,ρ + ρ x + ρx + x i { z, z } that is subject to the following two relationsi) y + 8 xy + 8 x y = − (30 ρ + 20 ρx + 6 x ) z − (3 x + 9 ρ ) z − z . ii) yz = 2 ρz . (cid:3) Over the field R of reals the monomials ρ r z can be replaced by r y r z bythe relation ii). Therefore, Theorem 5.2 implies the following presentationof the cohomology with real coefficients. Corollary 5.3. H ∗ ( f M ; R ) = R [ x, y, z ] / (cid:10) x ; g , g , g (cid:11) , where g = 2 z + (6 x + 9 y ) z + (15 y + 20 xy + 12 x ) z + 8 y + 16 xy +16 x y ; g = ( y + 2 xy + 4 x y + 8 x ) z ; g = y + 4 xy + 10 x y + 20 x y . (cid:3) The map s : C × C → Sym ( C ) ⊂ C ⊗ C by s ( u, v ) = u ⊗ v satisfiesthe relations s ( λu, v ) = s ( u, λv ) = λs ( u, v ), λ ∈ C , hence gives rise to thesmooth map on the quotients ϕ : P × P → P = P ( Sym ( C )).Clearly, ϕ restricts to an embedding on the diagonal ∆ = P ⊂ P × P , andis 2 to 1 on the complement P × P \ ∆. In what follows we set X = Im ϕ | ∆, X = Im ϕ ⊂ P .19eometrically, the manifold P = P ( Sym ( C )) is the space of quadrics on P ; the map ϕ is ϕ ( l , l ) = L ∪ L , ( l , l ) ∈ P × P ,with L i ⊂ P the hyperplane perpendicular to the line l i ∈ P , i = 1 , X r ⊂ P consists of the degenerate quadrics with rank ≤ r , r = 1 , e P be the blow–up of P along X with exceptional divisor E = P ( γ X ), and let X ⊂ e P be the strict transformation of X in e P . Theblow–up e N of e P along X is the variety of complete quadrics on P .The invariants of the first blow up e P can be easily calculated. Theorem 5.4.
Let u = e ( λ ) ∈ H ∗ ( P ), x = e ( λ ) ∈ H ∗ ( P ) and v = e ( λ E ) ∈ H ∗ ( E ) . Then (5.7) H ∗ ( e P ) = Z [ u ] / (cid:10) u (cid:11) ⊕ Z [ x ] / (cid:10) x (cid:11) { v, v , · · · , v } with v + 16 v x + 110 v x + 420 v x + 8 u = 0; uv = 2 xv .(5.8) C ( e P ) = 1 + (10 u + 5 v ) + (45 u + 42 uv + 9 v )+(120 u + 154 u v + 58 uv + 5 v ) + · · · where in the formula (5.8), the terms with order ≥ are not needed insequel, hence can be omitted. Proof.
In views of H ∗ ( P ) = Z [ x ] / (cid:10) x (cid:11) and H ∗ ( P ) = Z [ u ] / (cid:10) x (cid:11) one has i ∗ X ( u ) = 2 x ; C ( P ) = (1 + x ) , C ( P ) = (1 + u ) .From C ( γ X ) = i ∗ X C ( P ) ∪ C ( P ) − one gets that C ( γ X ) = 1 + 16 x + 110 x +420 x .The formulae (5.7) and (5.8) are shown by Theorems 4.1 and 4.5. (cid:3) Let G , be the Grassmannian of 2–planes on C , and let η be the canon-ical 2–plane bundle on G , . The projective bundle associated to the sym-metric product Sym ( η ) ⊂ η ⊗ η is denoted by P ( Sym ( η )) → G , .In view of the obvious characterization { ( l , l ) ∈ P × P | l ⊥ l } of themanifold P ( α ) in (5.1) one has the embedding20 : P ( α ) → P ( Sym ( η )) by g ( l , l ) = ( h l , l i , l ),where h l , l i ⊂ C is the 2–plane spanned by the orthonormal lines l , l ,and the line l is viewed as a degenerate conic of rank 1 on the plane h l , l i .Let i X : X → e P be the strict transformation of X ⊂ P in e P . It hasessentially been shown by Vainsencher [31] thata) the manifold X is diffeomorphic to P ( Sym ( η ));b) there is a diffeomorphism G : P ( Sym ( α )) → E ( ⊂ e P ) over theidentity of the base space P so that the following diagram commutes(5.9) P ( α ) i → P ( Sym ( α )) g ↓ ↓ GX = P ( Sym ( η )) i X → e P ,where i is the map given in Section 5.1. Let ω X ∈ H ( e P ) be the Poincar`edual of the cycle class i X ∗ [ X ] ∈ H ( e P ). Lemma 5.5.
