CHARACTERISTIC CLASSES AND SINGULAR VARIETIES Jean-Paul BRASSELET CNRS Marseille Preamble These notes are prepared for a course given during the School on Singularities in Geometry, Topology, Foliations and Dynamics 2014, A celebration of the 60th birthday of Pepe Seade. A first version of the course was given during the XVII Encontro Brasileira de Topologia, August 2nd to 6th, 2010 in PUC, Rio de Janeiro, Brasil. Notes of this course are partially taken from a book in preparation. Thanks to the reader for providing comments, critics etc... in order to improve the final version. The first three chapters are covered as well by the course given by Pepe Seade. They are provided in order to fix notations and results that are useful for the following. Contents 1 Euler-Poincar´ e characteristic 6 1.1 Combinatorial definition ............................ 6 1.2 Manifolds - Poincar´ e isomorphism ....................... 7 1.3 Pseudomanifolds ................................ 9 1.4 The genus of surfaces .............................. 10 1.5 Betti numbers .................................. 11 2 Poincar´ e-Hopf Theorem (smooth case) 13 2.1 The index of a vector field............................ 13 2.2 The index - Definition by obstruction theory ................. 15 2.3 Relation with the Gauss map ......................... 17 2.4 Poincar´ e-Hopf Theorem ............................ 18 2.4.1 The smooth case without boundary .................. 19 2.4.2 Consequences of Poincar´ e-Hopf Theorem ............... 20 2.4.3 The smooth case with boundary .................... 21 3 Characteristic classes : the smooth case 23 3.1 Fibre bundles................................... 23 3.1.1 Vector bundles ............................. 24 3.1.2 Fibre bundles .............................. 24 1
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CHARACTERISTIC CLASSESAND SINGULAR VARIETIES
Jean-Paul BRASSELETCNRS Marseille
Preamble
These notes are prepared for a course given during the School on Singularities in Geometry,Topology, Foliations and Dynamics 2014, A celebration of the 60th birthday of Pepe Seade.A first version of the course was given during the XVII Encontro Brasileira de Topologia,August 2nd to 6th, 2010 in PUC, Rio de Janeiro, Brasil. Notes of this course are partiallytaken from a book in preparation. Thanks to the reader for providing comments, criticsetc... in order to improve the final version.
The first three chapters are covered as well by the course given by Pepe Seade. Theyare provided in order to fix notations and results that are useful for the following.
The Euler-Poincare characteristic is the first characteristic class that has been intro-duced. The Poincare-Hopf theorem says that, if X is a compact manifold and v a con-tinuous vector field with a finite number of isolated singularities ak with indices I(v, ak),then
χ(X) =∑
I(v, ak) .
That means that the Euler-Poincare characteristic is a measure of the obstruction to theconstruction of a non-zero vector field tangent to X.
Later on, Severi and Todd (1935) defined characteristic cycles of projective varietiesusing polar varieties.
In his famous paper, Chern (1946) defined characteristic classes for hermitian mani-folds in several ways, in particular as the measure of the obstruction to the constructionof complex r-frames tangent to the manifold, generalising the Poincare-Hopf theorem.
The Chern classes are represented by algebraic cycles which coincide with Todd cycles.
During several years, the attractiveness of the axiomatic properties of Chern classescaused the viewpoints of obstruction theory and polar varieties to be somewhat forgotten.It is interesting to see that these viewpoints came back on the scene with the question ofdefining characteristic classes for singular varieties.
There are in fact various definitions of characteristic classes for singular varieties. Inthe real case, there is a combinatorial definition, which simplifies the problem. In thecomplex case, the situation is more complicated (and certainly more interesting !), dueto the fact that there is no combinatorial definition of Chern classes. The obstructiontheory point of view, in the smooth case, is based on the existence of the tangent bundle.If one wants to use the obstruction theory point of view in the singular case, one has tofind a substitute to the tangent bundle. There are various candidates to substitute thetangent bundle and each of them leads to a different definition of Chern class for singularvarieties. In particular, one has the following three substitutes.
a) If X is a singular complex analytic variety, equipped with a Whitney stratificationand embedded in a smooth complex analytic manifold M one can consider the union oftangent bundles to the strata, that is a subspace E of the tangent bundle to M . Thespace E is not a bundle but it generalises the notion of tangent bundle in the followingsense: A section of E over X is a section v of TM |X such that in each point x ∈ X, thenv(x) belongs to the tangent space of the stratum containing x. Such a section is calleda stratified vector field over X. To consider E as a substitute to the tangent bundle ofX and to use obstruction theory is the M.H. Schwartz point of view (1965), for definingChern classes of analytic complex varieties.
b) A second possibility is to consider, for x singular point of X, the space of allpossible limits of tangent vector spaces Txn(Xreg) where xn is a sequence of points in theregular part Xreg of X converging to x ∈ X. That point of view leads to the notion
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of Mather classes, which are an ingredient in the MacPherson definition for classes ofalgebraic complex varieties (1974). Another main ingredient for the definition of theseclasses is the notion of Euler local obstruction that we study in Section 6.5.
c) A third possibility for a substitute to the tangent bundle is the following: Let ussuppose that there exists a normal bundle N to X in M , that is the case of local completeintersections for example. Then, one can consider the virtual bundle TM |X \ N as asubstitute to the tangent bundle of X. That point of view is the one of Fulton (1980).
There are relations between the classes obtained by the previous constructions. Firstof all, the Schwartz and MacPherson classes coincide, via Alexander duality (1979, [B-S]).
The relation between Mather classes on one side and Schwartz-MacPherson classeson the other side follows form the MacPherson’s definition itself: His construction usesMather classes, taking into account the local complexity of the singular locus along Whit-ney strata. This is the role of the local Euler obstruction.
A natural question arised to compare the Schwartz-MacPherson and the Fulton classes.A result of Suwa [Su1] shows that in the case of isolated singularities, the difference ofthese classes is given (up to sign) by the sum of the Milnor numbers in the singular points.It was natural to call Milnor classes the difference in the general case. This differencehas been described by several authors using different methods (P. Aluffi, J.P. Brasselet-D.Lehmann-J. Seade-T. Suwa, A. Parusinski-P. Pragacz and S. Yokura).
In the case of manifolds, the Todd genus and the L-genus are degree 0 elements ofthe Todd class and the Thom-Hirzebruch L-class. Both of them are related to the Chernclass via the Chern roots and F. Hirzebruch gave a way to unify these three theoriesof characteristic classes by using the so-called multiplicative series. In the same waythat the MacPherson construction generalises the Chern class, the Todd class and theThom-Hirzebruch L-class have been generalised as natural transformations respectivelyby Baum-Fulton-MacPherson and by Cappell-Shaneson. The problem is that the threetransformations are defined on different groups on the singular complex algebraic varietyX: namely group of constructible functions, Grothendieck group of coherent sheaves,group of constructible self-dual sheaves. One way to unify the three theories, in thesingular case, is to use the motivic theory and the Grothendieck relative group of algebraicvarieties over X. That has been performed by J.-P. Brasselet, J. Schurmann and S. Yokuraand is explained in the section 7.3.
The author thanks the Scientific and Organising Committees of the School and Work-shop on Singularities in geometry, topology, foliations and dynamics, Cuernavaca 2014,for providing oportunity to give a course during the School.
Aubagne, August 18th, 2014
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1 Euler-Poincare characteristic
In this section, the varieties we consider are possibly singular varieties.
1.1 Combinatorial definition
History of characteristic classes begins with the discovery of the so-called Euler formula, byLeonhard Euler around 1750 : Let P be a 2-dimensional polyhedron in R3, homeomorphicto the sphere S2, one has
k0 − k1 + k2 = 2
where k0 is the number of vertices in P , k1 is the number of segments and k2 the numberof faces. That is the case for the tetraedron: 4 − 6 + 4, for the cube (with diagonals onthe faces): 8− 18 + 12.
According to different authors, that formula was first proven by Euler himself, byLegendre in 1794 or by Cauchy. In fact, it seems that this formula was already known byR. Descartes (around 1620) and even by Archimedes.
Simon Antoine-Jean Lhuilier, a Swiss mathematician, gave (in 1812) a slight gener-alization of Euler’s formula taking into account orientable 2-dimensional polyhedra withholes. The number g of holes is called genus (see Definition 1.16). Lhuilier’s formula is
k0 − k1 + k2 = 2− 2g,
where g is the genus. Thus one obtains 0 for a torus-like polyhedron.For a general 2-dimensional polyhedron P in R3, the alternative sum
χ(P ) = k0 − k1 + k2
is called Euler characteristic of P .H. Poincare [Po2] generalized the result in 1893 for finite polyhedra P of higher dimen-
sions and proved the so-called Poincare-Hopf Theorem, which is the bridge to differentialgeometry. One defines
Definition 1.1 Let us denote by ki the number of i-dimensional simplices of a finiten-dimensional polyhedron P in Rm, the Euler-Poincare characteristic of the polyhedronP is defined by
χ(P ) =n∑i=0
(−1)iki.
Let us remind some elementary definitions:
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Definition 1.2 A (finite) simplicial complex K is a collection of simplexes in some eu-clidean space such that
• if s ∈ K then every face of s belongs to K,
• if s, t ∈ K, then s ∩ t is either empty or is a common face of s and t.
Definition 1.3 Let us denote by K a (finite) simplicial complex in Rm. The union ofsimplexes in K is a compact subspace of Rm denoted by |K| = P and called geometricrealisation of K, or polyhedron associated to K.
Definition 1.4 A topological space X is triangulable (or a polyhedron) if there exists asimplicial complex K and a homeomorphism h : |K| → X. Such a pair (K,h), or simplythe simplicial complex K, is called a triangulation of X.
The Poincare’s result is the following:
Theorem 1.5 (Poincare, [Po2]) Let us consider two triangulations (K1, h1) and (K2, h2)of a (compact) topological space X, then one has χ(K1) = χ(K2).
Theorem 1.5 implies that Euler-Poincare characteristic is a topological invariant of thespace X. The result makes sense for the following definition.
Definition 1.6 The Euler-Poincare characteristic of the triangulable space X, denotedby χ(X) is defined as χ(K) for a triangulation (K,h) of X.
Examples 1.7 The Euler-Poincare characteristic of the sphere is χ(Sn) = 1 + (−1)n, ofthe 2-dimensional real torus T is χ(T ) = 0, of the pinched torus is χ(T ) = 1.
The Euler-Poincare characteristic of the complex projective space is χ(CPn) = n+ 1.
1.2 Manifolds - Poincare isomorphism
Let us recall the definition of topological manifold:
Definition 1.8 (Topological manifold) A Hausdorff space is called a (topological) m-manifold if each point x in M admits a neighbourhood Ux homeomorphic to a ball Bm ⊂Rm through a homeomorphism φ : Ux → Bm such that φ(x) = 0 and the boundary of Ux,called the link of x, is homeomorphic to the sphere Sm−1.
Let us denote by M a m-manifold and by (K) a triangulation of M . A dual celldecomposition of M is obtained in the following way:
Let us consider a barycentric subdivision (K ′) of (K). The barycenter of a simplexσ ∈ K will be denoted by σ. Every simplex in K ′ can be written as
(σi1 , σi2 , . . . , σip)
where σi1 < σi2 < · · · < σip . Here the symbol σ < σ′ means that σ is a face of σ′.The dual cell of a simplex σ, denoted by d(σ), is the set of all (closed) simplexes τ in
(K ′) such that τ ∩σ = σ. That is the set of simplexes on the form (σ, σi1 , . . . , σik) withσ < σi1 < · · · < σik .
The dual cells satisfy the nice properties:
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A
B
C
C0
B
C
E
A
A0B0
C0
D0
E0
F 0G0
H0
K0
L0
A
BB
AC0
D0
E0
(a) Dual of the triangle ABCis the barycenter C ′
A
B
C
C0
B
C
E
A
A0B0
C0
D0
E0
F 0G0
H0
K0
L0
A
BB
AC0
D0
E0
(b) Dual of AC is the 1-dimensional cell A′B′C ′
A
B
C
C0
B
C
E
A
A0B0
C0
D0
E0
F 0G0
H0
K0
L0
A
BB
AC0
D0
E0
(c) Dual of the vertex A′ isthe 2-dimensional cell A
Figure 1: Dual cells
Lemma 1.9 1. The dual cell is a cell, homeomorphic to a ball and its boundary ishomeomorphic to the corresponding sphere.
2. If σ is a k-simplex, then d(σ) is a (m− k)-cell.
3. The set of dual cells provide a cell decomposition of M , called dual cell decompositionassociated to the barycentric subdivision (K ′) of (K).
The unique intersection point σ = d(σ) ∩ σ is the barycenter of σ that we will denotealso sometimes by d = d(σ).
Let us assume M = |K| oriented, that is all m-simplices are given a compatibleorientation. One gives to every cell d(σ) the orientation such that orientation of σ followedby orientation of d(σ) is orientation of M .
Let us fix some notations:
• We denote by d∗(σ) the elementary (D)-cochain whose value is 1 at the cell d(σ)and 0 at other cells of (D).
• We denote by C(K)i the groups of simplicial K-chains with integer coefficients and
by Ci(D) the groups of simplicial D-cochains with integer coefficients.
Let us consider a compact oriented m-dimensional manifold, then one has, for everyk, a chain isomorphism:
Cm−k(D) (M ;Z) −→ C
(K)k (M ;Z), (1.10)
that one defines on the elementary elements as
d∗(σ) 7→ σ
and extends linearly.The following Theorem is one of the possible forms of the Poincare duality:
Theorem 1.11 (Poincare isomorphism) Let M be a compact oriented m-dimensionalmanifold, the morphism (1.10) induces, for every k, an isomorphism
Hm−k(M ;Z) −→ Hk(M ;Z) ,
which is the cap-product with the fundamental class [M ] ∈ Hm(M ;Z).
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1.3 Pseudomanifolds
The spaces we will consider are pseudomanifolds. This corresponds to the spaces called n-circuit by Poincare and Lefschetz. In fact, the notion of pseudomanifold differs accordingto the authors.
Definition 1.12 (Pseudomanifold - Combinatorial definition) One says that the poly-hedron |K| is an n-pseudomanifold if the simplicial complex K satisfies the followingproperties:
(i) dimK = n, i.e. the maximal dimension of simplexes in K is n.
(ii) Each simplex is face of a n-simplex.
(iii) Each n− 1-simplex is face of exactly two n-simplexes.
The notion of “simplicial simple n-circuit” (Lefschetz [Le], Poincare) corresponds tothe one of pseudomanifold with the following additional connexity property
(iv) The set of the n and n− 1-simplexes is connected.
The property means that |K| \ |K(n−2)| is connected. Equivalently, given two n simplexesσ and τ in K, there exists a sequence of n-simplexes σ = σ1, σ2, . . . , σr = τ such thatσi ∩ σi+1 is an (n− 1)-simplex.
If properties (i) to (iv) are verified, we will say that |K| is a simple n-pseudomanifold.
The topological definition of pseudomanifolds, which is equivalent to the combinatorialone in the case of triangulable topological space, goes as follows:
Definition 1.13 (Pseudomanifold - Topological definition) One says that the (paracom-pact, Hausdorff) topological space X is an n-pseudomanifold if there is a subset Σ ⊂ Xsuch that:
(i′) dimX = n.
(ii′) X \ Σ is a n-topological manifold dense in X.
(iii′) dim Σ ≤ n− 2.
The property (iv) is equivalent to the following connexity property
(iv′) The set X \ Σ is connected.
If properties (i′) to (iv′) are verified, we will say that X is a simple n-pseudomanifold.In the triangulated case, one can take Σ = |K(n−2)|.
Example 1.14 The pinched torus, the suspension of the torus, a Thom space, a complexalgebraic variety are examples of simple pseudomanifolds.
Let K be a triangulation of a connected homology n-manifold (over Z), then |K| isan n-pseudomanifold.
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A
a
b
(a) Pinched torus. The linkof the singular point A isunion of two circles.
(b) Suspension of the torus.The link of the vertex A is atorus
(c) Suspension of the torus(similar arrows should beidentified at each level)
Figure 2: Exemples of pseudomanifolds
Not all n-pseudomanifolds are homology n-manifolds. The pinched torus and thesuspension of the torus are pseudomanifolds, they are not (homological) n-manifolds.
Proposition 1.15 An oriented simple n-pseudomanifold X admits a fundamental class[X] ∈ Hn(X).
Proof: Let us consider a triangulation (K) of X. It is easy to verify that the sumof oriented n-simplices is a cycle, called a fundamental cycle. Its homology class is thefundamental class of X.
