Complex cobordisms and singular manifolds arising from Chern classes
aa r X i v : . [ m a t h . A T ] J u l Complex cobordisms and singular manifoldsarising from Chern classes.
A.Kustarev.November 19, 2018
Abstract
This paper deals with the question of J.Morava on existenceof canonical complex cobordism class of singular submanifold. Wepresent several solutions of this question for X r ( ξ ) – the set of pointswhere dim ξ − r + 1 generic sections of a complex vector bundle ξ are linearly dependent. The corresponding complex cobordism classes Q r ( ξ ) and P r ( ξ ) tend to have many nice properties, such as deformedsum formula, but they don’t coincide with Chern classes c Ur ( ξ ). Theyalso have relation to the theory of IH -small resolutions. A well-known question of Steenrod ([2]) asks if a given homology class insome cellular complex may be realized as an image of fundamental class ofsome manifold. As shown by R.Thom ([3]), the Steenrod problem alwayshas the positive solution in Z -homology. But if the ring of coefficients is Z ,there are counterexamples to the problem, though one can still show that anarbitrary class taken with large multiplicity may be realized as the image offundamental class of a manifold.Let Y ⊂ M be a complex semialgebraic subset, that is, locally determinedby algebraic equations, in a manifold M . (In this paper M is compact, even-dimensional, oriented manifold of real dimension 2 m with no boundary).Then the set of singular points of Y has real codimension at least two in Y ,so the homology (and dual cohomology) class [ Y ] ∈ H ∗ ( M, Z ) is well-defined. Question (J. Morava).
Does there exist a canonical complex cobor-dism class [ Y ] ∈ U ∗ ( M ) related to Y ?1he complex cobordism ring U ∗ ( M ) is generated by ”singular bordisms” –maps of smooth manifolds X → M with complex structure in stable normalbundle (for more details see [5] or [6]). Clearly one can find at least oneresolution of Y by applying Hironaka’s theorem on singular varieties, butusually it gives very uneffective solution of the problem. One has to look forefficient canonic resolutions depending on definition of Y .In the present paper we deal with the following situation. Let ξ be an n -dimensional complex vector bundle over M . One can consider the set X r ( ξ ) ⊂ M where generic n − r + 1 sections of ξ are linearly dependent.Then, according to well-known definition going to Pontryagin ([7]), we have[ X r ( ξ )] = c r ( ξ ) in integral cohomology. In this paper we give some explicitsolutions of Morava question for Y = X r ( ξ ) and investigate the correspondingcobordism classes.We begin with description of several geometric constructions of coho-mological Chern classes, based on well-known constructions in singularitytheory and Thom polynomials. Here ”geometric” means ”representing dualhomology class as an image of fundamental class of some canonically de-fined manifold”. It is known that top Chern class (Euler class) can alwaysbe represented by even an embedded submanifold (as follows from Thomtransversality theorem). Using only notions of Euler class and pushforward(Gysin) homomorphism we show that Steenrod problem for c r ( ξ ) always hasthe positive solution. (So there are examples of integer cohomology classeswhich can’t be Chern classes of any complex vector bundle).Our first result (Prop. 3.3) says that Chern classes in complex cobordisms c Ur ( ξ ) don’t solve Morava question for X r ( ξ ). (The classes c Ur ( ξ ) ∈ U r areuniquely determined by the same four axioms, see [5]). So one should look forsome other cobordism characteristic classes, which are represented by maps X → X r ( ξ ) resolving the singularities of X r ( ξ ). These classes are named by Q r ( ξ ) and P r ( ξ ), we investigate their basic properties (Th. 4.2 and 7.2) –they satisfy many Chern classes axioms, except Whitney sum formula andtriviality in non-positive dimensions.The deformation of sum formula for classes Q r ( ξ ) may be found explictly(Th. 6.3), thus giving the possibility of their purely axiomatic definition (Th.6.4).The classes Q r ( ξ ) and P r ( ξ ) are different (Th. 7.3), though in case ofsmall dimensions of M (up to dim R = 8) they coincide. If we switch toalgebraic situation ( M is a nonsingular variety, ξ is a locally free sheaf), then X r ( ξ ) is a singular variety. Our resolutions turn to be IH -small resolutionsof X r ( ξ ) (see [8]) and, as shown in that paper, their complex elliptic generaare equal. That gives restrictions on P r ( ξ ) − Q r ( ξ ).The author is grateful to his advisor, Prof. V.M.Buchstaber, for attention2o his work and to Prof. M.E.Kazarian for valuable discussions. In this section we present some new definitions of cohomological Chernclasses. These definitions use only notions of Euler class and pushforwardhomomorphism, so the Poincare duals of c r ( ξ ) are represented as the im-ages of fundamental classes of nonsingular oriented manifolds, thus givingthe solution of Steenrod problem for c r ( ξ ).Let ξ be an n -dimensional complex vector bundle over a manifold M .A well-known definition of Chern classes (going to Pontryagin, [7]) dealswith set X r ( ξ ) – a set of points on M where ( n − r + 1) generic sectionsof ξ are not of maximal rank. The set X r ( ξ ) is a complex semialgebraicset, that is, locally determined by algebraic equations, so its homology anddual cohomology classes are well-defined since all the singularities have realcodimension at least two. The cohomology class [ X r ( ξ )] is, by definition, theChern class c r ( ξ ).In the further text we’ll consider a smooth generic section s of abundle Hom( C n − r +1 , ξ ) instead of ( n − r + 1) sections of ξ . The Steenrodproblem always has positive solution for classes c r ( ξ ) – we are going topresent some explicit constructions starting from the X r ( ξ )-definition of c r ( ξ ). They may also be regarded as the alternate definitions of Chern classes. Proposition 2.1.