The cohomology H ∗ ( X ) has the presentation (5.10) H ∗ ( X ) = Z [ c ,c ,t ] h c c − c ; c − c c ; t +3 t c + t (2 c +4 c )+2 c i , t = c ( λ Sym ( η ) ) , with respect to iti) the induced map i ∗ X is given by i ∗ X ( u ) = t , i ∗ X ( v ) = − c + t ) ;ii) C ( γ X ) = 1 − (9 c + 3 t ) + (30 c + 18 c t + 3 t − c ) − (32 c + 32 c t + 12 c t + 2 t ) ;iii) ω X = 10 u + 22 u v + 16 uv + 4 v . Proof.
With X = P ( Sym ( η )) the formula (5.10) comes from Lemma 3.7,together with the computation H ∗ ( G , ) = Z [ c ,c ] h c c − c ; c − c c i ; C ( Sym ( η )) = 1 + 3 c + (2 c + 4 c ) + 2 c ∈ H ( G , ).In addition, one gets from Lemma 3.10 that(5.11) C ( X ) = 1 + ( − c + 3 t ) + ( − c + 3 t + 4 c − c t )+( c − c t + 4 c t − c t + t ) + · · · It remains for us to show properties i), ii), iii).The proof of i) uses the commutivity of the diagram (5.9) in which thecohomologies of the four spaces P ( α ) , P ( Sym ( α )), e P and X involved areall known, see (5.2), (5.3), (5.7), (5.10). Based on the presentation (5.7) and(5.2) the method illustrated in [17] is applicable to show that, there is onlyone ring isomorphism 21 : H ∗ ( E ) → H ∗ ( P ( Sym ( α ))) with f ( x ) = x ,and that satisfies that f ( v ) = − x + y . It implies that(5.12) G ∗ ( u ) = 2 x ; G ∗ ( v ) = − x + y .By the definition of g one gets that(5.13) g ∗ ( c ) = − ( x + ρ ), g ∗ ( c ) = xρ , g ∗ ( t ) = 2 x .Granted with (5.12), (5.13), together with a) of Lemma 5.1, one obtains i)from the relation i ∗ ◦ G ∗ = g ∗ ◦ i ∗ X by (5.9).By the formulae for C ( e P ) and C ( X ) in (5.8) and (5.11) one obtains ii)from the relation C ( γ X ) = i ∗ X C ( e P ) ∪ C ( X ) − .Finally, in view of the group H ( e P ) given in (5.7) we can assume that ω X = au + by v + cyv + dv , a, b, c, d ∈ Z .From the formula in i) of Lemma 3.3 one gets that i ∗ X ω X = c ( γ X ) = − (32 c + 32 c t + 12 c t + 2 t )Coefficients comparison then yields that a = 10, b = 88, c = 32, d = 4. Itemiii) is verified by the relations 4 y v = u v , 2 yv = uv on H ∗ ( e P ) in (5.7). (cid:3) Let E ⊂ e N be the exceptional divisor corresponding to X ⊂ e P , and set w = e ( λ E ). Combining Lemmas 5.4 and 5.5 with Theorem 4.1 we get Theorem 5.6.