We show in section 5.2 that for an oriented pseudomaniflod, the cap product with thefundamental class [X] defines a homomorphism
Hn−k(X;Z) −→ Hk(X;Z) .
1.4 The genus of surfaces
Definition 1.16 The genus g of a connected surface is the integer representing the max-imum possible number of cuttings along closed simple curves without obtaining a discon-nected manifold.
Proposition 1.17 An orientable surface of genus g can be obtained from S2 by succes-sively attaching handles g times.
Let us describe the “attaching process” used in Proposition 1.17: Let us consider aconnected surface M and an embedding f : S0 × D2 → M \ ∂M . Image of f is a pair ofdisjoint disks in M . Cut out interior of these disks and glue in the cylinder D1 × S1 byf |S0×S1 . One says that the resulting surface M ′ is obtained from M attaching an handleby f :
M ′ =[M \ Intf(S0 × D2)
]∪f D1 × S1.
Among orientable surfaces, genus of the sphere is 0, genus of the torus is 1. TheLhuilier formula is χ(X) = 2− 2g (Proposition 1.26).
For non-orientable surfaces, one has the following results :
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Lemma 1.18 (see [Hirs]) Every non-orientable surface contains a Mobius strip.
Theorem 1.19 (see [Hirs]) For a compact connected non-orientable surface without bound-ary the genus is the unique integer g such that M contains g but not g+1 disjoint Mobiusstrips.
Proposition 1.20 A non-orientable surface of genus g ≤ 1 is diffeomorphic to the con-nected sum of g disjoint copies of RP2, real projective plane.
The real projective plane RP2 is a non-orientable surface of genus 1. One has χ(RP2) =1. The Klein bottle B is a non-orientable surface of genus 2, one has χ(B) = 0. Thesphere S2 with g real projective planes attached is a non-orientable surface of genus g.We will see (Proposition 1.26) that if X is a non-orientable surface of genus g, then onehas χ(X) = 2− g.
1.5 Betti numbers
In 1871, Betti [Be] defined numbers relative to 3-dimensional compact manifolds withoutboundary and announced a duality property.
A first statement of Poincare duality was provided by Henri Poincare in 1893 in termsof Betti numbers: The i-th and (n − i)-th Betti numbers of a closed (i.e. compact andwithout boundary) orientable n-manifold are equal. In his 1895 paper “Analysis Situs”[Po3], Poincare proved the theorem using a new tool: topological intersection theory.
In his danish dissertation thesis, 1898, Poul Heegaard [He] (french translation in [HeF])gives a counter-example to the version of Poincare duality. The Heegard paper forcedPoincare to be more precise.
In fact, Poincare had overlooked the possibility of the appearance of torsion in thehomology groups of a space. According to Poincare, the Heegard and Poincare definitionsof Betti numbers (in fact definitions of homologies) are not the same (see NDLR, page 161in [HeF]). The Betti definition does not take into account possibility to consider cycleswith coefficients.
In order to underline clearly this fact and to provide an indisputable proof, Poincarewrote the first complement to Analysis situs [Po4]. In the first two complements toAnalysis Situs [Po4, Po5], Poincare gave a new proof in terms of dual triangulations.
Let us denote by |K| a n -dimensional finite polyhedron and by Ci(K) the finitely gen-erated free abelian group whose generators are (oriented) i-simplexes of the triangulationK. The boundary operator is classically defined as a complex map ∂i : Ci(K)→ Ci−1(K).The subgroups of cycles and boundaries
are finitely generated, as subgroups of a finitely generated group. The homology groupsHi(K,Z) = Zi(K)/Bi(K) are also finitely generated, as quotient group of a finitely gen-erated one. One can write
Hi(K,Z) = Fi(K)⊕ Ti(K)
where Fi(K) is the free subgroup and Ti(K) the torsion subgroup of Hi(K,Z).
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Definition 1.21 The Betti numbers of |K| are defined as
βi(K) = rk (Hi(K,Z)) = rk (Fi(K)).
Equivalently, one can define the Betti number βi(K) as the dimension of the vector spaceHi(|K|;Q).
Noting that βi(K) = 0 if i > dim |K| = n, one has the Poincare Theorem:
Theorem 1.22 (Poincare Theorem) [Po3] Let |K| be a finite polyhedron in Rm, withBetti numbers βi(K), one has
χ(K) =n∑i=0
(−1)iβi(K).
Proof: From the long exact sequence
· · · −→ Ci+1(K)∂i+1−→ Ci(K)
∂i−→ Ci−1(K) −→ · · ·
one deduces the following equalities (where ni = rk (Ci(K)) and n = dim(|K|))n∑i=0
(−1)ini =n∑i=0
(−1)i(dimZi+dimBi−1) =n∑i=0
(−1)i(dimZi−dimBi) =n∑i=0
(−1)iβi(K).
The first one comes from the short exact sequence
0→ Zi → Ci → Bi−1 → 0 ,
the second one because B−1 = ∅ and Bn = ∅ and the third one from the Definition 1.21.The Theorem follows.
Alexander [Al] proved in 1915 that two triangulations (K,h) and (K ′, h′) of the sametopological space X have same Betti numbers βi(K) = βi(K
′) for every i. One can defineβi(X) as being βi(K) for any triangulation (K,h) of X and one has
χ(X) =n∑i=0
(−1)iβi(X).
This result proves that each Betti number is a topological invariant. In that sense, it ismore precise than the Poincare Theorem 1.5, which globally proves invariance of Euler-Poincare characteristic only.
Theorem 1.23 The Betti numbers of a compact orientable n-manifold M satisfy
βi(M) = βn−i(M) for i = 0, 1, . . . , n.
Corollary 1.24 If M is a compact orientable n-manifold with odd n, then χ(M) = 0.
Examples 1.25 If n is odd, the Euler-Poincare characteristic of the sphere Sn, the realprojective space RPn, a compact hypersurface in Rn+1, are zero.
Proposition 1.26 If X is an orientable (connected) surface of genus g, then β0(X) = 1,β1(X) = 2g and β2(X) = 1. One has χ(X) = 2− 2g.
In X is a non-orientable surface of (non-orientable) genus g, then β0(X) = 1, β1(X) =g − 1 and β2(X) = 0. One has χ(X) = 2− g.
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2 Poincare-Hopf Theorem (smooth case)
The Poincare-Hopf Theorem is important for many points of view: that is the first resultthat links two invariants from topology and differential geometry.
The Poincare-Hopf Theorem has been proved by Poincare [Po1] in 1885, in the 2-dimensional case, and by Hopf in 1927 [Ho] for higher dimensions. In between, partialresults had been proved by Brouwer and Hadamard. This result is the first apparitionof Euler-Poincare characteristic in differential topology, out of combinatorial topology. Itseems strange that Poincare extended the notion of Euler-Poincare characteristic fromdimension 2 to the general case but proved the Poincare-Hopf Theorem in dimension 2only without extending it in the general dimension.
The meaning of Poincare-Hopf Theorem is that Euler-Poincare characteristic is a mea-sure of the obstruction to constructing a continuous vector field tangent to the consideredmanifold, without singularity.
One of the motivations of the Poincare-Hopf Theorem is the study of differentialequations in terms of integral curves of an appropriate vector field. The singular pointsof the vector field are points of equilibrium in dynamical systems.
That is the reason for which Poincare-Hopf Theorem has many applications: mathe-matical economics, optimisation of communication systems, electrical engineering, appliedprobability (cooperative dynamical systems), statistical complexity, particle physics (elec-tromagnetic fields), structure of materials: stability of molecular complexes in chemistry,crystallography, graphics applications, astrophysics: magnetic fields, etc... The interestedreader should experience to search for “Poincare-Hopf Theorem” on his/her favorite websearch engine.
The first part of the chapter is devoted to various definitions of the index of a vectorfield in an isolated singularity.
2.1 The index of a vector field.
In this section, one gives different ways to define the index of a vector field in an isolatedsingular point. The obstruction theory definition will be useful for the following andprovides a geometrical meaning to characteristic classes. One can find generalisation ofthe theory to non-isolated singularities in [BLSS2] (see [BSS] for a systematic study).
The index of a vector field at an isolated singularity can be defined in various ways.We limit ourself to the “classical” ones (see [B3, BSS]).
In a first step, we consider vector fields in Euclidean space, then we will define theindex for vector fields tangent to a manifold.
Let Ω be an open subset in Rn with coordinates (x1, . . . , xn). Let
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v =n∑i=1
fi ∂/∂xi
be a vector field on Ω. The vector field is said to be continuous, smooth, analytic,according as its components f1, . . . , fn are continuous, smooth, analytic, respectively.A singularity a of v in Ω is a point where all of its components vanish, i.e., fi(a) = 0 forall i = 1, . . . , n. The singularity is isolated if at every point x near a there is at least onecomponent of v which is not zero.
Let v be a continuous vector field on Ω with an isolated singularity at a, and let B(a)be a small ball in Ω around a so that there is no other singularity of v within B(a). Letus define the Gauss map
γ : ∂B(a) = S(a) ∼= Sn−1 −→ Sn−1
by γ(x) = v(x)/‖v(x)‖.
Definition 2.1 The (local) index of v at a, denoted by I(v, a), is the degree of the Gaussmap γ : Sn−1 → Sn−1.
The local index does not depend on the choice of the small ball B(a), on the choiceof coordinates nor on the choice of orientations.
Remark 2.2 In the following, we will use also a different kind of singularities for avector field, that M.-H. Schwartz called second type singularities. Let us introduce thesesingularities.
Given a vector field v defined on the boundary S(a) of the ball B(a) of radius 1,centred in a, there are many ways to extend the vector field inside B(a). Two are themost natural. Let us denote by Sε(a) the sphere of radius ε, 0 < ε ≤ 1. If x ∈ S(a) thevector v(εx) at the point εx ∈ Sε(a) is defined either as v(εx) = εv(x) or as v(εx) = v(x).
In the first case, the vector field v will be 0 in a, that is the already defined singularitytype. We will call it, according to M.-H. Schwartz, singularity of first type.
In the second case the extension is not defined at a (see [Sc4]), but it defines a cycleκ(v) in the fibre TaRn of the tangent bundle to Rn. We will call it, again according toM.-H. Schwartz, singularity of second type.
Whatever the type of singularity, the index I(v, a) of v at the isolated singularity a iswell defined by the Definition 2.1.
Proposition 2.3 Let us consider a second type singularity, then the index of the cycleκ(v) in the punctured fibre TaRn \ 0 is equal to I(v, a).
Proof: Let us denote by s0 the zero section of the tangent vector bundle TRn. Thetangent bundle TRn is trivial over B(a), as well as the bundle T×Rn = TRn \ Ims0 (notanymore a vector bundle). The fibre of T×Rn at a is TaRn\0 ∼= Rn\0 and, restricted toB(a), the bundle is homeomorphic to B(a)×(Rn\0). The vector field v defines a sectionof T×Rn over S(a) whose image by the second projection B(a)× (Rn \ 0)→ Rn \ 0is equal to κ(v), by definition. One concludes by the Definition 2.1.
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Let us consider now a n-dimensional smooth manifold M . A continuous vector fieldon M is a section of its tangent bundle TM (see 3.1). Giving a local chart (Ua, φ) on M ,where φ : Ua → Bn, a vector field on M is locally expressed as above:
Let us denote by xi = xi φ the coordinate functions of φ, i.e. the local coordinatesin Ua. We denote by ∂/∂xi the tangent vector at x defined by
∂
∂xi(h) =
∂
∂xi(h φ−1)|φ(x)
for a C∞ function h : M → R.
Definition 2.4 Let us denote by x = (x1, . . . , xn) the local coordinates of the manifoldM in the open neighborood Ua, a vector field v can be written in terms of the basis ∂/∂xiof the tangent vector space TxM
v =n∑i=1
fi∂
∂xi. (2.5)
The functions (f1, . . . , fn) are called coordinates of the vector v in Ua. The vector fieldis said to be continuous, smooth, analytic, according as its components f1, . . . , fn arecontinuous, smooth, analytic, respectively.
For simplicity, in the following, we will identify coordinates in Ua and Bn, omitting φand we will denote xi for xi.
A singularity (“first type singularity”) a of the vector field v is a point in which allcoordinate fi vanish. One can also define “second type singularities” of a vector field vin the same way than in the Euclidean situation.
The index of the vector field v at a is well defined in both cases as the degree of theGauss map
γ : ∂φ−1(Bn) ∼= Sn−1 −→ Sn−1
2.2 The index - Definition by obstruction theory
The index can be also defined in the following way: Let M be a differentiable manifoldof dimension n. The tangent bundle to M , denoted by TM , is a real vector bundle (seesection 3.1) of rank n, whose fibre in a point x of M is the tangent space to M at x,denoted by Tx(M) and is isomorphic to Rn. The vector bundle TM is locally trivial, i.e.there is a covering of M by open subsets Ua such that the restriction of TM to eachUa is homeomorphic to Ua × Rn.
Let us denote by s0 the zero section of TM , we will consider the bundle (not any morea vector bundle) T×M = TM \ s0(M). Its fibre in a point x ∈M is T×x M
∼= Rn \ 0.Let us consider a ball B(a) centred in a, contained in an open chart Ua over which
TM is trivial and sufficiently small so that a is the only singular point of v in B(a). Onecan think of B(a) as an n-cell, in view of the generalisation we will perform later (3.15).The vector field v defines a section of TM without zero over S(a) = ∂B(a), hence a map
where pr2 is the second projection.One obtains a map
Sn−1 ∼= ∂B(a)pr2v−→ Rn \ 0
hence an element λ(v, a) in πn−1(Rn \ 0). One knows that this homotopy group is Z.The generator +1 can be interpreted in the following way:
Let us consider the radial vector field vrad, that is the vector field pointing outwardsthe ball B(a) along S(a), image of the vector field
∑ni=1 ∂/∂xi in Rn. Image of vrad in
Rn \ 0 is Sn−1 and the map pr2 vrad is the identity of Sn−1.
Lemma 2.7 There is a homotopy
ψ : S(a)× [0, 1]→ T×(B(a)|S(a)) ⊂ T×(M |S(a))
such that
∂Imψ = v(S(a))− I(v, a) · pr2 vrad(S(a)) (2.8)
Proof: Let us suppose without loss of generality that v is an unitary vector field onS(a). For t 6= 0, let us define in the fibre T×tx(M)
ψt(x) = ψ(t, x) = the unitary vector parallel to v(x) at the point tx.
If t goes to 0, then ψ0(x) is the unit vector in T×0 (M) parallel to v(x) and with origin 0.Therefore ψ0(S(a)) is a cycle in the fiber T×0 (M) whose index is I(v, 0). In the case of theradial vector field vrad on S(a), the cycle is the projection pr2 vrad(S(a)) over the fibreat 0 (by local triviality of the bundle) and it has index I(vrad, 0) = 1. One concludes theLemma.
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Proposition 2.9 The integer λ(v, a) equals to the index I(v, a) by the isomorphismπn−1(Rn \ 0) ∼= Z.
By classical homotopy theory, the map
Sn−1 ∼= ∂B(a)pr2v−→ Rn \ 0
extends to a mapB(a) −→ Rn \ 0
if and only if the element λ(v, a) is zero in πn−1(Rn \ 0), i.e. the vector field v extendswithin the ball B(a) if and only if the index I(v, a) is zero.
∂B(a) ∼= Sn−1 −−→ Rn \ 0y ?
B(a) ∼= Bn(2.10)
That construction is the basis of obstruction theory, it will be generalised in chapter 3.
Remark 2.11 Let v be a vector field defined in a neighborhood U of a ∈ Rn and let Bbe a n-ball containing a such that a is an isolated singularity of v in B. Then, the indexof v is determined by the behavior of v on the boundary ∂B of the ball, independently ofwhat happens inside the ball.
2.3 Relation with the Gauss map
Let N a compact k-manifold with boundary in Rk. The Gauss map
g : ∂N → Sk−1
assigns to each x ∈ ∂N the outward unit normal vector at x. The degree of the Gaussmap is well defined as the class of g(∂N) in Hk−1(Sk−1) ∼= Z.
Lemma 2.12 (Hopf) ([Mi1], §6, Lemma 3) If v is a smooth vector field on N withisolated singularities ai and v points outward of N along the boundary, then the sum ofindices
∑I(v, ai) equals the degree of the Gauss mapping from ∂N to Sk−1.
Proof: For each singular point ai one considers a small (closed) ball B(ai) withcenter ai and which do not intersect each other. The vector field v has no singularity inW = N \
⋃iB(ai).
Let us consider a compact n-manifold without boundary M ⊂ Rk. Let Nε denote theclosed ε-neighbourhood of M (i.e. the set of all x ∈ Rk with ‖x−y‖ < ε for some y ∈M).For ε sufficiently small, Nε is a smooth manifold with boundary.