Let γ be a tautological bundle over projectivization M × C P n − r of a trivial ( n − r + 1) -dimensional bundle C n − r +1 and p : M × C P n − r → M – a projection map. Then r -th Chern class c r ( ξ ) ofbundle ξ equals to p ! e ( γ ∗ ⊗ p ∗ ξ ) .Proof 1. The ( n − r + 1) generic sections of ξ over M define the section s Q of bundle Hom( γ, p ∗ ξ ) over M × C P n − r by taking the composition: γ → C n − r +1 → p ∗ ξ . The set X Qr ( ξ ) of points on M × C P n − r where s Q vanishes is anonsingular submanifold by Thom transversality theorem and its cohomologyclass coincides with e (Hom( γ, p ∗ ξ )) = c n (Hom( γ, p ∗ ξ )).Note that p ( X Qr ( ξ )) = X r ( ξ ) and if rank of s drops by l in point x ∈ M ,then p − ( x ) ∩ X Qr ( ξ ) = C P l − . The generic point x ∈ X r ( ξ ) has l = 1 and p − ( x ) ∩ X Qr ( ξ ) is just a single point. So X Qr ( ξ ) is a nonsingular resolutionof X r ( ξ ) and p ∗ ([ X Qr ( ξ )]) = [ X r ( ξ )]. (cid:3) Proof 2.
We can calculate p ! e ( γ ∗ ⊗ p ∗ ξ ) explicitly using splitting principle.If t = c ( γ ∗ ) and t . . . t n are Chern roots of ξ , so c ( ξ ) = (1 + t ) . . . (1 + t n ),3hen e ( γ ∗ ⊗ p ∗ ξ ) = ( t + t ) . . . ( t + t n ). Pushforward homomorphism p ! : H ∗ ( M × C P n − r ) → H ∗ ( M ) acts by formula p ! ( t n − r ) = 1, p ! ( t i ) = 0 , i = n − r .It follows that p ! e ( γ ∗ ⊗ p ∗ ξ ) is r -th elementary symmetric polynomial in t . . . t n , which is c r ( ξ ). (cid:3) Now we define the ”dual” resolution X Pr ( ξ ) of X r ( ξ ). It corresponds tosection s ∗ : ξ ∗ → C n − r +1 dual to s : C n − r +1 → ξ . Proposition 2.2.