Let e N be the variety of complete quadrics on P . Then (5.16) H ∗ ( e N ) = Z [ u ] / (cid:10) u (cid:11) ⊕ Z [ y ] / (cid:10) y (cid:11) { v, v , · · · , v }⊕ Z [ c ,c ,t ] h c c − c ; c − c c ; t +3 t c + t (2 c +4 c )+2 c i { w, w } that is subject to the following relationsi) v + 16 v y + 110 v y + 420 v y + 8 u = 0; ii) uv = 2 yv ; uw = tw , vw = − c + t ) w ; iii) u + 22 u v + 16 uv + 4 v = − (30 c + 18 c t + 3 t − c ) w +(9 c + 3 t ) w − w . (cid:3) Corollary 5.7. H ∗ ( e N ; R ) = R [ u, v, w ] / h g , , g , , g , , g , , g i with , = − u − u v − u v − uv + 2(2 u + v ) h − u + v ) h +2(2 u + v ) h ; g , = 8 u + 4 u v − u v − uv − v − (16 u + 14 u v − v +2 uv ) h + 6(2 u + uv ) h − uh ; g , = (2 w − u − v ) h ; g , = u v ; g = 16 u + 105 u v + 55 u v + 16 uv + 2 v , where h = 3 u + 2 v + w . Proof.
By the relations in ii) of Theorem 5.6 and with the real field R ascoefficients of cohomology one has y r v = r u r v ; t r w = u r w , c r w = ( − r ( v + u ) r w and that, by the relation iii) of Theorem 5.6, c w = ((30 c + 18 c t + 3 t ) w − (9 c + 3 t ) w + w − (10 u +22 u v + 16 uv + 4 v )) c w = 9 twc c + w c − tw c + 2 t wc + 4 wc c − w c c .These imply that the algebra H ∗ ( e N ; R ) is generated by u, v, w . The relations g , , g , , g , , g , , g in Corollary 5.7 are obtained, respectively, from thefollowing relations in Theorem 5.6:(2 c c − c ) w = 0;( t + 3 t c + t (2 c + 4 c ) + 2 c ) w = 0;( c − c c ) w = 0; y v = 0; v + 16 v y + 110 v y + 420 v y + 8 u = 0. (cid:3) The diagram (5.9), as well as the proof of Theorem 5.6, implies that
Corollary 5.8.
The variety f M of complete conics on P is the strict trans-formation of the subvariety P ( Sym ( α )) ⊂ e P in e N .The induced map of the inclusion i f M : f M → e N is given by i ∗ f M ( u ) = 2 x, i ∗ f M ( v ) = − x + y, i ∗ f M ( w ) = z . (cid:3) .3 The problem of characteristics ([23, Chapter 6], [32]) We shall assume the reader’s familiarity with the cup product approach tothe intersection theory of submanifolds [12, § r smooth subvarieties N i in asmooth projective variety M that satisfy the dimension constraintΣ dim N i = ( r −
1) dim M ,let I ( M ; N , · · · , N r ) be the cardinality of the set ∩ ≤ i ≤ r N i of intersectionpoints, counted with multiplicities. A fundamental concern of projectivegeometry is Problem 5.9.
Find the number I ( M ; N , · · · , N r ) when the embeddings N i ⊂ M are in general position. Assume that the real cohomology H ∗ ( M ; R ) of the ambient space M hasbeen presented as a quotient of a free polynomial algebra R [ x , · · · , x k ] as(5.17) H ∗ ( M ; R ) = R [ x , · · · , x k ] / h g , · · · , g m i , g i ∈ R [ x , · · · , x k ],where h g , · · · , g m i is the ideal generated by a set { g , · · · , g m } of homoge-neous polynomials. Assume that n = dim R M , [ M ] ∈ H n ( M ) is the orienta-tion class, and that R [ x , · · · , x k ] n ⊂ R [ x , · · · , x k ] is the subspace spannedby the set of monomials in x , · · · , x k with degree n . The characteristic of M with respect to the generating set { x , · · · , x k } is the linear map R M : R [ x , · · · , x k ] n → R by R M h := h h, [ M ] i ,where h , i is the Kronecker pairing. Then a solution to Problem 5.9 is(5.18) I ( M ; N , · · · , N r ) = R M α · · · · · α r (e.g. [32], [12, § α i ∈ R [ x , · · · , x k ] is a representative of the Poincar`e dual of the cycleclass [ N i ] ∈ H ∗ ( M ). By (5.18) effective computability of the characteristicsis of fundamental importance to the intersection theory [9, 23, 32].We emphasis at this point that once the set { g , · · · , g m } of relations in(5.17) is made explicit, the problem of evaluating the function R M can beeasily mechanized by certain build-in functions of Mathematica . Let G be aGr¨obner basis of the ideal generated by the set { g , · · · , g m } of polynomials.Take a polynomial h ∈ R [ x , · · · , x k ] n with R M h = 1 as a reference . Algorithm 5.10:
Characteristic tep 1 . Call GroebnerBasis[ , ] to compute G from the set { g , · · · , g m } of relations; Step 2.