Theorem 2.13 Let v be a vector field with isolated singularities ai, on a compact manifoldwithout boundary M ⊂ Rk, the index sum
∑I(v, ai) is equal to the degree of the Gauss
mappingg : ∂Nε → Sk−1,
where Nε denotes the closed ε-neighbourhood of M in Rk.
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We reproduce the proof due to Milnor [Mi1], §6, Theorem 1. That proof has beendelivered in December 1963 in lectures in University of Virginia. The procedure used byMilnor is the same as the one developed independently and at the same time by M.-H.Schwartz [Sc1], in her definition of radial extension in the framework of stratified singularvarieties. More precisely, the idea is to extend a vector field v defined on the manifoldM with index I(v, a;M) at the isolated singularity a, as a vector field w in the ambientspace Rk that has also an isolated singularity at a with the same index I(w, a;Rk) =I(v, a;M). The principle is to sum the parallel extension of v in a neighbourhood of awith a transversal vector field.
Proof: For x ∈ Nε, let r(x) be the closest point of M . The vector x − r(x) isperpendicular to the tangent space of M at r(x), for otherwise, r(x) would not be theclosest point of M . If ε is sufficiently small, then the restriction r(x) is smooth and welldefined.
We consider the squared distance function (for the Euclidean metric in Rk):
φ(x) = ‖x− r(x)‖2
whose gradient vector field is
gradφ(x) = 2(x− r(x)).
On one hand, the gradient vector field is a vector field defined in Nε that is zero along M ,that is transverse to ∂Nε going outward and that increases with the distance to M . Foreach point x at the level surface ∂Nε = φ−1(ε2), the outward unit normal vector, calledtransversal vector, is given by
g(x) = gradφ(x)/‖gradφ(x)‖ = (x− r(x))/ε.
On the other hand, in each point x ∈ Nε, the vector v1(x) = v(r(x)) is a parallelextension of v.
Extend v to a vector field w on the neighbourhood Nε by setting
w(x) = (x− r(x)) + v1(x).
The vector field w points outward along the boundary ∂Nε, since the inner productw(x) · g(x) is equal to ε > 0. In fact w vanish only at the zeros of v in M . That is clearbecause the two summands (x− r(x)) and v1(x) are orthogonal.
Now, the index of w at the zero a, computed in Rk is equal to the index of v at a,computed in M and, according to the Lemma 2.12, the index sum
∑I(v, a) is equal to
the degree of g which proves the theorem.
The Theorem is another way to see that if M is compact, the sum∑I(v, ai) for all
singularities of v does not depend on v. We will see (Theorem 5.12) that, with suitablevector fields, the result extends to the case of singular variety M .
2.4 Poincare-Hopf Theorem
There are many ways to prove Poincare-Hopf Theorem. They correspond to the differentviewpoints and definitions of the index. The interested reader can consult [Li], [Mi1](Hopf and Gauss map), [GP] (Lefschetz fix points theory), [Hirs] (Intersection numbers).
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2.4.1 The smooth case without boundary
Theorem 2.14 [Poincare-Hopf Theorem] Let M be a compact differentiable manifold,and let v be a continuous vector field on M with finitely isolated singularities ai. One has
χ(M) =∑i
I(v, ai)
Proof: Firstly we prove the Theorem in the orientable case, then in the non-orientablecase. We will follow the Milnor proof which is close to the generalisation to singularvarieties that we will provide in the next chapters.
1) Orientable case.The idea of the proof is the following: In a first step, one shows that the sum of
indices of a continuous tangent vector field with isolated singularities does not dependof the choice of the vector field. The second step of the proof consists in describing aparticular vector field for which the sum of indices is equal to χ(M).
For the first step, Theorem 2.13 provides directly the result.For the second step, such a vector field is given for instance by the gradient field
associated to a Morse function. Another possibility is to consider the Hopf vector fieldH of which we recall the construction (see [Ste], p. 202). Let us consider a triangulationK of M and a barycentric subdivision K ′ of K. The Hopf vector field will be tangent tosimplexes of K ′, with a singularity in every vertex of K ′, i.e. in every barycenter of K.On every simplex [σ, τ ] of K ′, where σ and τ are barycenters of σ and τ , with σ < τ , thevector field H is going in the direction from σ to τ . For example it is going outward allvertices of K. One complete with a vector field whose integral curves are given (in the2-simplex) in Figure 4. The higher dimensional case is easy to understand.
Figure 4: Hopf vector field
The Hopf vector field H has a singularity of index (−1)i in the barycenter of everyi-simplex of K. The sum of indices of H in all singularities is
∑ni=0(−1)iki where ki is
the number of i-dimensional simplexes of K, so it is equal to χ(M).
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2) The non-orientable case:
Let us consider the oriented double covering π : M → M . On one hand, if v is acontinuous vector field on M with isolated singular points ai of index I(v; ai), then on
can define a lifting v of v which is a continuous vector field on M with isolated singularpoints aji , j = 1, 2 such that π(aji ) = ai. As π is a local homeomorphism, one hasI(v; aji ) = I(v; ai) for j = 1, 2. One obtains
∑i,j I(v; aji ) = 2
∑i I(v; ai). On the other
hand, one has χ(M) = 2χ(M) (use suitable triangulations). One conclude the Poincare-Hopf Theorem :
χ(M) = 1/2 · χ(M) = 1/2∑i,j
I(v; aji ) =∑i
I(v; ai).
2.4.2 Consequences of Poincare-Hopf Theorem
As an important consequence of the Poincare-Hopf Theorem, one has the following
Corollary 2.15 Let M be a compact smooth manifold, if χ(M) 6= 0, then any continuousvector field tangent to the manifold M admits at least a singular point. Reciprocally, everycompact manifold such that χ(M) = 0 admits a continuous tangent vector field withoutsingularities.
The unitary sphere Sn with odd n satisfies χ(Sn) = 0 and admits continuous tangentvector fields without singularities. If n is even, χ(Sn) = 2 and in that case every continuousvector field tangent to Sn admits at least one singularity.
Corollary 2.16 Every compact odd dimensional manifold admits a continuous tangentvector field without singularity.
The torus and the Klein bottle are the only one compact 2-dimensional surface ad-mitting a non-zero continuous tangent vector field.
Lemma 2.17 For even-dimensional hypersurfaces, the Euler-Poincare characteristic χ(M)equals twice the degree of the Gauss map γ.
Proof: Take the projection π : Sn → RPn and a regular value p ∈ RPn of thecomposed map π γ : M → RPn. Take a differentiable vector field w on Sn withisolated singularities in a, b = π−1(p) of indices +1. The vector field v on M such thatv(x) = w(γ(x)) has a finite number of isolated singularities a1, . . . , ar = γ−1(a) andb1, . . . , bs = γ−1(b). One one hand, one has deg(γ) =
∑ri=1 I(v; ai) =
∑sj=1 I(v; bj), on
the other hand χ(M) =∑r
i=1 I(v; ai) +∑s
j=1 I(v; bj). That gives the Lemma.
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2.4.3 The smooth case with boundary
Let M be an oriented manifold with boundary, one has a similar theorem:
Theorem 2.18 [Poincare-Hopf Theorem with boundary] Let M be a compact manifoldwith boundary ∂M embedded in an oriented differentiable manifold N . Let v be a non-singular continuous vector field tangent to N , strictly pointing outwards (resp. inwards)M along the boundary ∂M . Then:
1. v can be extended to the interior of M as a vector field tangent to M with finitelymany isolated singularities ai.
2. The total index of v in M is independent of the way we extend it to the interior ofM . In other words, the total index of v is fully determined by its behaviour near theboundary.
3. If v is everywhere transverse to the boundary and pointing outwards from M , thenone has
χ(M) =∑i
I(v, ai). (2.19)
If v is everywhere transverse to ∂M and pointing inwards then
χ(intM) = χ(M)− χ(∂M) =∑i
I(v, ai). (2.20)
Proof: The first statement is proved by obstruction theory (section 2.2). The vectorfield can be extended without singularities to the (n− 1)-skeleton of M . Then we extendit to the n-cells introducing (if necessary) a singular point for each n-cell.
The second statement is also a general result in obstruction theory, that can be ob-tained (for instance) as a consequence of statement 3 (see also 3.3 and [Ste]).
A proof of the third statement goes in the following way: Like in Theorem 2.13,on consider the closed ε-neighbourhood of M , denoted by Nε. If the vector field ispointing outward along ∂M , then it can be extended over the neighbourhood Nε so thatthe extended one points outward along ∂Nε. The extension w is defined as before byw(x) = (x− r(x)) + v(r(x)) and is a continuous vector field near ∂M . In this case, Nε isnot necessarily of class C∞, but only a C1-manifold. Nevertheless, the same argument asin the case “without boundary” can be carried out (see [Mi1] §6), that gives (2.19).
If the vector field is pointing inward along ∂M , one can extend v inside M with finitelymany isolated singularities ai of index I(v, ai).
One proceeds to the following construction: the boundary ∂M admits a neighbourhood∂M × [0, 1] in M and one can extend this neighbourhood as ∂M × [0, 2]. Let us call M ′
the new manifold M ∪ (∂M × [0, 2]). One has χ(M ′) = χ(M) and ∂M ′ ∼= ∂M . Let uscall C the collar ∂M × [1, 2]. One has χ(C) = χ(∂M).
At the level C1 = ∂M×1, one has the vector field v pointing inward M and outwardC. At the level C2 = ∂M × 2, one considers any vector field v′ pointing outward M ′
along ∂M ′. Let us call w the vector field defined on ∂C which is equal to v and v′ onC1 and C2 respectively. The vector field w is defined on the boundary of C and pointing
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outward C along the boundary. By (2.19) on C, one can extend w inside C with finitelymany isolated singularities bj and one has
χ(C) = χ(∂M) =∑j
I(w, bj).
On M ′ one consider the vector field v′, which is v on M and w on C. It has isolatedsingularities ai and bj and it is pointing outward M ′. Again one can apply (2.19) (on M ′)and one has
χ(M) = χ(M ′) =∑i
I(v, ai) +∑
I(w, bj) =∑i
I(v, ai) + χ(∂M)
and the result.
Corollary 2.21 Let us suppose M is odd-dimensional, then
χ(∂M) = 2 · χ(M)
Proof: Let us denote by v a vector field pointing outwards D along the boundary,like in 2.19 and let us consider the vector field w = −v. Then, w has same singularitiesthan v and, as M is odd-dimensional, in each singularity ai, one has I(w, ai) = −I(v, ai).Equations 2.19 and 2.20 provide the result.
Corollary 2.22 Let us denote by M ⊂ Rk a compact manifold without boundary in anodd-dimensional Euclidean space and by Nε the closed ε-neighbourhood of M in Rk. Thenone has
χ(∂Nε) = 2 · χ(M)
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3 Characteristic classes : the smooth case
In 1935 and independently, Stiefel, who was student of Hopf, and Whitney defined char-acteristic classes in cohomology for real manifolds. Stiefel considers the obstruction pointof view (for the construction of r-frames tangent to the manifold), computing homotopygroups of so-called Stiefel manifolds. Whitney considers sphere bundles on a manifold Mand defines cohomology classes with coefficients in Z/2 = Z/2Z. The Stiefel and Whitneymethods are similar and represent the basis of obstruction theory. We call Stiefel-Whitneyclasses of a vector bundle or of the associated sphere bundle, the classes obtained in thatway.
In 1942 Pontrjagyn defined classes for Grassmannian manifolds, using a decompositionof these manifolds in terms of Schubert varieties, due to Ehresmann.
In his fundamental 1946 paper [Ch], Chern gave several constructions of characteristicclasses for Hermitian Manifolds. The paper provides basement for the relationship be-tween obstruction theory, Schubert varieties, differential forms, transgression, etc... Wewill briefly recall some of these definitions, either because they extend to the case ofsingular varieties, or because they will be useful for the following.
Contribution of Wu Wen Tsun in the history of characteristic classes is important.Among results, he proved the product formula for Stiefel-Whitney and Chern classes, hegave a simple formulation of the decomposition of the Grassmann manifold of orientedvector subspaces and he extended the definition of Chern classes for any complex vectorspace on any finite simplicial complex.
As it happens often in Mathematics, one object, here the Chern classes, has (at least)two definitions: the geometric definition allows to understand the signification of classes,but it is difficult to proceed to effective computations in this context. The axiomaticdefinition provides easy ways to compute effectively the classes but is less suitable tounderstanding the origin and the meaning of the classes (see [MS, Hu]).
A trivial bundle is induced from a map to a point, so all its characteristic classes (exceptthe zero dimensional one) should be zero. More generally, equality of all characteristicclasses of two bundles is a necessary (and in some circumstances sufficient) test for theirequivalence. That is one of the important uses of characteristic classes.
The interested reader will find all wished references in the Dieudonne book [Di], 3,IV.
3.1 Fibre bundles.
In this section, we will denote by K either the real field R or the complex field C. Weprovide elementary definitions and properties of vector and fibre bundles, as well as a
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series of examples in the real and complex situations. The reader will find in the literaturesuitable references for more definitions and properties (see for instance [Hirz] and [Hu]).
3.1.1 Vector bundles
Definition 3.1 A vector bundle E, over the field K, with base X and rank n is a topo-logical space E, with a continuous map π : E → X, the projection, such that for everypoint x ∈ X, the fibre Ex = π−1(x) is a vector space of rank n over K.
A vector bundle satisfies the local triviality condition: for every point x ∈ X, there isan open neighbourhood Ux in X and a homeomorphism φ : π−1(Ux) → Ux × Kn whichinduces for every y ∈ Ux an isomorphism π−1(y)→ Kn.
A trivial bundle is a bundle for which one has “global” triviality, i.e. one can takeUx = X in the previous condition.
Given a vector bundle E over X, one define in a natural way the dual vector bundleE∗ and the bundle of k-vectors ΛkE whose fibres are respectively (Kn)∗ and ΛkKn.
Vector bundles are special cases of fibre bundles that we recall now.
3.1.2 Fibre bundles
Let F a topological space and G a topological group which acts effectively and contin-uously on F . That means there is a continuous map G × F → F such that one hasg1 · (g2 · a) = (g1g2) · a for g1, g2 ∈ G and a ∈ F , and e · a = a if e is the identity elementin G. The action is effective means that if g · a = a for some a ∈ F then g = e.
Definition 3.2 A topological space E with a continuous projection π : E → X, is calleda fibre bundle with fibre F and structure group G if G acts effectively and continuouslyon F and there are a system of coordinates (Ui, φi) on X and continuous functions gij :Ui ∩ Uj → G such that:
• Ui is an open covering of X and φi : π−1(Ui) → Ui × F is a homeomorphismidentifying π−1(x) with the fibre x × F ,
• (φi φ−1j )(x, a) = (x, gij(x) · a) for all x ∈ Ui ∩ Uj and a ∈ F .
The fonctions gij are called transition functions. They satisfy
gij gjk gki = id for all i, j, k,
hence, they define a cocycle in Z1(X,G), then an element in H1(X,G). It is well knownthat isomorphism classes of fibre bundles over X with fibre F and structural group G are ina one-one correspondence with the elements of H1(X,G). The trivial bundle correspondsto the element 1 ∈ H1(X,G). Fibre bundles in the same isomorphism class ξ ∈ H1(X,G)are said associated bundles.
The fibre bundle is said differentiable if X is a differentiable manifold and G a real Liegroup, the gij being differentiable functions. The fibre bundle is said complex analytic ifX is a complex manifold and G a complex Lie group, the gij being holomorphic functions.
A section of the fibre bundle E is a continuous application s : X → E such that, forevery point x ∈ X, one has s(x) ∈ Ex = π−1(x).
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3.1.3 Examples of fibre bundles - real case
In order to provide examples of real vector bundles and fibre bundles, we will use thefollowing spaces:
The real projective space RPn is the space of lines through the origin of Rn+1. TheGrassman manifold Gr(Rn) is the space of all vector subspaces of dimension r of Rn.
Let R∞ be the vector space of all infinite sequences (x1, x2, . . .) whose elements xi arereal numbers, a finite number of them being nonzero. The infinite Grassmannian manifoldGr(R∞) is the set of all r-dimensional subspaces in R∞, i.e. the direct limit of the naturalsequence of inclusions
Gr(Rr) ⊂ Gr(Rr+1) ⊂ Gr(Rr+2) ⊂ · · ·
We consider onGr(R∞) the topology for which closed subsets are those whose intersectionswith all Gr(Rr+k) are closed.