Let γ be a tautological r -dimensional bundle over thegrassmanization Gr r ( ξ ∗ ) of ξ ∗ and p : Gr r ( ξ ∗ ) → M – a projection map.Then the Chern class c r ( ξ ) equals to p ! e (( γ ∗ ) n − r +1 ) . The proof is analogous to the proof 1 of Prop. 2.1. The section s ∗ ofHom( ξ, C n − r +1 ) determines the section s ∗ P of Hom( γ, C n − r +1 ). If the rank of s in the point x ∈ M drops by l , then the rank of s ∗ drops by r + l − x .If x is a generic point of X r ( ξ ), the rank in x drops by r and the preimage p − ( x ) ∩ X Pr ( ξ ) is a single point. The set X Pr ( ξ ) of points on Gr r ( ξ ∗ ) where s ∗ P vanishes is a nonsingular resolution of X r ( ξ ), so the image of a fundamentalclass p ∗ ([ X Pr ( ξ )]) equals to [ X r ( ξ )]. (cid:3) If r >
1, then X Qr ( ξ ) and X Pr ( ξ ) are different resolutions of X r ( ξ ). Thepreimage of point x ∈ X r ( ξ ) where dim ker s = l is C P l − in X Qr ( ξ ) and Gr r ( C r + l − ) in X Pr ( ξ ). If r = 1, then C P l − = Gr r ( C r + l − ) but the corre-sponding resolutions are different – their complex cobordism classes differ aswe show later.One can also define the Chern classes in cohomology using the transferhomomorphism: Proposition 2.3 ([9])
Let γ be a r -dimensional tautological bundleover grassmanization Gr r ( ξ ) and t : H ∗ ( Gr r ( ξ )) → H ∗ ( M ) a transferhomomorphism. Then t ( e ( γ )) = c r ( ξ ) . Cohomological transfer homomophism t may be viewed as the compo-sition of taking product with Euler class of fiber tangent bundle τ and apushforward homomorphism. In other words, we have c r ( ξ ) = p ! ( e ( γ ) e ( τ )),so the Poincare dual of c r ( ξ ) is the image of a fundamental class of non-singular manifold – a transverse intersection of Poincare duals of e ( γ ) and e ( τ ).The classes c Ur ( ξ ) and t ( e U ( γ )) also coincide in complex cobordisms, asshown in [9]. But as we show later, there are (very simple) examples when theset X ( ξ ) is nonsingular but its cobordism class is not equal to c U ( ξ ). Hence,the construction in Prop. 2.3 does not give the resolution of singularities of X r ( ξ ) in general. 4e finish this section with just one more geometric construction, whichworks only for c ( ξ ). Proposition 2.4
Let det ξ = Λ n ξ be a determinant line bundle for ξ .Then c ( ξ ) = c (det ξ ) = e (det ξ ) . The easiest way to prove this statement is to use splitting principle: if t . . . t n are Chern roots of ξ , then c (Λ n ξ ) = c ( ξ ) = t + · · · + t n . (cid:3) Since det ξ is a linear bundle, the Poincare dual to its first Chern classmay be represented by oriented nonsingular submanifold of codimension two.We obtain another geometric realization of c ( ξ ) which has no analogue for c r ( ξ ) if r > As we said before, the class c r ( ξ ) ∈ H r ( M m , Z ) may be defined as Poincaredual to the set X r ( ξ ) where the generic section s of bundle Hom( C n − r +1 , ξ )is of non-maximal rank.In this section we consider the case when X r ( ξ ) is a nonsingular manifoldfor any generic section s . This condition always holds if n = 1, r = n (thiscorresponds to Euler class) or r = 1 and m < Proposition 3.1.
Suppose X r ( ξ ) is nonsingular. Then its normalbundle carries canonical complex structure. Let U ⊂ M be a neighbourhood of a point x ∈ X r ( ξ ) such that ξ istrivial over U . Then n − r + 1 sections of ξ over U are determined by a C ∞ -map f : U → ( C n ) n − r +1 . Hence X r ( ξ ) is the set of points where vanish r corner minors of the corresponding matrix n × ( n − r + 1). So X r ( ξ ) is locallydetermined by r complex equations – it’s an intersection of r nonsingularmanifolds of (real) codimension two with complex structures in their normalbundle. The corresponding complex structure is preserved under change oftrivialization. (cid:3) Example 3.2.
Let ξ = O (1) ⊕ O (1) be a 2-dimensional bundle over M = C P , r = 1.Consider two generic holomorphic sections of ξ over C P – they have form( a z + a z + a z , b z + b z + b z ) and ( c z + c z + c z , d z + d z + d z ),where z , z , z are homogenous coordinates on C P . Then the condition that5ections are linearly dependent means that a z + a z + a z b z + b z + b z = c z + c z + c z d z + d z + d z . So X ( ξ ) is a nonsingular plane curve of degree two.It is known that Chern classes c Ur ( ξ ) may also be defined in complexcobordisms – they’re uniquely determined by the same four axioms thatdefine cohomological Chern classes. It turns out that if r < n then cobordismclasses corresponding to nonsingular X r ( ξ )’s are not necessarily equal to c Ur ( ξ ). Proposition 3.3.