For a h ∈ R [ x , · · · , x k ] n call PolynomialReduce[ , ] tocompute the residue h ( a ) of the difference h − a · h module G with a ∈ R an indeterminacy; Step 3. R M h := a , where a is the solution to h ( a ) = 0. (cid:3) Note that the geometric fact H n ( M ) = R implies that the residue h ( a )obtained in step 2 is always linear in a . Example 5.11.
Let f M be the variety of complete conics on P . Withrespect to the presentation of the algebra H ∗ ( f M ; R ) in Corollary 5.3 all thecharacteristic numbers R f M x r y s z t , r + s + t = 8, generated by Character-istic are tabulated below (with the symbol R f M being omitted), where themonomial x y is an obvious reference: x y = 1 x y = − xy = 6 y = − x y z = 0 x y z = 0 xy z = 0 y z = 0 x y z = 0 x y z = 0 xy z = 0 y z = 0 x y z = − x y z = 8 xy z = 0 y z = 0 x yz = 18 x y z = − xy z = − y z = 0 x z = − x yz = 34 xy z = 124 y z = 24 x z = 0 xyz = − y z = − xz = 890 yz = 620 z = − Example 5.12.
Let e N the variety of complete quadrics. With respect tothe presentation of H ∗ ( e N ; R ) in Corollary 5.8, all the characteristic numbers R e N u r v s w t with r + s + t = 9 generated by Characteristics are tabulatedbelow, where the monomial u is an obvious reference. u = 1 u w = 60 u v = 64 uv w = 760 u v = 0 u v = 0 u v w = 0 uvw = − u w = 0 u v w = 0 u v w = 0 uw = 1610 u v = 0 u v w = 0 u v w = − v = 996 u vw = 0 u v w = 0 u v w = 384 v w = 0 u w = 0 u vw = 0 u v w = − v w = 0 u v = 0 u w = − u vw = 0 v w = − u v w = 0 u v = − u w = 830 v w = 960 u vw = 0 u v w = 0 uv = − v w = − u w = − u v w = 0 uv w = 0 v w = − u v = 0 u v w = 32 uv w = 0 v w = 1160 u v w = 0 u v w = − uv w = 320 vw = 1820 u v w = 0 u vw = 408 uv w = − w = − u vw = 0 u w = − uv w = 43225 xample 5.13. For a compact connected Lie group G with a maximal torus T , presentations of the integral cohomology ring H ∗ ( G/T ) by certain Schu-bert classes on
G/T has been obtained by Duan and Zhao in [5, 6]. Basedon these presentations, the package ”Characteristic” has been applied in [7,Section 5.3] to compute the structure of the mod p cohomology H ∗ ( G ; F p )the Lie group G as a module over the Steenrod algebra A p . (cid:3) In order to avoid unnecessary repetition, calculation in this section will befacilitated with certain lower degree relations that can be found in the stan-dard reference books [11, 8]. Given a hypersurface V of a smooth projectivevariety N let { V } ∈ H ( N ) denote the Poincar`e dual of the oriented cycleclass [ V ] ∈ H ∗ ( N ).For a line l ⊂ P (resp. a plane L ⊂ P ) let V l ⊂ M = P ( Sym ( α )) (resp. V L ⊂ M ) be the hypersurface of conics meeting the line l (resp. tangent tothe plane L ). Then with respect to the presentation of the group H ( M ) in(5.2) one can show easily that { V l } = 2 x + y (resp. { V L } = 2 x + 2 y ).Moreover, let e V l (resp. e V L ) be the strict transformation of V l (resp. V L ) inthe blow-up f M of M along P ( α ). Then the calculation in [11, p.754] impliesthat(5.19) n e V l o = 2 x + y ; n e V L o = 2 x + 2 y + z in H ( f M ).On the other hand, for a generic quadric S ⊂ P let e V S ⊂ f M be thestrict transformation of the variety V S ⊂ M of conics tangent to S . It hasbeen shown in [8, p.192] that n e V S o = 2 n e V l o + 2 n e V L o (= 8 x + 6 y + 2 z ).By the Characteristic (alternatively, by Example 5.11) one gets n e V S o = (8 x + 6 y + 2 z ) = 4 , , Proposition 5.14.