Examples of real vector bundles are given by:
1. the tangent bundle TM to a differentiable manifold M . That is the set of all pairs(x, v) such that x ∈ M and v is a vector tangent to M at the point x, i.e. anelement of TxM . If M is an n-manifold, then TM is a real vector bundle with rankn over M , the fibre is Rn.
In particular, one has the bundle TSn tangent to the sphere Sn, that is a trivialbundle if n = 1, a non trivial bundle if n = 2. One has also the bundle TRPntangent to RPn.
2. the normal bundle to a differentiable n-manifold M embedded in Rn+k. That is theset of all pairs (x, v) ∈ M × Rn+k such that v is orthogonal to the tangent spaceTxM ∼= Rn in Tx(Rn+k) ∼= Rn+k.
3. the canonical bundle over RPn also called tautological bundle and denoted by γn1 :
γn1 → RPn (3.3)
This line bundle is the set of all pairs (λ, v) where λ is an element of RPn, i.e. aline passing through the origin of Rn+1 and v a vector of λ. The canonical bundleis not trivial, and this fact is the basis for the axiomatic definition of characteristicclasses.
4. the canonical bundle γnr over the Grassman manifold Gr(Rn). That is the set of allpairs (P, v) where P is an element of Gr(Rn) and v a vector in P . One has thebundle projection
γnr → Gr(Rn)
and γnr is a vector bundle with rank r.
The bundle is also called universal bundle for vector bundles of rank r. That meansthat every bundle ξ with rank r over a (paracompact) topological space X is iso-morphic to f ∗(γnr ) for some f : X → Gr(Rn) with sufficiently large n.
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5. the universal bundleγr → Gr(R∞)
set of all pairs (P, v) where P is an element of Gr(R∞) and v a vector of P . It isuniversal for all rank r-vector bundles.
In the case r = 1, that is the bundle γ1 → RP∞.
6. the Stiefel manifold, denoted by Vr(Rn) is the set of r-frames in Rn, that is the setof ordered r-uples (v1, . . . , vr) of r linearly independent vectors in Rn. (see Steenrod[Ste] where this manifold is denoted by V ′r,n).
One has a homotopy Vr(Rn) ∼= Vr,n = O(n)/O(n− r).The natural map Vr,n → Gr(Rn) is a principal fibre bundle, i.e. the fibre O(r)coincides with the structural group. That is an universal bundle for fibre bundleswhose basis has dimension ≤ n− r − 1.
The vector bundle γnr → Gr(Rn) is a bundle associated to Vr,n → Gr(Rn) with fibreRr.
7. the bundle Vr(TM) of r-frames tangent to a n-differentiable manifold M , i.e. theset of all pairs (x, (v1, . . . , vr)) where x is a point of M and (v1, . . . , vr) is a r-framein the fibre TxM over x. That is the fibre bundle over M whose fibre at x is themanifold Vr(TxM) of all r-frames in TxM . The fibre is the Stiefel manifold Vr(Rn).
Note that a section of this bundle in Stiefel manifolds is a r-uple of linearly inde-pendent sections of the vector bundle TM .
3.1.4 Examples of fibre bundles - complex case
One considers the complex projective space CPn whose homogeneous coordinates will bedenoted by (x0 : x1 : . . . : xn). The projective space is covered by open subsets Uii=0,...n
homeomorphic to Cn and whose coordinates are (x0, x1, . . . , xi−1, 1, xi+1, . . . , xn).We will consider the complex Grassmannian manifolds Gr(Cn) and Gr(C∞) in a similar
way than in the real case.
1. the complex tangent bundle TM to a complex n-dimensional manifold M is a fibrebundle over M . Each fibre TxM has a complex structure and is isomorphic to Cn.
In particular, one has the tangent bundle TCPn to CPn.
2. the canonical bundle γn1 over CPn. also called tautological or universal bundle anddenoted by O(−1) in algebraic geometry:
γn1 → CPn (3.4)
This line bundle is the set of all pairs (λ, v) where λ is an element of CPn, i.e.a complex line passing through the origin of Cn+1 and v a vector in λ. That is thefibre over λ is the line λ.
γn1 = (λ, v) ∈ CPn × Cn+1|v ∈ λ.
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With the previous homogeneous coordinates in CPn, the transition functions of γn1in Ui ∩ Uj are defined by xi/xj.
3. the “hyperplane” bundle H over CPn, dual of the canonical bundle. It is denotedby O(1) in algebraic geometry. With the previous homogeneous coordinates, thetransition functions of the hyperplane bundle in Ui ∩ Uj are defined by (xi/xj)
−1.
The hyperplane x0 = 0 with the induced orientation, is CPn−1, that is a 2(n − 1)-cycle in H2(n−1)(CPn;Z). The Poincare dual cohomology class hn is a generator ofH2(CPn;Z). We will see that c(H) = 1 + hn.
4. the universal bundleγnr → Gr(Cn)
is the set of all pairs (P, v) where P is an element of Gr(Cn) and v a vector of P .That is a vector bundle of rank r over Gr(Cn).
Every complex vector bundle ξ with rank r over a (paracompact) topological spaceX is isomorphic to f ∗(γnr ) for some f : X → Gr(Cn) with sufficiently large n.
5. the universal bundleγr → Gr(C∞)
is the set of all pairs (P, v) where P is an element of Gr(C∞) and v a vector ofP . In particular, one has the bundle γ1 → CP∞.
The bundle γr is universal for all rank r-vector bundles.
6. One defines the Stiefel manifold Vr(Cn) which is the set of r-frames in Cn, that isthe set of ordered r-uples (v1, . . . , vr) of C-linearly independent vectors in Cn (seeSteenrod [Ste] where the Stiefel manifold is denoted by W ′
r,n).
One has a homotopy
Vr(Cn) ∼= Wr,n = U(n)/U(n− r).
The fibre bundle Wr,n → Gr(Cn) is a principal bundle with fibre and structuralgroup U(r).
The fibre bundle Wr,n → Gr(Cn) is an universal bundle for bundles which basis hasdimension ≤ 2(n− r).The vector bundle γnr → Gr(Cn) is a bundle associated to Wr,n → Gr(Cn) with fibreCr.
7. One define the bundle of complex r-frames tangent to the complex n-manifold M ,i.e. the set of all pairs (x, (v1, . . . , vr)) where x is a point of M and (v1, . . . , vr) isa r-frame in the fibre TxM over x. That is the fibre bundle whose fibre at x is themanifold Vr(TxM) consisting of all complex r-frames in TxM . The “typical” fibreis the Stiefel manifold Vr(Cn).
Note that a section of this bundle in complex Stiefel manifolds is a r-uple of C-linearly independent sections of the complex vector bundle TM .
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3.2 General obstruction theory
Let us recall the idea of the construction of characteristic classes by obstruction theory,following Steenrod [Ste], part III.
We have seen that the meaning of Poincare-Hopf Theorem is that the Euler-Poincarecharacteristic of a manifold M is a measure of the obstruction for the construction of avector field tangent to M . In a more general way, the aim of the obstruction theory isto define invariants providing a measure of the obstruction to the construction of linearlyindependent sections of vector bundles. In a more precise way, the objective is to answerto questions of the following type:
Let E be a vector bundle of rank n on a variety X and fix r such that 1 ≤ r ≤ n, isit possible to construct r sections of E, linearly independent everywhere?
It is obviously possible to define such sections on the 0-skeleton of a triangulation ofX. So, the question becomes the following:
Let us consider a triangulation of X. Performing the construction of r independentsections by increasing dimension of the simplexes, up to what dimension can we proceed?Arriving to this obstruction dimension, is it possible to evaluate the obstruction?
At that point, let us make a comment: Classical obstruction theory uses a triangulationof the considered space. In the following we will use a slightly different viewpoint, takinginto account the fact that we want to deal with the singular case. It appears that in thesingular case, the good decomposition to be taken into account for the construction of thesections is not a triangulation of the space but a dual cell decomposition in the ambientspace. That is the reason for which, we will work on a cell decomposition, already in thenon-singular case.
Let us consider a (simplicial or cellular) complex K and a subcomplex L. We willdenote by X = |K| and Y = |L| the respective geometric realisations. The q-skeleton ofK is denoted by Kq, that is the subcomplex consisting of all simplexes (or cells) whosedimension is less or equal to q. Let us denote Xq = |Kq| the associated space.
We consider a fibre bundle E with basis X and fibre F . To consider a section definedin a trivialisation open subset for the bundle provides a map with target F , as we alreadyseen, see for instance (3.16).
Aim of obstruction theory is to describe the problem of extension of maps f : Y → Fto all of X, by successive extensions of the map from Xq to Xq+1. Let us suppose thatthe function f : X → F is already known on Xp−1 and let us denote it by fp−1. Let dp
an oriented p-cell, fp−1 is well defined on the boundary ∂dp and determines an element[fp−1|∂dp ] ∈ πp−1(F ).
Definition 3.5 The relative cochain denoted by c(fp−1) ∈ Cp(K,L; πp−1(F )) and definedby
c(fp−1)(dp) = [fp−1|∂dp ] ∈ πp−1(F ) (3.6)
is called obstruction cochain (for the extension of fp−1 to Xp).
The function fp−1 can be extended to Xp if and only if c(fp−1) = 0. In particular,if πi(F ) = 0 for i = 1, . . . , j − 1, then every function fY : Y → F can be extended tofj : Xj → F .
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Lemma 3.7 If fp−1 is homotopic to gp−1, then c(fp−1) = c(gp−1).
Proof: In fact, as fp−1|∂dp ∼= gp−1|∂dp , one has [fp−1|∂dp ] = [gp−1|∂dp ].
Theorem 3.8 c(fp−1) is a cocycle.
Proof: Let τ p+1 a (p+ 1)-cell. One has to show that δ[c(fp−1)](τ p+1) = 0. One has
δ[c(fp−1)](τ p+1) = c(fp−1)[∂τ p+1] = c(fp−1)(∑
[τ p+1 : dpi ]dpi ) =∑
[τ p+1 : dpi ]c(fp−1)(dpi ) =∑
[τ p+1 : dpi ][fp−1|∂dpi ]
where the sum is taken on all cells dpi which are faces of τ p+1. Let us suppose that incidenceof all faces dpi of τ p+1 with τ p+1 is positive, then denoting fp−1|∂dpi = αi, one has
∑αi = 0,
and the result. If incidence is not positive, then [τ p+1 : dpi ][fp−1|∂dpi ] = αi is the elementof πp−1(F ) obtained from the function fp−1 restricted to the boundary of the face dpi withthe orientation induced from τ p+1 and one has
∑[τ p+1 : dpi ][fp−1|∂dpi ] =
∑αi = 0.
3.2.1 The difference cochain
Let fp−1 and gp−1 two extensions on Xp−1 of the same fp−2 : Xp−2 → F . We intend toprovide a “measure” of their difference. Let dp−1 a (p− 1)-cell in (K). The two functionsfp−1|dp−1 and gp−1|dp−1 coincide on the boundary ∂dp−1. The cell dp−1 is homeomorphicboth to the north hemisphere Dp−1
+ , and to the south hemisphere Dp−1− , of the sphere
Sp−1. One can interpret fp−1|dp−1 , resp. gp−1|dp−1 , as a function of Dp−1+ , resp. Dp−1
− in F .These functions coincide on the equator Sp−2, homeomorphic to ∂dp−1, hence they definea function γ : Sp−1 → F . Annulation of the homotopy class [γ] ∈ πp−1(F ) is a necessaryand sufficient condition to deform gp−1|dp−1 in fp−1|dp−1 .
Definition 3.9 The difference cochain d(fp−1, gp−1) ∈ Cp−1(K,L; πp−1(F )) is defined by
d(fp−1, gp−1)(dp−1) = (−1)p[γ] ∈ πp−1(F ).
The difference cochain is a relative cochain of K modulo L. It vanishes if and only iffp−1
∼= gp−1 relatively to Xp−2. If hp−1 is a third extension of fp−2 then one has
d(fp−1, hp−1) = d(fp−1, gp−1) + d(gp−1, hp−1).
If fp−1 is an extension of fp−2 and cp−1 ∈ Cp−1(K,L; πp−1(F )) is a relative cochain, thenthere is an extension gp−1 of fp−2 such that d(fp−1, gp−1) = cp−1.
Theorem 3.10 One has
δd(fp−1, gp−1) = c(fp−1)− c(gp−1).
That means that the difference of the obstruction cocycles of two extensions of fp−2
is a coboundary.
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Lemma 3.11 If fp−2 can be extended as a function fp−1 : Xp−1 → F , then all obstructioncocycles c(fp−1) for extension of fp−2 to Xp−1 belong to the same cohomology class
c(f) ∈ Hp(K,L; πp−1(F )).
Theorem 3.12 Let fp−1 : Xp−1 → F , then fp−2 extends to fp : Xp → F if and only ifc(f) = 0.
3.2.2 The obstruction class
Here we are using cohomology with local coefficients Hp(K; πp−1(F )), i.e. bundle ofabelian groups which associate to each point x of X the coefficient group πp−1(Fx).
Let us suppose that πi(F ) = 0 for i ≤ p − 2. Then one can construct a fonction fiwithout singularity for 1 ≤ i ≤ p− 1.
Definition 3.13 Let us suppose that πi(F ) = 0 for i ≤ p− 2. The primary obstructionclass is the class of the obstruction cocycle [c(fp−1)], that is
c(f) ∈ Hp(K,L; πp−1(F )).
Let us remark that in general, the system of coefficients πp−1(F ) is twisted.
3.3 Case of the tangent bundle
We study the particular case of the (real) tangent bundle to a differentiable smoothmanifold or the (complex) tangent bundle to an analytic complex manifold. We willdenote by K the field R or C, according to the situation.
Let M be a manifold of dimension n, over K, endowed with an euclidean (or hermitian)metric. The tangent bundle to M , denoted by TM , is a vector bundle of rank n over K,whose fibre in a point x of M is the tangent vector space to M in x, denoted by Tx(M)and is isomorphic to Kn. The vector bundle TM is locally trivial, i.e. there is a coveringof M by open subsets such that the restriction of TM to U is isomorphic to U ×Kn.
The objective is to evaluate the obstruction to the construction of r sections of TMlinearly independent (over K) in each point, i.e. an r-frame:
Definition 3.14 An r-field on a subset A of M is a set v(r) = v1, . . . , vr of r continuousvector fields tangent to M , defined on A. A singular point of v(r) is a point where thevectors (vi) fail to be linearly independent. A non-singular r-field is called an r-frame.
The r-frames are sections of the fibre bundle Vr(TM) over M . That is the fibre bundleassociated to TM and whose fibre at the point x of M is the set of r-frames of Tx(M).The fibre is the Stiefel manifold denoted by Vr(Kn) that we described in 3.1.3 in the realcase and in 3.1.4 in the complex case.
To construct r linearly independent sections of TM over a subset A of M is equivalentto construct a section of Vr(TM) over A.
Let us consider the following situation: (K) is a cell decomposition of M sufficientlysmall so that every cell d is included in an open subset U over which Vr(TM) is trivial.
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One remarks that trivialisation open sets for Vr(TM) are the same that the ones of T (M).There exists always such a cell decomposition.
Let us consider the following question:Let us suppose that one has a section v(r) of Vr(TM) on the boundary ∂d of the k-cell
d. Is it possible to extend this section in the interior of d ? Is the answer is no, can oneevaluate the obstruction for such an extension ?
In order to answer the question, we need to define the notion of index of an r-field in asingular point and we need some notions and results on general obstruction theory. Thatis aim of the following sections. Then we will apply these results to the real and complexcase, that is to define Stiefel-Whitney and Chern classes.
3.3.1 Index of a r-frame
Let us consider an r-field v(r) defined on the boundary ∂d of a k-cell d of the cell decom-position (D) of M . In the same way than in 2.2, v(r) is a section of the bundle Vr(TM),defined on the boundary of d. It provides a map
∂dv(r)
−→ Vr(TM)|U ∼= U × Vr(Kn)pr2−→ Vr(Kn), (3.15)
where pr2 is the second projection.
Figure 5: Obstruction theory
One obtains a map
Sk−1 ∼= ∂dpr2v(r)
−→ Vr(Kn) (3.16)
which defines an element of πk−1(Vr(Kn)) denoted by [ξ(v(r), d)].Let us suppose that [ξ(v(r), d)] = 0, then, by classical homotopy theory, the map
Sk−1 → Vr(Kn) defined on the boundary Sk−1 of the ball Bk can be extended inside the
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ball. In another words, if [ξ(v(r), d)] = 0, then the map ∂d → Vr(Kn), i.e. the r-frame,can be extended inside the cell d. One can resume the situation by the following diagramsimilar to (2.10).