Classes c Ur ( ξ ) and [ X r ( ξ )] are not equal when M = C P , r = 1 , ξ = O (1) ⊕ O (1) . We’ll calculate these classes explicitly. Denote by t = c U ( O (1)) thecobordism class of the line C P ⊂ C P , then, by Whitney sum formula, c U ( O (1) ⊕ O (1)) = c U ( O (1)) + c U ( O (1)).In our example X ( ξ ) is a plane quadric, so its cobordismclass equals to e U ( O (2)) = c U ( O (2)). This class is the sum oftwo classes c U ( ξ ) in the formal group of geometric cobordisms: c U ( ν ⊕ η ) = c U ( ν ) + c U ( η ) − [ C P ] c U ( ν ) c U ( η ) + . . . ([4]). So[ X ( ξ )] = 2 t − [ C P ] t , which is not 2 t . (cid:3) Problem 3.4.
Construct the classes A r ( ξ ) ∈ U r ( M ) satisfying the fol-lowing properties:1. Classes A r ( ξ ) are functorial.2. A r ( ξ ) = [ X r ( ξ )] if X r ( ξ ) is nonsingular.3. If X r ( ξ ) is singular then A r ( ξ ) is realized by complex-oriented map ([5])of a manifold resolving singularities of X r ( ξ ) . Also we expect some more nice properties from A r ( ξ ) – for example, theycould satisfy some Chern classes axioms.By functoriality property, class A r ( ξ ) is a formal series in classes c U ( ξ ) , c U ( ξ ) , . . . over ring U ∗ ( pt ) = Z [ a , a , . . . ] , deg a i = − i . The aug-mentation map U ∗ ( BU ) → H ∗ ( BU, Z ) acts as follows: all c Ui ( ξ )’s map to c i ( ξ )’s and a i ’s – to zero. From the second and third condition it follows that A r ( ξ ) has the form c Ur ( ξ ) + . . . , where the part ”dots” maps to zero underaugmentation map.The geometric constructions from the previous section may easily bespread to the complex cobordisms. The most efficient construction are classes Q r ( ξ ), which are defined by using the projectivization of trivial bundle of rank n − r + 1. In the next sections we investigate some of their properties andshow that they satisfy deformed Whitney sum formula.6ence, the problem of constructing characteristic classes using singularitycycles makes us vary the system of corresponding axioms. Classes Q r ( ξ ) don’tonly possess deformed Whitney sum formula, they also may be non-trivialin negative dimension. Q r ( ξ ) and their basic properties. In this section we define characteristic classes Q r ( ξ ) ∈ U r ( M ) and establishsome of their properties. Recall that M is an oriented manifold and ξ – acomplex vector bundle over M .Consider the projectivization M × C P n − r of a trivial vector bundle C n − r +1 over M and its tautological bundle γ . Then the section s determines thesection s Q of a bundle Hom( γ, p ∗ ξ ) (here p : M × C P n − r → M is a projectionmap) by taking the composition γ → C n − r +1 → p ∗ ξ .The set X Qr ( ξ ) of points on M × C P n − r where s Q vanishes is a nonsingularsubmanifold, its complex cobordism class is well-defined and coincides withEuler class e U ( γ ∗ ⊗ p ∗ ξ ). Definition 4.1. Q r ( ξ ) = p ! e U ( γ ∗ ⊗ p ∗ ξ ) . Theorem 4.2.
1. Classes Q r ( ξ ) are functorial.2. Classes Q r ( ξ ) satisfy the dimension axiom: Q r ( ξ ) = 0 if r > n .3. Classes Q r ( ξ ) are normalized: Q n ( ξ ) = e U ( ξ ) = c Un ( ξ ) .4. If X r ( ξ ) is a nonsingular manifold in M , then Q r ( ξ ) = [ X r ( ξ )] .5. The Whitney sum formula fails for Q r ( ξ ) so they don’t coincide with c Ur ( ξ ) . The functoriality property follows from construction. If r > n , then n − r + 1 is non-positive and class Q r ( ξ ) corresponds to an empty map to M , so Q r ( ξ ) is zero.If r = n , then M × C P n − r = M , the bundle γ ∗ is trivial and p is anidentity map. So Q r ( ξ ) = p ! e U ( γ ⊗ p ∗ ξ ) = e U ( γ ∗ ⊗ ξ ) = e U ( ξ ). This provesthe third statement.The class Q r ( ξ ) is, by definition, realized by map of a nonsingular mani-fold X Qr ( ξ ) to M , resolving the singularities of X r ( ξ ). If X r ( ξ ) is nonsingularitself, then the corresponding map is identical and Q r ( ξ ) = [ X r ( ξ )].To prove the last statement it’s enough to find a counterexample to Whit-ney sum formula. We’ll do it in the next section. (cid:3) Examples: bundles over pro jective spaces.