Given quadrics in the space P in general position,there are , , conics tangent to all of them. (cid:3) As in Section 5.2 let e N be variety of complete quadrics on P . For apoint p ∈ P (resp. a line l ⊂ P ; a plane L ⊂ P ) let W p ⊂ e N (resp. W l , W L ⊂ e N ) be the strict transformation the subvariety on P of the quadricscontaining p (resp. tangent to the line l ; tangent to the plane L ). Clearly,in view of the presentation of the group H ( e N ) by (5.16) one has26 W p } = u in H ( e N ).Moreover, with respect to the embedding i f M : f M → e N in Corollary 5.8 onehas W l ∩ f M = e V l , W L ∩ f M = e V L .Corollary 5.8, together with (5.19), implies that { W l } = 2 u + v ; { W L } = 3 u + 2 v + w in H ( e N ).On the other hand, for a generic quadric S ⊂ P let W S ⊂ e N be thestrict transformation of the subvariety on P of the quadrics tangent to S .It was shown in [8, p.192] that { W S } = 2 { W p } + 2 { W l } + 2 { W L } (= 12 u + 6 v + 2 w ).By the Characteristic (alternatively, by Example 5.12) one gets { W S } = (12 u + 6 v + 2 w ) = 666 , , Proposition 5.15.
Given quadrics in the space P in general position,there are , , quadrics tangent to all of them. (cid:3) Remark 5.16.
In the notation µ, ν, ̺ of Schubert [23, §
20] (resp. [23, § µ = x , ν = 2 x + y , ̺ = 2 x + 2 y + z (resp. µ = u , ν = 2 u + v , ̺ = 3 u + 2 v + w ).These allow one to recover all the characteristic numbers µ r ν s ̺ t , r + s + t = 8(resp. r + s + t = 9) in Schubert ’s book [23, p.95] (resp. [23, p.105]) fromthe numbers tabulated in Example 5.11 (resp. Example 5.12). (cid:3) Motivated by the 8 conics and 9 quadrics problems the varieties of completeconics and quadrics on the 3–space P have been generalized to the varieties V n,r of complete quadric r –folds on the projective n –space P n which havebeen studied by many authors during the history, e.g [3, 8, 11, 16, 25, 27, 30,31]. To our knowledge presentations of their cohomologies remains unknowneven for the initial cases ( n, r ) = (3 , , (3 , X ⊂ M of the blow–up is ”closed”, namely, connected, compact and withoutboundaries . Indeed, the main results of Section 4 hold for the general caseswhere X is connected, and satisfies the following two conditionsi) the map i X : X → M embeds X as a closed subspace of M ,ii) if X has non–empty boundaries ∂X = ∅ , then ∂M = ∅ and X meets ∂M transversely along ∂X .For such an embedding X ⊂ M the existence theorem on tubular neighbor-hood [20, p.115] assures one with all the results of Sections 2 and 3 requiredby establishing Theorems 4.1, 4.3 and 4.4 for the blow–up f M in the generalsituation. References [1] M. Atiyah, K-theory, New York-Amsterdam: W.A. Benjamin, Inc.(1967).[2] P. Aluffi, Chern classes of blow-ups, Math. Proc. Cambridge Philos.Soc. 148 (2010), no. 2, 227–242.[3] C. De Concini, C. Procesi, Complete symmetric varieties, Invariant the-ory, Proc. 1st 1982 Sess. C.I.M.E., Lect. Notes Math. 996, 1-44 (1983).[4] H. Duan, The degree of a Schubert variety, Adv. Math., 180(2003),112-133.[5] H. Duan, Xuezhi Zhao, The Chow rings of generalized Grassmannians,Found. Math. Comput. Vol.10, no.3(2010), 245–274.[6] H. Duan, Xuezhi Zhao, Schubert presentation of the integral cohomol-ogy ring of the flag manifolds
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