∂d ∼= Sk−1 −−→ Vr(Kn)y ?
d ∼= Bk
This means that there is no obstruction to the extension of the section v(r) inside d. Thishappens for example in the case πk−1(Vr(Kn)) = 0.
In order to answer to the previous question, we need to know the homotopy groupsof Vr(Kn). The homotopy groups πk−1(Vr(Kn)) have been computed by Stiefel and byWhitney (see [Sti]) in the cases K = R and C. One has the following result:
Let Vr(Rn) be the Stiefel manifold of r-frames in Rn, one has:
πi(Vr(Rn)) =
0 for i < n− rZ for i = n− r even or i = n− 1 if r = 1
Z2 for i = n− r odd and r > 1
(3.17)
For the Stiefel manifold Vr(Cn) of (complex) r-frames in Cn, one has:
πi(Vr(Cn)) =
0 for i < 2n− 2r + 1
Z for i = 2n− 2r + 1(3.18)
A generator of the first non-zero homotopy group can be described in the followingway. Let us give it in the real framework, then in the complex one.
In the real case, let us denote p = n−r+1, one describes a generator of πn−r(Vr(Rn)).Let us fix a (r−1)-frame in Rn. It defines a (r−1)-subspace of Rn whose complementaryis a real space Rp. The unit sphere in Rp denoted by Sp−1 is oriented. Let us consider,for each point w of the sphere, a r-frame consisting of the vector w and the fixed (r− 1)-frame, one obtains an element of Vr(Rn). The induced map from the oriented sphere Sp−1
to Vr(Rn) defines a generator of πp−1(Vr(Rn)).In the complex case, let us denote 2p = 2(n − r + 1), one describes a generator of
π2p−1(Vr(Cn)) ∼= Z. Let us fix a (r − 1)-frame in Cn. It defines a (r − 1)-subspace of Cn
whose complementary is a complex space Cp. The unit sphere in Cp denoted by S2p−1 isoriented, the orientation being induced by the natural one of Cp. Let us consider, for eachpoint w of the sphere, a r-frame consisting of the vector w and the fixed (r − 1)-frame,one obtains an element of Vr(Cn). The induced map from the oriented sphere S2p−1 toVr(Cn) defines a generator of π2p−1(Vr(Cn)).
One obtain the following result:
Proposition 3.19 – Real case. Let us consider an r-frame v(r) defined on the boundaryof the k-cell d.
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• If k < n− r+ 1, one has [ξ(v(r), d)] = 0 and one can extend the r-frame defined on ∂dinside d without singularity.
• If r = 1 and k = n, one can extend the vector field v(1) = v defined on ∂d inside d withan isolated singularity in the barycenter d of the cell, with index [ξ(v, d)] = I(v, d).That is the index we defined in Definition 2.1.
• If r > 1 and k = n− r + 1, then one can extend the r-frame v(r) defined on ∂d insided with an isolated singularity at the barycenter d of the cell. In that case, [ξ(v(r), d)]is an integer if k is odd and an integer mod 2 if k is even. Reducing modulo 2,one obtains an index I(v(r), d) that measures the obstruction to the extension of v(r)
inside the k-cell d.
The dimension p = n − r + 1 is called the obstruction dimension for the construction ofa r-frame tangent to M .
Proposition 3.20 – Complex case. Let us consider a complex r-frame v(r) defined onthe boundary ∂d of the k-cell d.
• If k < 2(n− r + 1), one has [ξ(v(r), d)] = 0 and one can extend the r-frame v(r) insided without singularity.
• If k = 2(n−r+1), then one can extend the r-frame v(r) inside d with an isolated singu-larity at the barycenter d of the cell. In that case, one obtain an index [ξ(v(r), d)] ∈ Zthat we define as I(v(r), d). The index measures the obstruction to the extension ofv(r), defined on the boundary ∂d, inside d.
The dimension 2p = 2(n− r+ 1) is called the obstruction dimension for the constructionof a complex r-frame tangent to M .
3.4 Applications: Stiefel-Whitney and Chern classes
Let us apply the previous construction to the cases of r-fields tangent to a manifold, inthe real and the complex case.
3.4.1 Stiefel-Whitney classes
The Stiefel-Whitney classes have been defined by obstruction theory (see [Sti],[Wh1]). Infact, Whitney used the same strategy than Stiefel, applying it to arbitrary sphere bundles.We use the Steenrod construction ([Ste], part III).
The pth Stiefel-Whitney class of M , denoted by wp(M), is defined as the primaryobstruction to constructing an r-frame over M , that is a section of Vr(TM) or a set of rlinearly independent vector fields tangent to M , with p = n− r + 1. More precisely, oneperforms the following construction:
Using the result in (3.17) one can construct an r-frame by choosing any r-frame v(r)
on the 0-skeleton of the cell decomposition (D), then extending it without zeroes till theobstruction dimension p = n − r + 1. That means that v(r) has no singularity on the
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(p − 1)-skeleton and isolated singularities on the p-skeleton of (D). Given the r-framev(r) on the boundary of each p-cell d, one can extend v(r) on d with a singularity at thebarycenter d of index
I(v(r), d) = [(v(r))p−1|∂dp ] ∈ πp−1(Vr(Rn)) =
Z for p = n− r + 1 odd or p = n if r = 1
Z2 for p = n− r + 1 even and r > 1,
using the notation in 3.6. In any case, there is a non trivial homomorphism from πp−1(F )
to Z2. hence we can reduce the coefficients modulo 2 obtaining I(v(r), d) ∈ Z2.One can define the p-cochain
∑I(v(r), d) d∗ in Cp(D,Z2), its value on each p-cell d
is I(v(r), d). According to Theorem 3.8, the cochain is in fact a cocycle and defines anelement wp(M) in Hp(M ;Z2). The Definition 3.13 provides the following:
Definition 3.21 The p-th Stiefel-Whitney class of M , denoted by wp(M) ∈ Hp(M ;Z2)is the class of the primary obstruction cocycle corresponding to constructing an r-frametangent to M .
By the general obstruction theory, the obtained classes do not depend on the choiceswe make in the construction.
In the particular case r = 1, one can use integer coefficients. The evaluation ofwm(M) ∈ Hm(M ;Z) on the fundamental class [M ] of M is the Euler-Poincare character-istic of M .
Let us suppose that the cell decomposition (D) is obtained by duality of a triangulation(K) of M . Each p-cell d = d(σ) in (D) is dual of an (r−1)-simplex σ in (K). By Poincareduality (cap-product by the fundamental class),
Hm−r+1(M ;Z2) −→ Hr−1(M ;Z2)
the image of d∗ is σ and image of wp(M) is the so-called (r− 1)-homology Siefel-Whitneyclass, denoted by wr−1(M). A cycle representing wr−1(M) is given (mod 2) by∑
dimσ=r−1
I(v(r), d(σ))σ.
In fact, Stiefel-Whitney classes can be defined in a combinatorial way [HT].
3.4.2 Chern classes
The definition of Chern classes by obstruction theory in the complex case is similar to thereal case, even simpler.
Let M denote an analytic complex manifold and TM the complex tangent bundleto M . The pth Chern class of M , denoted by cp(M), will be defined as the primaryobstruction to constructing a complex r-frame over M , that is a section of Vr(TM) or aset of r linearly independent vector fields tangent to M , with p = n− r + 1.
Using the result in (3.18) one can construct an r-frame by choosing any r-frame v(r)
on the 0-skeleton of the cell decomposition (D), then extending it without zeroes till theobstruction dimension
2p = 2(n− r + 1). (3.22)
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That means that v(r) has no singularity on the (2p−1)-skeleton and isolated singularitieson the 2p-skeleton of (D). Given the r-frame v(r) on the boundary of each 2p-cell d, onecan extend v(r) on d with a singularity at the barycenter d of index
I(v(r), d) = [(v(r))2p−1|∂d2p ] ∈ π2p−1(Vr(Cn)) = Z
using the notation in 3.6.
The generators of π2p−1(Vr(Cn)) being consistent (see [Ste]), one can define the 2p-
cochain∑I(v(r), d) d∗ in C2p(D,Z), its value on each 2p-cell d is I(v(r), d). this defines a
cochain
γ ∈ C2q(M ; π2q−1(Wr,m)) ,
by γ(d) = I(v(r), d), for each 2q-cell d, and then extend it by linearity.
One can define the 2p-cochain∑I(v(r), d) d∗ ∈ C2p(K,Z) (3.22)
whose value on each 2p-cell d is I(v(r), d). According to Theorem 3.8, the cochain is infact a cocycle and defines an element cp(M) in H2p(M ;Z). The Definition 3.13 providesthe following:
Definition 3.23 The p-th Chern class of M , denoted by cp(M) ∈ H2p(M ;Z) is theclass of the primary obstruction cocycle corresponding to constructing a complex r-frametangent to M .
By the general obstruction theory, the obtained classes do not depend on the choiceswe make in the construction.
In the particular case r = 1, the evaluation of cm(M) on the fundamental class [M ] ofM yields the Euler-Poincare characteristic of M .
Note that cm(M) coincides with the Euler class of the underlying real tangent bundleTRM , so these classes are natural generalization of the Euler class.
Let us suppose that the cell decomposition (D) is obtained by duality of a triangulation(K) of M . Each 2p-cell d = d(σ) in (D) is dual of an 2(r − 1)-simplex σ in (K). ByPoincare duality (cap-product by the fundamental class),
H2(m−r+1)(M ;Z) −→ H2(r−1)(M ;Z)
the image of d∗ is σ and image of cp(M) is the so-called 2(r − 1)-homology Chern class,denoted by cr−1(M). A cycle representing cr−1(M) is given by∑
dimσ=2(r−1)
I(v(r), d(σ))σ. (3.23)
As we will see in the next sections, there is no cohomology Chern class in the case ofsingular varieties, but expression (3.23) will generalise for suitable frames we will define.
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3.5 Axiomatic definition
Definition 3.24 [Theorem] To each complex vector bundle ξ of rank n over a space M ,one can associate a class c(ξ) = 1 + c1(ξ) + · · · + cn(ξ), where ci(ξ) ∈ H2i(M ;Z) andci(ξ) = 0 if i > n, satisfying the following properties:
1. (Naturality) For each f : Y →M , then f ∗(c(ξ)) = c(f ∗(ξ)).
2. (Whitney sum) If ξ and η are two bundles over M , then
c(ξ ⊕ η) = c(ξ) ∪ c(η).
3. the class c1(γ11) ∈ H2(CP1;Z) of the canonical line bundle over CP1 (see 3.4) is non
zero.
Modulo naturality, the last axiom is equivalent to the following ones:
4′. Let γn1 be the canonical bundle over CPn, then c1(γn1 ) = hn is a generator ofH2(CPn;Z) (see 3.4).
4′′. Let γ1 be the canonical line bundle over CP∞, then c1(γ1) is a generator of thepolynomial ring H∗(CP∞;Z).
Proposition 3.25 If ξ is a trivial bundle, then ci(ξ) = 0 for i ≥ 1.
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4 Hirzebruch theory
In this section, one explicits the Hirzebruch theory that provides a way to unify, in thecase of manifolds, the three theories of characteristic classes: the Chern class, the Toddclass and the Thom-Hirzebruch class.
4.1 The arithmetic genus
Let gi be the number of C-linearly independent holomorphic differential i-forms on then-dimensional complex algebraic manifold X. The number gi equals the Hodge numberhi,0(M).
• g0 is the number of linearly independent holomorphic functions, i.e. the number ofconnected components of X,
• gn is called geometric genus of X,
• g1 is called irregularity of X,
Definition 4.1 [Arithmetic Genus] The arithmetic genus of X, denoted by χa(X) isdefined as :
χa(X) :=n∑i=0
(−1)igi
Example 4.2 Let X be a complex algebraic curve, i.e. a compact Riemann surface. Xis homeomorphic to a sphere with g handles. Then g0 = 1 and g1 = gn = g.
The arithmetic genus of X is:
χa(X) = 1− g
4.2 The Todd genus
The Todd genus T (X) has been defined (by Todd) in terms of Eger-Todd fundamentalclasses (polar varieties), using Severi results. The Eger-Todd classes are homologicalChern classes of X.
Todd “proved” thatT (X) = χa(X).
In fact, the Todd proof uses a Severi Lemma which has never been completely proved.The Todd result has been proved by Hirzebruch, using other methods.
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4.3 The signature
Definition 4.3 [Thom-Hirzebruch] Let M be a (real) compact oriented 4k-dimensionalmanifold. Then the map
H2k(M ;R)×H2k(M ;R) −→ R, (x, y) 7→ 〈x ∪ y, [M ]〉 ∈ R
defines a bilinear form on the vector space H2k(M ;R).The index (or signature) of M , denoted by sign(M), is defined as the index of this
form, i.e. the number of positive eigenvalues minus the number of negative eigenvalues.
4.4 Hirzebruch Theory
The Hirzebruch theory uses multiplicative series and Chern root. let us define (or recall)these two ingredients.
4.4.1 Hirzebruch Series
For y ∈ R, let us define the Hirzebruch multiplicative series (for a review on multiplicativeseries, see Hirzebruch [Hirz]):
Qy(α) :=α(1 + y)
1− e−α(1+y)− αy ∈ Q[y][[α]]
One has the particular cases:
• Q−1(α) = 1 + α y = −1
• Q0(α) = α
1− e−α y = 0
• Q1(α) = αtanhα
y = 1
4.5 Characteristic Classes of Manifolds
Let X be a complex manifold with dimension dimCX = n, let us denote by
c∗(TX) =n∑j=0
cj(TX), cj(TX) ∈ H2j(X;Z)
the total Chern class of the (complex) tangent bundle TX.
Definition 4.4 The Chern roots αi of the complex manifold X are elements in H2(X;Z)defined by:
n∑j=0
cj(TX) tj =n∏i=1
(1 + αit).
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For each y ∈ R, one defines the Todd-Hirzebruch class: td(y)(TX) :=n∏i=1
Qy(αi)
td(y)(TX) =
c∗(TX) =n∏i=1
(1 + αi) y = −1
Chern class,
td∗(TX) =n∏i=1
( αi1−e−αi ) y = 0
Todd class,
L∗(TX) =n∏i=1
( αitanhαi
) y = 1
Thom-Hirzebruch L-class.
Remark 4.5 The previous equalities can be considered as definition of the Todd classand the Thom-Hirzebruch L-class for the reader who is not already aware of these notions.
4.6 The χy-characteristic
Let X be a complex projective manifold.
Definition 4.6 For each y ∈ R, one defines the χy-characteristic of X by
χy(X) :=<∞∑p=0
(<∞∑i=0
(−1)i dimCHi(X,∧pT ∗X
)· yp.
One has the particular cases:
• y = −1 χ−1(X) = χ(X), Euler-Poincare characteristic of X (by Hodge theory),
• y = 0 χ0(X) = χa(X), arithmetic genus of X (by definition),
• y = 1 χ1(X) = sign(X), signature of X (by Hodge theory).
One has the following table of invariants:
y χy(X) td(y)(TX)
-1 χ(X) c∗(TX)
0 χa(X) td∗(TX)
1 sign(X) L∗(TX)
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4.7 Hirzebruch Riemann-Roch Theorem
Theorem 4.7 (Hirzebruch Riemann-Roch Theorem) One has:
χy(X) =
∫X
td(y)(TX) ∩ [X] ∈ Q[y].
In the three particular cases, one obtains:
• χ(X) =∫Xc∗(TX) ∩ [X] y = −1
Euler - Poincare characteristic of X
Poincare-Hopf Theorem,
• χa(X) =∫Xtd∗(TX) ∩ [X] y = 0
arithmetic genus of X
Hirzebruch-Riemann-Roch Theorem,
• sign(X) =∫XL∗(TX) ∩ [X] y = 1
signature of X
Hirzebruch signature Theorem.
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5 Singular varieties
A singular variety is a variety which contains singular points, that is points for which theproperty in Definition 1.8 is not satisfied. Examples of singular varieties are the following:The pinched torus: the pinched point a does not admit any neighbourhood satisfying theproperty 1.8. In that case, the link of an “elementary neighbourhood” of a is the union oftwo not connected circles. Another example is provided by the suspension of the torus.The two points a and b of the suspension of the torus are singular points, in that case,the link of a (or b) is a torus, it is not a sphere.
In order to extend the notion of characteristic classes to singular varieties, it is neces-sary to know the local structure of the singular variety. That is given by the structure ofstratified space and by suitable definition of triangulation on the variety.
5.1 Stratifications
Definition 5.1 Let X be a topological space, we denote by X a filtration of X by closedsubsets
A topological stratification of X is the data of a filtration X of X such that each differenceVk = Xk − Xk−1 is either empty or a topological manifold of pure dimension k. Theconnected components of the Vk are called the strata.