Example 5.1.
Let ξ be an arbitrary linear bundle over M = C P , e U ( ξ ) = c U ( ξ ) = u . We’ll calculate Q r ( ξ ) for r = 1 , , − r = 1, then n − r + 1 = 1, M × C P n − r = M = C P and γ is trivial.So Q ( ξ ) = p ! e U ( γ ∗ ⊗ p ∗ ξ ) = e U ( γ ⊗ ξ ) = e U ( ξ ) = u .If r = 0, then M × C P n − r = C P × C P , p is a projection map tofirst C P . Denote e U ( γ ∗ ) by t . Then the class e U ( γ ∗ ⊗ p ∗ ξ ) has the form t + u − [ C P ] tu + . . . ([4]), where every monomial in ( . . . ) is divisible byeither t or u .The pushforward homomorphism p ! , corresponding to a map M × C P k → M is a U ∗ ( M )-module homomorphism given by formula p ! ( t i ) = [ C P k − i ] if i k and p ! ( t i ) = 0 if i > k . In our case p ! ( t + u − [ C P ] tu + . . . ) =1 − [ C P ] u + [ C P ] u + . . . because u = 0 in U ∗ ( C P ). So Q ( ξ ) = 1 forevery linear bundle ξ over C P .Finally, if r = − n − r + 1 = 3 and p : C P × C P → C P is againa projection map. In this case p ! ( t ) = 1, p ! ( t ) = [ C P ], p ! (1) = [ C P ] and Q − ( ξ ) = p ! e U ( γ ∗ ⊗ p ∗ ξ ) = p ! ( t + u − [ C P ] tu + . . . ) = [ C P ] + ([ C P ] − [ C P ] ) u = 0 ∈ U − ( C P ). We see that classes Q r ( ξ ) and c Ur ( ξ ) are notequal because c Ur ( ξ ) = 0 if r < Example 5.2.
Let ξ be a bundle of rank k > C P . We’ll showthat again Q ( ξ ) is 1.Let t . . . t k be Chern roots of ξ , then Q ( ξ ) = p ! e U ( γ ∗ ⊗ p ∗ ξ ) = p ! ( F ( t, t ) . . . F ( t, t k )) = p ! (( t + t − [ C P ] tt + . . . ) . . . ( t + t k − [ C P ] tt k + . . . )),where p : C P × C P k → C P is a projection map.In our case p ! ( t k ) = 1, p ! ( t k − ) = [ C P ] and t i · t j = 0 in U ∗ ( C P ). So Q ( ξ ) = p ! ( t k + t k − ( t + . . . + t k ) − [ C P ] t k ( t + . . . + t k )) = 1 − [ C P ]( t + . . . + t k ) + [ C P ]( t + . . . + t k ) + · · · = 1.We see that in case of the bundle over C P classes Q ( ξ ) and Q ( ξ )coincide with c U ( ξ ) and c U ( ξ ) respectively. Example 5.3.
Let us show that Q ( O (1)) = 1 + u ([ C P ] − [ C P ])in U ( C P ). If p : C P × C P → C P is a projection, t = e U ( γ ∗ ), u = e U ( p ∗ O (1)), then Q ( O (1)) = p ! e U ( γ ∗ ⊗ p ∗ ξ ) = p ! ( t + u − [ C P ] tu + ([ C P ] − [ C P ]) tu + . . . ), where any monomial in ( . . . ) divides either by t or u . So Q ( ξ ) = 1 + [ C P ] u − [ C P ] u + ([ C P ] − [ C P ]) u = 1 + ([ C P ] − [ C P ]) u in U ( C P ).We may now give the example of bundles ξ and η such that classes Q r ( ξ ⊕ η ), Q i ( ξ ) and Q j ( η ) don’t satisfy Whitney sum formula. Consider M = C P , ξ = η = O (1). Then Q ( O (1) ⊕ O (1)) is equal to cobordism class of anonsingular quadric in C P , which is 2 u − [ C P ] u , u = e U ( O (1)). If weassume that Whitney sum formula holds for classes Q r ( · ), then Q ( O (1) ⊕ (1)) = 2 Q ( O (1)) Q ( O (1)) = 2 u = 2 u − [ C P ] u . This finishes the proof ofTh. 4.2.As we said before, in general the classes Q r ( ξ ) are formal series in c Uk ( ξ )with coefficients in U ∗ ( pt ) (this follows from functoriality property). Forexample, if ξ is a 2-dimensional bundle, then the class Q ( ξ ) has the form Q ( ξ ) = c U ( ξ ) + [ C P ] c U ( ξ ) + ([ C P ] − [ C P ]) c U ( ξ ) c U ( ξ ) + . . . (which may be computed by splitting principle). Q r ( ξ ) . In this section we calculate deformed sum formula for classes Q r ( ξ ).Let ξ be a complex vector bundle of rank k and t . . . t k – its Chern roots.Consider the product F ( t, t ) · . . . · F ( t, t k ) where F ( u, v ) is a formal groupin geometric cobordisms. Then we can write k Y i =1 F ( t, t i ) = ∞ X j =0 t j Φ k − j ( t , . . . , t k ) , where Φ r ( t , . . . , t k ) is a symmetric formal series in t , . . . , t k with coefficientsin U ∗ ( pt ). It is easy to see that Φ r ( t , . . . , t k ) = Φ r ( ξ ) is a homogenouscharacteristic class of dimension 2 r .Classes Φ r ( ξ ) naturally arise in the following well-known construction: if ξ is a complex vector bundle of rank n over base X , then one can considerthe bundle ξ ⊗ γ over X × C P ∞ ( γ is a tautological linear bundle over C P ∞ ). The ring of complex cobordisms U ∗ ( X × C P ∞ ) is isomorphic toring of formal series U ∗ ( X )[[ u ]], where deg u = 2. This means that theclass c n ( ξ ⊗ γ ) may be written in the form Φ n ( ξ )+ u Φ n − ( ξ )+ u Φ n − ( ξ )+ . . . . Proposition 6.1.
Classes Φ r ( ξ ) satisfy the following properties:1. Φ r ( ξ ) = 0 if r > n .2. Classes Φ r ( ξ ) are functorial.3. Classes Φ r ( ξ ) satisfy Whitney sum formula Φ r ( ξ ⊕ η ) = ∞ P j = −∞ Φ j ( ξ ) · Φ r − j ( η ) (which is well-defined by property 1).4. Class Φ n ( ξ ) is equal to Euler class (top Chern class) e U ( ξ ) = c Un ( ξ ) ofbundle ξ . ξ ⊕ η ) ⊗ γ = ( ξ ⊗ γ ) ⊕ ( η ⊗ γ ).By using upper-triangular change of variables one can express Q r ( ξ )through Φ r ( ξ ) (and vice versa). Proposition 6.2. Q r ( ξ ) = n − r P k =0 Φ r + k ( ξ )[ C P k ] . As we said before, the pushforward homomorphism p ! : H ∗ ( M × C P k ) → H ∗ ( M ) is a U ∗ ( pt )-module homomorphism given by formula p ! ( t i ) = [ C P k − i ]if i k , p ! ( t i ) = 0 otherwise. So one can obtain Q r ( ξ ) by taking c n ( γ ∗ ⊗ p ∗ ξ )= Φ n ( ξ ) + t Φ n − ( ξ ) + . . . and substitution t i → [ C P n − r − i ]. (cid:3) The transition matrix Φ r ( ξ ) → Q r ( ξ ) is upper-triangular; its i -th diago-nal is containing only [ C P i ]’s. The inverse matrix is also upper-triangular:its i -th diagonal consists of elements [ M i ] ∈ U − i ( pt ), where generatingfunction 1 + P [ M i ] x i is equal to P [ C P i ] x i . Theorem 6.3.
Consider classes [ M i ] ∈ U − i ( pt ) with generating func-tion P [ M i ] x i = P [ C P i ] x i . Then for any complex vector bundles ξ , η and any r we have Q r ( ξ ⊕ η ) = X i,j,k,l ∈ Z Q l + i ( ξ ) Q r − l + k + j [ M i ][ M j ][ C P k ] . (if i < , then [ M i ] = [ C P i ] = 0 ). The proof is straightforward: Q r ( ξ ⊕ η ) = P k ∈ Z Φ r + k ( ξ ⊕ η )[ C P k ] = P k,l ∈ Z Φ l ( ξ )Φ r − l + k ( η )[ C P k ] = P k,l,i,j ∈ Z Q l + i ( ξ ) Q r − l + k + j [ M j ][ C P k ]. (cid:3) If we set [ M i ] = [ C P i ] = 0 for all positive i (this corresponds toaugmentation map U ∗ ( pt ) → H ∗ ( pt )), then our sum formula turns intostandart Whitney sum formula for classes c r ( ξ ) – as it should be. Theorem 6.4.