The stratifications that we will consider will be locally finite partitions of X into locallyclosed submanifolds, the strata, satisfying the frontier condition:
Vk ∩ V j 6= ∅ ⇒ Vk ⊂ V j
Let X be a closed subset of a differentiable manifold M . A differentiable stratificationof X is a topological stratification X of X such that each stratum in Vk is a differentiablesubmanifold of M .
In order to work with, the considered stratification should satisfy conditions whichprecise the way the strata are glued together. On one hand, there are many ways todefine these conditions, according to the specific problem. On the other hand, givenconditions on the stratification, one has to know what kind of singular variety admits astratification satisfying these conditions. In the following, one considers the stratificationswhich will be useful for the construction of characteristic classes. One refer to [B3, Tr]for more information on the different types of stratifications.
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5.1.1 Whitney stratifications
As we have seen, on a singular variety, there is no longer tangent space in the singularpoints. One way to find a substitute for the tangent bundle is to stratify the singularvariety in submanifolds. One can proceed the following construction: If X is a singularanalytic variety, equipped with a stratification and embedded in a smooth analytic mani-fold M , one can consider the union of tangent bundles to the strata, that is a subspace Eof the tangent bundle to M . The space E is not a bundle but it generalizes the notion oftangent bundle in the following sense: A section of E over X is a section v of TM |X suchthat at each point x ∈ X the vector v(x) belongs to the tangent space of the stratumcontaining x. Such a section is called a stratified vector field over X. To consider E asthe substitute for the tangent bundle of X and to use obstruction theory is the M.-H.Schwartz point of view (1965, [Sc2]) in the case of analytic varieties.
When one consider stratification of singular varieties, it is natural to ask for conditionswith which the strata glue together. The so-called Whitney conditions are those whichallow one to proceed to the construction of radial extension of vector fields. According toa result of Whitney, every analytic variety can be equipped with a Whitney stratification.
Definition 5.3 One says that the Whitney conditions are satisfied for a stratification if,for any pair of strata (Vi, Vj) such that Vi is in the closure of Vj, one has:a) if (xn) is a sequence of points in Vj with limit y ∈ Vi and if the sequence of tangentspaces Txn(Vj) admits a limit T (in the suitable grassmanian space) when n goes to +∞,then Ty(Vi) is included in T .b) if (xn) is a sequence of points in Vj with limit y ∈ Vi and if (yn) is a sequence of pointsin Vi with limit y, such that the sequence of tangent spaces Txn(Vj) admits a limit T forn going to +∞ and such that the sequence of directions xn yn admits a limit λ when ngoes to +∞, then λ lies in T .
(a) Condition (b) Condition
Figure 6: Whitney conditions
Example The conditions (a) and (b) are not satisfied for the stratification of the cone Xconsisting of a generatix D = Vi and Vj = X \D. This is clear taking for (xn) a sequenceof points going to the vertex y of the cone, along a generatrix (different from D), and for
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yn a sequence of points such that the segment xnyn has always the same direction. Addingthe vertex of the cone as a supplementary 0-dimensional stratum, the new stratificationsatisfies the Whitney conditions.
Figure 7: Whitney conditions on the cone
Example Let V be the variety whose equation in C3 is y2 + x3 − t2x2 = 0, stratified bythe horizontal axis W = Vi and Vj = V − Vi, then the (a)-condition is satisfied but not(b). Adding the vertex of Cn as a new stratum, the Whitney conditions are verified.
Figure 8: Whitney conditions
Let us consider a neighborhood of a point y ∈ Vi, and a sufficiently samlll Let Dε
a disk in M transverse to Vi at y and let us consider at x ∈ Vj ∩ D the vector v(x)parallel to v(y), For epsilon sufficiently small, angle(v(x), Tx(Vj)) is small (less than ε.The projection of v(x) on Tx(Vj)) does not vanish.
5.2 Poincare homomorphism
Let us recall (see [B1]) that the Poincare homomorphism can be described in the followingway. Let us suppose that the oriented singular n-dimensional variety X is a subvarietyof a P.L. oriented m-dimensional manifold M . Any stratification of X defines a strati-fication of M adding M − X as regular stratum. Let us denote by (K) a locally finitetriangulation of M compatible with the stratification and by (K ′) a barycentric subdivi-sion of (K). The chain (or cochain) complexes relatively to (K) or (K ′) will be denoted
by C(K′)∗ (X), C∗(K)(X) for example.
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Providing to all n-dimensional simplexes of (K ′) the orientation of Xreg, the sum of
these simplexes is a cycle in C(K′)n (X). Its class in Hn(X) is the fundamental class of X,
denoted by [X].For every (n − i)-simplex σ in (K), the dual cell of σ in M , denoted by d(σ) has
dimension m− (n− i). It is transverse to X, i.e. to every stratum Xn−α −Xn−α−1 of X.The intersection d(σ) ∩X is an oriented i-dimensional (K ′)-chain in X.
Let us define the Poincare homomorphism
Hn−i(X)→ Hi(X)
given by the chain map
Cn−i(K) (X)→ C
(K′)i (X)
which maps the elementary (n− i)-cochain d∗ = d∗(σ), dual of the simplex σ in K, to thei-chain ξ = d(σ) ∩X of K ′. In other words, ξ is the cap-product of σ∗ by [X] (see [B1]).
5.2.1 Alexander isomorphism
The Alexander isomorphism (see [B1]) Hm−i(M,M − X) → Hi(X) is induced by theisomorphism:
Cm−i(D) (M,M − T )→ C
(K)i (X)
which associates to a D-cochain (dm−i)∗ such that dm−i∩X 6= ∅ the K-chain σi such thatdm−i = d(σi).
5.3 Poincare-Hopf Theorem: The singular case
If X is a singular variety, the Poincare-Hopf Theorem fails to be true. The main reasonis that there is no longer tangent space at each singularity. The definition of the index ofa vector field in one of its singular points takes sense on a smooth m-manifold only. Inparticular the singular point must have a neighbourhood isomorphic to the ball Bm andwhose boundary is isomorphic to the sphere Sm−1. Let us consider the example of thepinched torus X in R3. The pinched point a is a singular point of X, in fact it constitutesthe singular part of the pinched torus. The only ‘natural’ way to define an index of avector field at the point a is to consider a vector field v defined in a ball B3(a) centered ina, in R3, with an isolated singularity at a, such that if x ∈ X \a, then v(x) is tangent tothe smooth manifold X \ a and such that v does not have other singularities in B3(a).
Let us consider two examples of such vector fields:a) The vector field tangent to the parallels of the torus T (see Figure 9a) determines,
on the pinched torus X, a vector field v pointing inward the ball B3(a) along one of thetwo unlinked circles, which are the intersection of ∂B3(a) and X, and pointing outwardthe ball along the other circle. On the one hand, this vector field, defined on ∂B3(a)∩X, isthe restriction of a vector field w defined on ∂B3(a) with index 0 at a. On the other hand,there is no more singularity of v on X \ a. In this case, the Poincare-Hopf Theorem isnot satisfied: one has
χ(X) = 1 6= 0 = I(w, a).
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b) Let us consider a vector field v pointing outward the ball B3(a) along ∂B3(a) andtangent to X along the restriction ∂B3(a) ∩ X (see Figure 9b) . This vector field hasindex +1 at a. It is orthogonal to the two meridians ∂B3(a) ∩X and it can be extendedon the pinched torus as a continuous vector field without other singularity. In fact, onecan define an extension v such that, on each meridian, the angle of v(x) with the tangentspace to the meridian is constant and this angle goes down continuously as the distanceto a grows till being 0 for the meridian opposed to a. In this case, the Poincare-HopfTheorem is valid:
χ(X) = 1 = I(v, a).
(a) The vector field v tangentto the parallels (b) The radial vector field
Figure 9: Vector fields on the pinched torus
The radial vector field in case (b) is the first example of M.-H. Schwartz’s radial vectorfield, of which we will make a systematic study.
If X is a singular variety, the Poincare-Hopf Theorem fails to be true, the main reasonis that there is no more tangent space in singular points. The definition of the index of avector field at one of its singular points takes sense on a smooth manifold only, the reasonbeing that the link of a point is a sphere.
In order to obtain a Poincare-Hopf Theorem, one can think to consider a stratificationof the singular variety (see section 5.1), i.e. a decomposition of the singular variety intosmooth manifolds and consider continuous vector fields which are stratified.
Definition 5.4 A stratified vector field v on X is a (continuous) section of the tangentbundle of M , T (M), such that, for every x ∈ X, then one has v(x) ∈ T (Vi(x)) where Vi(x)
is the stratum containing x.
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Proposition 5.5 [Sc4] If the stratification satisfies Whitney conditions (a) and (b), thenthere exists on X a stratified vector field v with isolated singularities ak. The indexI(v|Vi(ak)
, ak), defined in the stratum Vi(x) containing x, is well defined.
One could define the index of a stratified vector field v at a singular point a situatedin the stratum Vi as the index of the restriction I(v|Vi , a). The natural generalization ofthe Poincare-Hopf Theorem to singular varieties would be the following formula:
χ(X) =∑
ak∈Sing(v)
I(v|Vi(ak), ak). (5.6)
In general, the formula(5.6) is not true. Let us provide the (counter)-example givenin [Sc4, 6.2.1]:
Example 5.7 In a first step, in R2 with coordinates (x, y), one considers the (closed)balls centered at the origin, B with radius 1 and D with radius 2. One has χ(D) = +1.
Figure 10: Counter exemple for Poincare-Hopf
Inside the ball B, one considers the continuous vector field v1(x, y) = (|x|, y). One hasv1(0) = 0, the point 0 is a singularity of v1 with index I(v1, 0) = 0.
On the boundary ∂D, one considers the vector field v2(x, y) = (x, y) pointing radiallyoutwards. One can extend v2 inside D as a continuous vector field v which is v2 along∂D, v1 inside B and which is tangent to the y-axis Y along Y . The vector field definedby
v(x, y) =
(2|x| − x+ (x− |x|)
√x2 + y2, y
)on D \B
v1(x, y) = (|x|, y) inside B
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satisfies the conditions.The vector field v has an isolated singular point of index 0 at 0 and another isolated
singular point at a = (−3/2, 0) ∈ D\B. By Poincare-Hopf Theorem with boundary(2.4.3),one has
χ(D) = +1 = I(v, 0) + I(v, a),
that implies I(v, a) = +1.Let us remark that while I(v, 0) = 0, one has I(v|Y , 0) = +1.Now fold the picture along the y-axis, as a (differentiable) singular surface x2− z3 = 0
in R3. In that case, D becomes a singular variety ∆, with boundary and stratified by Yand ∆\Y . The vector field v in D defines a stratified vector field, still denoted by v in ∆.It has two isolated singular points: 0 and a. One has I(v|Y , 0) = +1 and I(v, a) = +1.The formula (5.6) would be written:
χ(∆) = +1 6= I(v, a) + I(v|Y , 0) = 1 + 1 = 2.
Let us remark that the vector field v is not “radial” at the singular point 0, it is notpointing outwards the unit ball centered at 0.
The main result of M.H. Schwartz is to provide, for singular varieties, an explicitconstruction of certain vector fields, called radial vector fields for which the Poincare-Hopf theorem is still valid.
Moreover, in the same way that the radial vector fields allow to recover the Poincare-Hopf Theorem, the construction of characteristic classes for singular varieties will consistin a construction of vector frames adapted to the singular situation and generalising thenotion of radial vector fields.
5.3.1 Radial extension process
One gives a description of the local radial extension process. This will be used for theglobal process in the next section.
The local radial extension, defined by M.-H. Schwartz, is similar to the one defined byMilnor in order to prove Theorem 2.13.
Let X be a singular variety embedded in an m-dimensional manifold M . Let Vα ⊂ Xbe a stratum, B(a) ⊂ Vα a neighbourhood of the point a in Vα and v a vector field definedon B(a) with an isolated singularity at a.
One can construct two vector fields:
1. Parallel Extension. We will denote by N(a) a tubular neighbourhood homeomorphicto B(a) × Dk, where Dk is a disk transverse to Vα and k = m − dimVα. Let us considerthe parallel extension v of v in the tube N(a). Let Vβ be a stratum such that a ∈ Vβ. Ata point x ∈ N(a) the parallel extension v(x) is not necessarily tangent to Vβ. However,the Whitney condition (a) guarantees that if N(a) is sufficiently small, then the anglebetween Ta(Vα) and Tx(Vβ) is small. That implies that the orthogonal projection of v(x)onto Tx(Vβ) does not vanish. Of course, considering for each stratum the projection ofthe parallel extension on the tangent space to the stratum at the given point does notprovide a continuous vector field. In order to obtain a continuous vector field, one has
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to consider a slight modification of the construction in the neighbourhood of the strata,which is easy to understand, but complicated to describe in details. A good extensionwill be v(x) away from Vβ and continuously going to the projection of v(x) onto Tx(Vβ)when approaching Vβ, using a suitable partition of unity. The construction of such anextension is correctly and entirely described in M.-H. Schwartz’s book [Sc4]. In fact, onehas to work simultaneously for all strata Vβ such that a ∈ Vβ, that complicates a detailedconstruction.
In conclusion, the Whitney condition (a) implies that one can proceed to the construc-tion of a stratified vector field, still denoted by v(x), which is a “parallel extension” of thegiven vector field on Vα, in a suitably small tubular neighbourhood around B(a) in M .
One observes that the singular locus of v corresponds to an (m−dimVα)-dimensionaldisk which is transversal to B(a) in M .
Figure 11: The local radial extension of a vector field
2. Transversal vector field. Let us consider the transversal vector field τ(x), as in theproof of Theorem 2.13. This vector field is essentially the gradient of the function “squareof the distance to Vα”, for an appropriate Riemannian metric. The vector field τ(x) is notnecessarily tangent to the strata Vβ such that a ∈ Vβ. However, the Whitney condition(b) guarantees that in a sufficiently small “tube” around B(a), the angle between τ(x)and Tx(Vβ) is small. That means that the orthogonal projection of τ(x) onto Tx(Vβ) doesnot vanish. In the same way as for the parallel extension, for each stratum one couldconsider the projection of τ(x) onto the tangent space to the stratum at the given point.However, this does not provide a continuous vector field. In order to obtain a continuousvector field, one has to consider a similar modification of the construction. The correctvector field is τ(x) away from Vβ and continuously going to the projection of τ(x) ontoTx(Vβ) when approaching Vβ. That construction is also completely described in M.-H.Schwartz’s book [Sc4], and one has to work simultaneously for all strata Vβ such thata ∈ Vβ.
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In the boundary of the tube N(a) ∼= B(a) × Dk the part B(a) × ∂Dk = B(a) × Sk−1
shall be called the horizontal part.In conclusion, one obtains a stratified “transversal” vector field, still denoted by τ(x),
which is zero along Vα and growing with the distance to Vα and which is pointing outwardthe horizontal part of the boundary of the tube N(a) provided that the tube is sufficientlysmall.
Definition 5.8 The radial extension of the vector field v defined on B(a) ⊂ Vα is thevector field v defined on the tube N(a) as the sum
v(x) = v(x) + τ(x).
Proposition 5.9 The radial extension v of the vector field v is transversal to the bound-aries of the tube N(a) around B(a), pointing outward the horizontal part of ∂N(a). Itsunique singularity inside N(a) is a, i.e. the same singularity as the initial vector field v.Moreover, the index of v in a, computed in the tube N(a), is the same as the index of vat a, computed in the manifold Vα, namely we have
I(v, a;M) = I(v, a;Vα).
This property, i.e., the above equality, is the main property of the radial extension,that is precisely the property which allows one to prove the Poincare-Hopf Theorem forsingular varieties.
5.3.2 Poincare-Hopf Theorem for singular varieties.
In this section one proceeds to the construction of a “global” radial extension of a vectorfield, which M.-H. Schwartz called radial vector field and one shows the following:
Theorem 5.10 Let X ⊂M be a (compact) singular variety embedded in a manifold M .One can construct on X a (stratified) radial vector field, in the sense of M.-H. Schwartz.That is a vector field v defined in a tubular neighbourhood Nε(X) of X in M , pointingoutward Nε(X) along the boundary. It has finitely many isolated singularities ai in Nε(X),all situated in X, and one has
I(v, ai;M) = I(v|Vα(i), ai;Vα(i)),
where Vα(i) is the stratum of X containing ai.
Proof: The “global” construction of the radial vector field goes as follows:One consider a Whitney stratification on X as before and one adds the stratum M \X
in order to obtain a Whitney stratification of M .The aim of the process is to construct, by increasing induction on the dimension of
the strata of X, a stratified vector field v in a neighbourhood of X in M , with finitelymany isolated singularities ai in the strata Vα, such that if ai ∈ Vα, the index of v at ai isthe same, computed in Vα or in M .
By the induction process on the dimension of the strata, we will show the following:
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(P) For each stratum Vα, there is a neighbourhood Nαε around V α and a stratified
vector field v defined on Nαε , pointing outward Nα
ε along the boundary, with isolatedsingularities ai in V α and such that if Vβ is the stratum containing ai, then the indexof ai computed in Vβ is the same as the index of ai computed in Nα
ε , i.e. in M .Namely, we will denote
I(v, ai) = I(v, ai;Vβ) = I(v, ai;M).
The neighbourhood Nαε is the set of points in M with distance less than ε from points
in V α. That is not a fibre bundle over V α, but the following construction shows thatfor each stratum Vβ ⊂ V α, there is a neighbourhood Aβ of V β \ Vβ in Vβ such thatthe restriction of Nα
ε to Aβ is a fibre bundle with fibre a disk whose dimension is thecodimension of Vβ in M .
Let us show that induction property (P) is true for the lowest dimensional stratum.If the lowest dimensional strata in X are 0-dimensional ones, i.e. V0 is a set of finitely
many points ak, then one considers a radial vector field v in a ball Bε(ak) centered ateach of these points. According to the Bertini-Sard Lemma, the boundary ∂Bε(ak) istransverse to the strata Vγ containing ak in their closure (one takes for ε the smallest of εfor all ak). For each point x ∈ Vγ ∩ ∂Bε(ak), the radial vector field v(x) is not orthogonalto Tx(Vγ). One can deform the radial vector field to a stratified vector field v pointingoutward Bε(ak) along ∂Bε(ak). In this case, the index I(v, ak) is +1 and obviously onehas
χ(V0) =∑k
I(v, ak).
In this case, N0ε is the union of Bε(ak).
If the lowest dimensional stratum is a stratum Vα of dimension s > 0, then oneconstructs, by classical obstruction theory, a vector field v on Vα with finitely manyisolated singularities ai. We notice that Vα is a manifold without boundary and it has tobe compact if X is compact. According to the classical Poincare-Hopf Theorem 2.14, onehas
χ(Vα) =∑i
I(v, ai).
The extension process in a neighbourhood Nαε of Vα, described in the proof of Theorem
2.13, can be slightly deformed according to the local case (cf Proposition 5.9) in order toobtain a stratified vector field defined on Nα
ε , pointing outward Nαε along its boundary
and such that the index of v at each singularity ai is the same, whether it is computed inVα or in Nα
ε , i.e. in M .Let us now suppose that induction property (P) holds for all strata up to Vα (i.e. for
all strata whose dimension is less than or equal to dimVα) and let us call Vγ the followingone. One has to show that (P) holds for Vγ.
The vector field v is defined on Uγ = Vγ ∩Nαε and is pointing inward Vγ along ∂Uγ =
Uγ \Uγ. By classical obstruction theory, one can extend v inside Vγ, as a vector field stilldenoted by v, with finitely many isolated singularities aj in Vγ \ Uγ.
For t ∈]0, 1], let us denote by Nαtε the (open) neighbourhood of V α, which is the set of
points whose distance to V α is less than tε. The vector field v defined on Vγ \ Nαε/2 can
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Figure 12: The radial vector field
be extended, using the local extension process, as a vector field v′ defined in a tube N ′ηaround Vγ \Nα
ε/2 without other singularities than the points aj and such that
I(v, aj;Vγ) = I(v, aj;N′η).
Let us notice that N ′η is a disk bundle with the basis Vγ \Nαε/2 and the fiber a disk whose
dimension is the codimension of Vγ in M .On the intersection Nα
ε ∩N ′η, one has two vector fields: v defined on Nαε and v′ defined
on N ′η. They coincide on Vγ. The vector field defined by w = (2 − 2t)v + (2t − 1)v′ oneach Nα
tε, for t ∈ [1/2, 1], coincides with v on ∂Nαε/2 and with v′ on ∂N ′η.
The vector field defined as v on Nαε/2, v′ on N ′η \ Nα
ε and w on Nαε ∩ N ′η satisfies the
property (P) for Vγ with the neighbourhood Nγε defined as Nα
ε/2 ∪N ′η.At the last step, one denotes by Nε(X) the neighbourhood of X constructed by the in-
duction process. By construction, the stratified radial vector field v satisfies the statementof the theorem.
Theorem 5.11 (Poincare-Hopf Theorem for singular varieties) Let X ⊂ M be a (com-pact) singular variety embedded in a manifold M , and v be a stratified radial vector fielddefined in the neighbourhood Nε(X) of X (Theorem 5.10). Then one has
χ(X) =∑i
I(v, ai).
Proof: Using the stratified radial vector field constructed in the proof of Theorem5.10 and Theorem 2.18, one has
χ(Nε(X)) =∑i
I(v, ai).
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Note that X is a deformation retract of its neighbourhood Nε(X), that ends the proof.Here we remark that at each stage of the proof of Theorem 5.10 one has
χ(V α) =∑
Vβ⊂V α
∑ai∈Vβ
I(v, ai),
the first summand being for all strata Vβ in V α, including Vα.
In fact, our proof shows the more precise result:
Theorem 5.12 ([Sc4, Theoreme 6.2.2]) Let X be an analytic subset of the analytic man-ifold M and V/α a Whitney stratification of the pair (M,X). Let us denote by D acompact domain with a smooth boundary transverse to the strata. Let v (resp. v−) bea radial vector field pointing outwards (resp. inwards ∂D). There is a finite number ofzeroes ai of v (resp. v−) in D and we have :
χ(X ∩D) =∑
ai∈X∩D
I(v, ai) =∑
ai∈X∩D
I(v|Vα(ai), ai)
χ(X ∩D)− χ(X ∩ ∂D) =∑Vα⊂X
χ(Vα ∩ int(D)) =∑
ai∈X∩D
I(v−, ai)
=∑
ai∈X∩D
I(v−|Vα(ai), ai),
where, if dimVα(a) = 0, then by convention I(v|Vα(a), a) = +1.
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6 Schwartz and MacPherson classes
In the following, M will be a complex analytic manifold equipped with an analytic strat-ification Vi: for every stratum Vi, the closure Vi and the boundary Vi = Vi \ Vi areanalytic sets, union of strata. We denote by X ⊂ M a complex analytic compact subsetstratified by Vi.
The first definition of Chern class for singular varieties was given by M.H. Schwartz inthe preprint [Sc1] (Lille University), then in 1965 in two “Notes aux CRAS” [Sc2]. Herewe provide a sketch of the M.H. Schwartz construction.
6.1 Radial frames
Let X ⊂ M be a singular n-dimensional complex variety embedded in a complex m-dimensional manifold. Let us consider a Whitney stratification Vi of M such thatX is a union of strata and denote by (K) a triangulation of M compatible with thestratification, i.e. each open simplex is contained in a stratum.
The first observation by M.-H. Schwartz concerns the triangulations:One knows (3.22) that the primary obstruction dimension to the construction of an
r-frame tangent to M is 2p = 2(m − r + 1). In the same way, the primary obstructiondimension to the construction of an r-frame tangent to the 2s-dimensional stratum Vi is2(s−r+1). That means that if one intends to construct a stratified r-frame tangent to Xusing the triangulation (K), then one will obtain obstruction cocycles on the strata withdifferent dimensions according to the dimension of the considered stratum. Consideringthe triangulation (K), one cannot obtain a global cocycle with a well-defined dimension.
The M.-H. Schwartz observation is the following: Let us denote by (D) the dualcell decomposition of (K) associated to a barycentric subdivision (K ′) (see 1.9). Each(D)-cell is transverse to the strata. In particular, if d is a (D)-cell whose dimension2p = 2(m− r+ 1) is the obstruction dimension for the construction of an r-frame tangentto M and if Vi is a stratum of dimension 2s, then the dimension of the cell d ∩ Vi is
dim(d ∩ Vi) = 2(m− r + 1)− 2(m− s) = 2(s− r + 1)
that is precisely the obstruction dimension for the construction of an r-frame tangent toVi.
This observation leads naturally to the construction of a stratified vector field byinduction on the dimension of the strata, using the dual cell decomposition (D) and notthe triangulation (K).
The second observation by M.-H. Schwartz is that one has to consider stratified vectorfields and frames which are radial in the sense we explained in the previous section. Below,
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we provide an explicit construction of radial extension of frames.
The obstruction dimension for a r-frame, over M is equal to 2p = 2(m− r+ 1). Thatmeans that one can construct such a section, without singularity over the (2p−1)-skeletonD(2p−1) of D and with isolated singularities over D(2p).
The obstruction dimension for an r-frame tangent to V 2si is equal to 2q = 2(s− r+ 1).
That means that one considers strata such that s ≥ r − 1. As we know, the dual celldecomposition is transverse to the stratification, that means that if d2p is a 2p-dimensionalcell in D which intersects V 2s
i , then d2p∩V 2si is a 2q-dimensional ∆ complex. Let us denote
∆2q = d2p ∩ V 2si .
Let us consider a stratified r-frame v(r) = (v(r−1), vr), section of Er over ∆2q ⊂ V 2si ,
with isolated singularities which are zeroes of the last vector vr. One suppose that vr haslength less than 1. One can define in a tube Tε(∆
2q) of radius ε (see above) on one handthe parallel extension v(r) = (v(r−1), vr) of v(r) and on the other hand the radial transversevector field τ and one has:
Proposition 6.1 (Radial extension for a frame) If µ and ε are sufficiently small, theradial extension of v(r), defined by v(r) = (v(r−1), vr + τ) satisfies the following conditions:
1. the radial extension of vr satisfies the Proposition 5.9,
2. if the (r − 1)-frame v(r−1) has no singularity on ∆2q and if v(r) admits an isolatedsingularity in a ∈ ∆2q ∩K ⊂ V 2s
i which is a zero of vr, then v(r) = (v(r−1), vr + τ)satisfies the same properties in Tε(∆
2q).In that case, if the (r−1)-complex plane generated by v(r−1)(a) is linearly independentof the tangent plane T (∆2q, a) in T (V 2s
i , a), then the index of the extension v(r) in a,considered as an r-frame tangent to M is equal to the index of v(r) in a consideredas an r-frame tangent to V 2s
i .
3. In the same hypothesis than (ii), if q = 0 (i.e; s = r− 1), and if a = ∆0 ⊂ V 2si is a
zero of vr, then the index of v(r) in a is +1.
We will denote by I(v(r), a) the index of v(r) in the isolated singularity a.
6.1.1 Global radial extension
The “global” construction of vector fields by radial extension goes as follows: If the lowestdimensional stratum is a 0-dimensional one, i.e. a set of finitely many points ak, thenone consider a radial vector field v in a ball B(ak) centered in each of these points. If thelowest dimensional of strata is a stratum Vi of dimension 2s > 0, then one construct avector field v on Vi with isolated singularities. We notice that Vi is a manifold and it hasto be compact if X is compact; in this case the total Poincare-Hopf index of v on Vi isχ(Vi).
Now we go up by increasing dimensions of the strata containing Vi in their closure:if Vi ⊂ V j, then we extend v to a neighborhood of Vi in X as above, and then extendit further to all of Vj with isolated singularities. We proceed further in this way to geta stratified vector field v on all of X. Furthermore, by construction one has that if
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we consider the closed strata V j, then the vector field is pointing inwards Vj near itsboundary. Thus one has, by theorem , that the Poincare-Hopf index of v on each stratumVj is χ(Vj).
One will construct v(r) over the subsets A2qi = d2p ∩ V 2s
i , by increasing dimensions ofthe strata Vi. One will construct at each step over Ai and a tube Tε(Ai), neighbourhoodof Ai in D(2p−1). At the following steps, the vector field could be modified, but without atube Tε′(Ai) ⊂ Tε(Ai).
i) If V 2r−2i is a stratum whose real dimension is 2r − 2 = 2(m − p), the obstruction
dimension to the construction of a section of Vr(TVi) is zero. One takes any (r−1)-framev(r−1) tangent to V 2r−2
i in the vertices aj = ∆0j of ∆ located in d2p ∩ V 2r−2
i and vr zero inthese points.
One construct the radial extension of the r-vector in the tubes Tε(∆0j) as a r-frame
still denoted by v(r). According to Proposition 6.1 (iii), one has I(v(r), aj) = +1.ii) Let us suppose s > r − 1 and the construction already performed on all strata Vk
whose dimension is less than 2s. That means that the construction has been performedon the sets Ak and the tubes Tε(Ak). Let us consider a 2s-dimensional stratum Vi.
Let us denote Ai = Ai ∩ (Vi \ Vi). The r-frame is already constructed over Ai ∩Tε(Ai). One can extend it on the rest of Ai with isolated singularities denoted by ak anslocated in 2q-open cells ∆2q
k located outside T1(Ai) (see[Sc4]) and such that, over a ballbk neighbourhood of ak in ∆2q
k , one has:a) v(r−1)(x) generates a (r− 1)-complex plane P 2r−2(x) supplementary of T (∆2q
k , x) inT (V 2s
i , x),b) for x ∈ Bk \ ak, one has vr(x) tangent to ∆2q
k and it has length less than 1.At that stage, one has v(r) on Ai ∪ Tε(Ai). One can extend it in a tube Tε′(Ai) (with
ε′ < ε1 < ε), in such a way that one has:
• the r-frame does not change within a tube Tε1(Ai) ⊂ Tε(Ai),
• in Tε′(Ai) ⊂ Tε(Ai), the extension is the radial extension of v(r),
• in Tε(Ai) ⊂ Tε1(Ai), the obtained r-frame is linear combination of the r-framev(r)|Tε(Ai) already constructed and of the radial extension of v(r)|Ai .
The constructed r-frame satisfies the properties of
Theorem 6.2 [B-S], [Sc2], [Sc5] One can construct, on the 2p-skeleton (D)2p, a stratifiedr-frame v(r), called radial frame, whose singularities satisfy the following properties:
(i) v(r) has only isolated singular points, which are zeroes of the last vector vr. On(D)2p−1, the r-frame v(r) has no singular point and on (D)2p the (r − 1)-frame v(r−1) hasno singular point.
(ii) Let a ∈ Vi ∩ (D)2p be a singular point of v(r) in the 2s-dimensional stratum Vi.If s > r − 1, the index of v(r) at a, denoted by I(v(r), a), is the same as the index of therestriction of v(r) to Vi ∩ (D)2p considered as an r-frame tangent to Vi. If s = r− 1, thenI(v(r), a) = +1.
(iii) Inside a 2p-cell d which meets several strata, the only singularities of v(r) areinside the lowest dimensional one (in fact located in the barycenter of d).
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(iv) The r-frame v(r) is pointing outwards a (particular) regular neighborhood U of Xin M . It has no singularity on ∂U .
6.2 Schwartz classes
Let us denote by T the tubular neighborhood of X in M consisting of the (D)-cells whichmeet X. Let us recall that d∗(σ) is the elementary (D)-cochain whose value is 1 at d(σ)and 0 at all other cells. We can define a 2p-dimensional (D)-cochain in C2p(T , ∂T ) by:
c =∑
d(σ)∈Tdim d(σ)=2p
I(v(r), σ) d∗(σ).
In other words, the cochain c satisfies
〈c · d(σ)〉 = I(v(r), , σ).
Figure 13: The tubular neighborhood T
In a classical way the cochain is a cocycle, the obstruction cocycle (see [Sc4]) whoseclass cp(X) lies in
H2p(T , ∂T ) ∼= H2p(T , T \X) ∼= H2p(M,M \X),
where the first isomorphism is given by retraction along the rays of T and the second byexcision (by M \ T ).
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Definition 6.3 [Sc2],[Sc5] The p-th Schwartz class is the class
cp(X) ∈ H2p(M,M \X).
M.-H. Schwartz proved that the class does not depend of the different choices involved inits construction. The proof of this fact is now easier, using Theorem 6.24 below.
6.3 Nash transformation
Let M be an complex analytic manifold, of complex dimension m. Let X be a n-dimensional subanalytic complex variety, X ⊂ M . Let us denote by Σ = Xsing thesingular part of X and by Xreg = X \ Σ its regular part.
The Grassmanian manifold of complex n-planes in Cm is denoted by Gn(Cm). Let usconsider the Grassmann bundle of n (complex) planes in TM , denoted by G. The fibreGx over x ∈M is the set of n-planes in Tx(M), isomorphic to Gn(Cm). An element of Gis denoted by (x, P ) where x ∈M and P ∈ Gx.