Classes Q r ( ξ ) ∈ U r ( X ) of a complex vector bundle bun-dle ξ of rank n over X are uniquely determined by four axioms:1. Classes Q r ( ξ ) are functorial.2. Q r ( ξ ) = 0 if r > n .3. Classes Q r ( ξ ) satisfy the sum formula Q r ( ξ ⊕ η ) = X i,j,k,l ∈ Z Q l + i ( ξ ) Q r − l + k + j ( η )[ M i ][ M j ][ C P k ]10 . Q n ( ξ ) = e U ( ξ ) = c Un ( ξ ) . The proof is analogous to proof of uniqueness of cohomological Chernclasses. Axiom 4 may be reformulated in a more classical way: the class Q ( O (1)), where O (1) is a canonical bundle over C P , is equal to fundamentalclass u ∈ U ( C P ). P r ( ξ ) and I H -small resolutions.
The definition of classes P r ( ξ ) follows the construction from section 1. Givena generic section s : C n − r +1 → ξ (we denote by C the trivial linear bundle)one can consider the dual section s ∗ : ξ ∗ → C n − r +1 . If dim ker s = l in somepoint x ∈ M , then dim ker s ∗ = l + r − γ be an r -dimensional tautological bundle over Gr r ( ξ ∗ ) – r -thgrassmanization of ξ ∗ , p : Gr r ( ξ ∗ ) → M – a projection map. Then section s ∗ determines the section γ → p ∗ ξ ∗ → C n − r +1 of a bundle Hom( γ, C n − r +1 )over Gr r ( ξ ∗ ). Let X Pr ( ξ ) be a zero set of this new section. If dim ker s = l in x ∈ M , then p − ( x ) ∩ X Pr ( ξ ) = Gr r ( C l + r − ). The set X Pr ( ξ ) is anonsingular manifold and its cobordism class is equal to e U (Hom( γ, C n − r +1 ))= e U (( γ ∗ ) n − r +1 ). Definition 7.1. P r ( ξ ) = p ! e U (( γ ∗ ) n − r +1 ). Theorem 7.2.
1. Classes P r ( ξ ) are functorial.2. P r ( ξ ) = 0 if r > n .3. The classes P r ( ξ ) satisfy the normalization axiom: P n ( ξ ) = e U ( ξ ) .4. If set X r ( ξ ) is nonsingular then P r ( ξ ) = [ X r ( ξ )] .5. The classes P r ( ξ ) and c Ur ( ξ ) are not equal. The proofs are straightforward from definitions (except the last statementwhich follows from theorems below).We see that there exist at least two resolutions of singular manifold X r ( ξ ), corresponding to Q r ( ξ ) and P r ( ξ ) respectively. The classes P r ( ξ ) and Q r ( ξ ) differ – but only in high dimensions. Theorem 7.3. . If dim ξ = 2 , the class P ( ξ ) − Q ( ξ ) vanishes if dim R M and maybe nonzero if dim R M = 10 (recall that M is even-dimensional).2. If dim ξ = 2 , the class P ( ξ ) − Q ( ξ ) vanishes if dim R M and maybe nonzero if dim R M = 12 . This is proved by direct computation using generalized Riemann-Rochtheorem for Chern-Dold character ([1]). Chern-Dold character is a multi-plicative transform ch U : U ∗ ( X ) → H ∗ ( X, U ∗ ( pt )), which is an isomorphismfor X = BU ( n ). So if one wants to compare P r ( ξ ) and Q r ( ξ ), then it is betterto compare their transformations ch U P r ( ξ ) and ch U Q r ( ξ ) – the result shouldbe the same, but the computations will be much more easy. If p : X → Y isa smooth bundle, τ – its fiber tangent bundle carrying a complex structure,then (according to generalized Riemann-Roch theorem)ch U p U ! ( x ) = p H ! (ch U ( x ) · T ( τ )) , where p U ! and p H ! are pushrorward homomorphisms in cobordisms and coho-mology and T ( τ ) ∈ H ∗ ( X, U ∗ ( pt )) is generalized Todd class. T ( τ ) is mul-tiplicative: T ( η ⊕ ξ ) = T ( η ) T ( ξ ) and if γ is linear, then T ( γ ) = c ( γ ) g − ( c ( γ )) (see [1]), where g − is an exponential of formal group of complex cobordisms.The homomorphism p H ! is easier than p U ! , so we can compare ch U Q r ( ξ ) andch U P r ( ξ ) explicitly.Let dim ξ = 2 and r = 1. If dim M Q ( ξ ) = P ( ξ )because X ( ξ ) is nonsingular. The beginning of the row corresponding to Q ( ξ ) and P ( ξ ) has the form c U ( ξ ) − [ C P ] c U ( ξ ) + ([ C P ] − [ C P ]) c U ( ξ ) c U ( ξ ) + . . . If dim M = 8 or 10, then X ( ξ ) may be singular, but still there is nodifference between Q ( ξ ) and P ( ξ ). This is not surprising: the manifolds X P ( ξ ) and X Q ( ξ ) are ”made of equal parts”: the preimages of point x ∈ X ( ξ ) where dim ker s = l are diffeomorphic to C P l − in X P ( ξ ) and X Q ( ξ ).Not let M be a complex variety and ξ a holomorphic vector bundle,dim C M = 4, dim ξ = 2. Then X ( ξ ) is nonsingular in points of the comple-ment to several isolated points, and in these points X ( ξ ) is locally given byequation xy − zt = 0. Then X Q ( ξ ) and X P ( ξ ) correspond to two differentresolutions of isolated quadratic singularity.This corresponds to the following situation: the resolution f : X → Y of a singular variety Y is called IH -small if codim C Y i > i , where Y i = y ∈ Y | dim C f − ( y ) = i . Most singular varieties don’t possess IH -smallresolutions. As proved by B.Totaro ([8]), two different IH -small resolutions12f one singular variety have equal elliptic genera. So even if P ( ξ ) = Q ( ξ ),the classes [ X P ( ξ )] and [ X Q ( ξ )] have the same elliptic genus.Now let dim ξ = 3 and r = 2. If dim M
10 then X ( ξ ) is againnonsingular and Q ( ξ ) = P ( ξ ). The beginning of the row corresponding to Q ( ξ ) and P ( ξ ) has the form c U ( ξ ) − C P ] c U ( ξ ) + ([ C P ] − [ C P ]) c U ( ξ ) c U ( ξ ) + . . . The preimage in X Q ( ξ ) of point x ∈ X ( ξ ) where dim ker s = l is C P and in X P ( ξ ) it is Gr ( C ) = C P . So Euler characteristics of X P ( ξ ) and X Q ( ξ ) are not equal if X ( ξ ) is singular. The corresponding cobordismclasses Q ( ξ ) and P ( ξ ) also differ. Corollary 7.4.
Two resolutions of singular variety X r ( ξ ) correspondingto Q r ( ξ ) and P r ( ξ ) are not equivalent. If r >
1, this is obvious because Euler characteristics of correspondingmanifolds differ; the case r = 1 follows from the fact that classes P ( ξ ) and Q ( ξ ) aren’t equal. (cid:3) D ( ξ ) . Finally, in this section we consider the characteristic class D ( ξ ) = c U (det ξ ) ∈ U ( M ) of det ξ = Λ n ξ . It is functorial and equal to [ X ( ξ )] if X ( ξ ) is nonsingular. But our last condition (resolution of singularities of X ( ξ ) if it’s singular) fails for D ( ξ ) (example: O (1) ⊕ O ( −
1) over C P ).But if X ( ξ ) is nonsingular, the class D ( ξ ) is much easier to compute than Q ( ξ ) and (of course) P ( ξ ). Proposition 8.1.
We have ch U D ( ξ ) = ∞ X i =0 [ N i ] c ( ξ ) i +1 , where elements [ N i ] ∈ U ∗ ( pt ) ⊗ Q satisfy s ( i ) ([ N i ]) = 1 , s ω ([ N i ]) = 0 if ( ω ) = ( i ) , s ω are Landweber-Novikov operations in complex cobordism. We use splitting principle for the one last time. If ξ = η ⊕ · · · ⊕ η n is adirect sum of linear bundles, c U ( η i ) = t i , then c U ! (det ξ ) = g − ( g ( t ) + . . . + g ( t k )), where g ( x ) = ∞ P i =0 [ C P i ] i +1 x i +1 is a logarithm of formal group of complexcobordisms. 13he Chern-Dold character is a ring homomorphism, so ch U g − ( g ( t ) + . . . + g ( t n )) = g − ( g (ch U t ) + . . . + g (ch U t k )). But, as we said before, ch U ( x )regarded as formal series, coincides with g − ( x ) ([1]). So if ch U t i = g − ( s i ),then ch U g − ( g ( t ) + . . . + g ( t n )) = g − ( s + . . . + s n ) = g − ( c ( ξ )). (cid:3) References [1] V. M. Buchstaber.
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