On the regular part of X, one can define the Gauss map γ : Xreg −→ G by
γ(x) = (x, Tx(Xreg)).
Definition 6.4 The Nash transformation X is defined as the closure of the image of γin G.
G X = Imγ → G γ ↓ ν ↓ ↓
Xreg → M X → M
(6.5)
Figure 14: The Nash transformation of the cone
In general, X is not smooth, nevertheless, it is an analytic variety and the restrictionν : X → X of the bundle projection G→M is analytic.
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Let us denote by E the tautological bundle over G. The fibre EP at a point (x, P ) ∈ Gis the set of the vectors v in the n-plane P ∈ Gx.
EP = v(x) ∈ TxM : v(x) ∈ P
Let us define E = E|X , then E|Xregcan be identified with T (Xreg) where Xreg =
ν−1(Xreg) ∼= Xreg and
E = E ×G X = (v(x), x) ∈ E × X : v(x) ∈ x
x ∈ X is a n-complex plane in Tx(M) and x = ν(x).
One has a diagram:
E −−−→ Ey yX −−−→ G
ν
y yX −−−→ M
An element in E is written (x, P, v) where x ∈ X, P is a n-plane in ν−1(x) and v is avector in P . If x ∈ Xreg, then P = Tx(Xreg).
Let us denote by Vi a complex analytic stratification of (M,X) satisfying the Whit-ney conditions.
The following lemma is fundamental for the understanding of the geometrical definitionof the local Euler obstruction. We recall the proof which is a direct application of theWhitney condition (a).
Lemma 6.6 ([B-S, Proposition 9.1]) A stratified vector field v on A ⊂ X admits a canon-
ical lifting v on ν−1(A) as a section of E.
Eν∗−→ TM |X
v ↑↓ v ↑↓ ν∗(x, x, v(x)) = (x, v(x)).
Xν−→ X
Proof: Let us consider a stratified vector field v on A ⊂ X and a point x ∈ X. Letus denote x = ν(x).
(i) If x ∈ Xreg, then v(x) ∈ Tx(Xreg) = x with x = ν−1(x). We define v(x) =(x, Tx(Xreg), v(x)).
(ii) If x ∈ Vi, then v(x) ∈ Tx(Vi). Each x ∈ ν−1(x) is in the closure of the image of
γ, i.e. there is a sequence (xn) of points of Xreg such that x = lim xn, ν(xn) = xn ∈ Xreg
and xn = Txn(Wreg). Then one has lim(xn) = x and limTxn(Xreg) = x. By the Whitneycondition (a), one has Tx(Vi) ⊂ x and we can define v(x) = (x, x, v(x)).
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6.4 Mather classes
We introduce now two ingredients useful to define the MacPherson classes, namely theMather classes and the local Euler obstruction.
Mather classes are defined as Chern classes of the Nash bundle, more precisely:
Definition 6.7 Let X ⊂ M a singular algebraic complex subvariety of a complex alge-braic manifold M . The Mather class of X is defined by:
cM(X) = ν∗(c∗(E) ∩ [X])
where c∗(E) denotes the usual (total) Chern class of the bundle E in H∗(X) and the
cap-product with [X] is the Poincare duality homomorphism.
Let us recall that in general, Poincare homomorphism is not an isomorphism.The Mather classes do not satisfy the Deligne-Grothendieck’s conjecture that we state
later on (see Theorem 6.13).
6.5 Euler local obstruction
The notion of local Euler obstruction was defined originally by R. MacPherson [MP] in1974. Definitions equivalent to MacPherson’s have been given by several authors. Ithas been shown in [BDK] that the local invariant of singularities which appear in theKashiwara formula for the index of holonomic modules [Ka] is equal to the local Eulerobstruction.
The original definition, due to R. MacPherson [MP] uses differential form. We willintroduce the dual definition, using vector fields.
Notation Let us consider a bundle E over X ⊂ Cm and A ⊂ X with boundary ∂A. Letus suppose that s is a section of E defined over the ∂A. The obstruction cocycle to extendthe section s inside A will be denoted by Obs(s, E,A).
Let z = (z1, . . . , zm) be local coordinates in Cm around 0, such that zi(0) = 0, wedenote by Bε and Sε the ball and the sphere centered in 0 with radius ε in Cm. Let usdenote by Oν−1(Bε),ν−1(Sε) the orientation class (fundamental class)
Oν−1(Bε),ν−1(Sε) ∈ H2n(ν−1(Bε), ν−1(Sε);Z).
Let us recall that a radial vector field v in a neighborhood of the point 0 ∈ X is astratified vector field so that there exists ε0 > 0 such that for all ε, 0 < ε < ε0, the vectorv(x) is pointing outwards the ball Bε over the boundary Sε = ∂Bε. By the Bertini-Sardtheorem, Sε is transverse to the strata Vi if ε is small enough, so the definition takes sense.
Definition 6.8 [B-S] Let v be a stratified radial vector field over X ∩Sε and v the liftingof v on ν−1(X ∩ Sε). The local Euler obstruction Eu0(X) is the obstruction to extendv as a nowhere zero section of E over ν−1(X ∩ Bε), evaluated on the orientation classOν−1(Bε),ν−1(Sε):
Eu0(X) = Obs(v, E, ν−1(X ∩ Bε)).
The local Euler obstruction is independent of all choices involved.
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Figure 15: Local Euler obstruction
6.5.1 Properties of Euler local obstruction
Theorem 6.9 [MP],[GS] The Euler obstruction satisfies:
1. Eux(X) = 1 if x is a regular point of X;
2. Eux×x′(X ×X ′) = Eux(X) · Eux′(X ′);
3. If X is a curve, then Eux(X) is the multiplicity of X at x;
If X is the cone over a non singular plane curve of degree d and x is the vertex ofthe cone, then Eux(X) = 2d− d2;
4. If X is locally reducible at x and Xi are its irreducible components, then Eux(X) =∑Eux(Xi).
The following proposition has been proved by many authors, in particular see [B-S] :
Proposition 6.10 (Constructibility) ([MP],[B-S],[Du] and other authors): The local Eu-ler obstruction is constant along the strata of a Whitney stratification of X.
The main property of local Euler obstruction is the following Brasselet-Schwartz Pro-portionality Theorem [B-S]. Things being local, one can suppose we are in a ball Bε inCm local chart of M , with center a ∈ Vi ⊂ X.
Theorem 6.11 ([B-S], Theoreme 11.1) (Proportionality Theorem for vector fields). Letv be a stratified vector field obtained by radial extension (of a vector field on Vi) with anisolated singularity in the point a ∈ Vi, with index I(v, a) = I(v|Vi , a), then
Obs(v, E, ν−1(Bε ∩X)) = Eua(X) · I(v, a).
One defines the bundle of frames in E in an obvious way. We will denote by Er thebundle of r-frames in E.
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Theorem 6.12 ([B-S], Theoreme 11.1) (Proportionality Theorem for frames). Let vr bea radial r-frame with an isolated singularities at the barycenter of the 2p-cells d2p = d(σ)with index I(vr, σ) at σ = d2p ∩ σ. Then the obstruction to the extension of vr as a
section of Er on ν−1(d2p ∩X) is
Obs(vr, Er, ν−1(d2p ∩X)) = Euσ(X) · I(vr, σ).
6.6 MacPherson classes
Let us recall firstly some basic definitions.A constructible set in a variety X is a subset obtained by finitely many unions, in-
tersections and complements of subvarieties. A constructible function α : X → Z is afunction such that α−1(n) is a constructible set for all n. The constructible functionson X form a group denoted by F(X). If A ⊂ X is a subvariety, we denote by 1A thecharacteristic function whose value is 1 over A and 0 elsewhere.
If X is triangulable, α is a constructible function if and only if there is a triangula-tion (K) of X such that α is constant on the interior of each simplex of (K). Such atriangulation of X is called α-adapted.
The correspondence F : X → F(X) defines a contravariant functor when consideringthe usual pull-back f ∗ : F(Y ) → F(X) for a morphism f : X → Y . One interesting factis that it can be made a covariant functor when considering the pushforward defined oncharacteristic functions by:
f∗(1A)(y) = χ(f−1(y) ∩ A), y ∈ Y
for a morphism f : X → Y , and linearly extended to elements of F(X). The followingresult was conjectured by Deligne and Grothendieck in 1969.
Theorem 6.13 [MP] Let F be the covariant functor of constructible functions and letH∗( ;Z) be the usual covariant Z-homology functor. Then there exists a unique naturaltransformation
c∗ : F→ H∗( ;Z)
satisfying c∗(1X) = c∗(X) ∩ [X] if X is a manifold.
The theorem means that for every algebraic complex variety, one has a natural trans-formation c∗ : F(X)→ H∗(X;Z) satisfying the following properties:
1. c∗(α + β) = c∗(α) + c∗(β) for α and β in F(X),
2. c∗(f∗α) = f∗(c∗(α)) for f : X → Y morphism of algebraic varieties and α ∈ F(X),
3. c∗(1X) = c∗(X) ∩ [X] if X is a manifold.
The MacPherson’s construction uses both the constructions of Mather classes andlocal Euler obstruction.
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Proposition 6.14 There is a isomorphism T between algebraic cycles on X and con-structible functions, given by
T (∑
niVi)(p) =∑
niEup(Vi)
For a Whitney stratification, we have the following lemma:
Lemma 6.15 [MP] There are integers nα such that, for every point x ∈ X, we have:∑α
nαEux(Vα) = 1.
Proof: The proof is an easy exercise.
Definition 6.16 [MP] The MacPherson class of X is defined by
c∗(X) = c∗(1X) =∑α
nα i∗cM(Vα)
where i denotes the inclusion Vα → X.
Note that we have the following relation : cM(X) = c∗(EuX).In particular, the Chern classes of an algebraic variety are represented by algebraic
cycles.
6.7 Schwartz and MacPherson classes
In this section, we show the Theorem 6.24 that states that Schwartz and MacPhersonclasses correspond via Alexander duality (see section 5.2.1).
Proposition 6.17 The are a simplicial triangulation K of M compatible with the strat-ification and a cellular decomposition K of the Grassman bundle G compatible with thestrata ν−1(Vi) such that:
1. equipped with the triangulation K (resp. with the decomposition K), M (resp. G)is a combinatorial variety,
2. the triangulation of M and decomposition of G are C1-differentiable,
3. for each cell σβ, then ν(σβ) is a simplex σα,
4. the restriction of ν to each cell σβ has constant rank.
One says that the K-cell σβ is horizontal if one has dim ν(σβ) = dim σβ. In that case,the restriction of ν on the (open) cell σβ is a diffeomorphism on ν(σβ).
In the sequel, one defines a simplicial subdivision ∆ of K. Image of a ∆-simplex willnot be a ∆-simplex but will satisfy a property that is given in the following Proposition(see [B-S]).
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Proposition 6.18 . One can construct a simplicial subdivision ∆ of K satisfying thefollowing conditions:
• (i) Every open cell σβ contains one and only one ∆-vertex denoted by aβ. Let aα bethe barycenter of σα = ν(σβ), then aβ is contained in the inverse image ν−1(aα),
• (ii) Let σβ be a ∆-cell with image σα, and σαi a simplex in the boundary of σα. Letus denote by d(σαi) the dual cell of σαi, then σβ ∩ ν−1(σαi) is a ∆-complex.
From the construction of ∆, one has the following results:
Proposition 6.19 Let σmα a m-dimensional K-simplex, d(σmα ) the dual cell in M . Letus denote by σmβ the horizontal K-cells whose image is σmα and d(σhβ) their dual cells inG. Then one has
Closure of ν−1(d(σmα ) ∩X) = Closure of⋃β
d(σhβ) ∩ X. (6.20)
Corollary 6.21 For each q-dimensional cell d(σα), one has ν−1(d(σα)) is a q-dimensional∆-complex.
Let σ2r−2α be a (2r − 2)-dimensional K-simplex, d(σ2r−2
α ) its dual cell whose inter-section with X is 2p-dimensional. It results from Corollary 6.21 that ν−1(d(σα)) is a2p-dimensional ∆-complex. One can proves the following Theorem:
Theorem 6.22 Let c be a ∆-cocycle of the 2p-Chern class cp(E) of the Nash bundle E.Let us denote kα = 〈c, ν−1(D2p
α )〉. Then the Mather class cM,r−1(X) contains the cycle∑σ2r−2α ⊂X
kασ2r−2α
Proof: Let us consider the ∆-cycle w2n which is the sum of all simplexes δ2n with theorientation induced by the one of X. The class of w2n in H2n(X) is the fundamental class[X]. Let c be a cocycle in C2p
∆(X) representing the Chern class cp(E), then a cycle of the
Mather class cM,r−1(X) = ν∗(cp(E) ∩ [X]) is given by
ν∗(c ∩ w2n). (6.23)
Let us recall the result in [B1]: The Poincare morphism C2p
∆(X) → C2r−2,K(X), cap-
product by w2n is composed of the Alexander isomorphism ([B1], §3) and the Thomhomomorphism, dual of intersection with X. In another words, if d2t
β = d(σ2r−2β ) is the
dual cell, in G of the (2r − 2)-cell σ2r−2β , then one has
c ∩ w2n =∑
σ2r−2β ⊂X
νβσ2r−2β where νβ = 〈c · d2t
β ∩ X〉.
(see [B1], formulae (7), (8) and diagram (16)).
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Taking the image by ν∗ of that cycle, the contributions of horizontal cells σ2r−2β are
the only one which do not vanish, the others having an image whose dimension is lessthan 2r − 2. The cycle 6.23 is homologous to the cycle
ν∗(∑β
νβσ2r−2β ) =
∑σ2r−2α ⊂X
kασ2r−2α
where kα =∑νβ =
∑〈c · d2t
β ∩ X〉, the sum being extended on indices β such that σ2r−2β
is horizontal with image σ2r−2α . One obtains the result, using Proposition 6.19.
The following result has been proved in [B-S]:
Theorem 6.24 [B-S] The MacPherson class is the image of the Schwartz class by theAlexander duality isomorphism
H2(m−r+1)(M,M \X)∼=−→ H2(r−1)(X).
Proof: Using the notations of section 6.2, the r-frame vr determines a cocycle of theM.H. Schwartz class:
c =∑
d2qα ∩X 6=∅
να(d2qα )∗ with να =
∑aj∈d2q
α ∩X
I(vr, aj). (6.25)
It determines also a cocycle c of the Chern class cp(E) such that
< c.ν−1(d2qα ∩X) >= Euaα(X)µα,
where aα is any point of σ2r−2α , simplex whose the cell d2q
α is dual, i.e. d2qα = d(σ2r−2
α ).Theorem 6.22 implies that the Chern-Mather class cMr−1(X) is represented by the co-
cycle: ∑σ2r−2α ⊂X
Euaα(X)µασ2r−2α
where µα is the coefficient of the cocycle c relatively to the cell d2qα dual of σ2r−2
α andwhere aα is any point of σ2r−2
α .Let us write the previous result for each V i: The Chern-Mather class cMr−1(V i) is
represented by the cocycle: ∑σ2r−2α ⊂V i
Euaα(V i)µασ2r−2α .
By definition 6.16, the MacPherson class c∗(X) is represented by the cocycle:∑i
ni∑
σ2r−2α ⊂V i
Euaα(V i)µασ2r−2α .
In this expression, the coefficient of µασ2r−2α is
Cα =∑i∈Iα
niEuaα(V i) = 1, with Iα = i : σ2r−2α ⊂ V i = i : aα ∈ V i.
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One obtains a cycle of the MacPherson class of X of the form:
γ =∑
σ2r−2α ⊂X
µασ2r−2α .
Let us recall (5.2.1) that the Alexander isomorphism H2q(M,M −X)→ H2r−2(X) isinduced by the isomorphism:
C2q(D)(M,M − T )→ C2r−2,(K)(X)
which associates to a (D)-cochain (d2qα )∗ such that d2q
α ∩X 6= ∅ the (K)-chain σ2r−2α such
that d2qα = d(σ2r−2
α ). By this isomorphism, the cycle γ is image of the cocycle of theM.H.Schwartz class (cf 6.25), which proves the theorem.
We observe that we determined a cocycle of the MacPherson class. In fact, one hasthe following corollary:
Corollary 6.26 Let (K) be a simplicial triangulation of M compatible with a Whitneystratification and vr a r-radial frame defined on, the 2q-squeleton D(2q) of a cellular de-composition (D) dual of (K). The (r− 1)-st MacPherson class cr−1(X) is represented bythe cycle ∑
σ∈X
I(v(r), σ) σ
where dimσ = 2(r − 1).
6.8 Schwartz-MacPherson classes for projective cones
In this section, we show the following result, due to Barthel, Brasselet and Fieseler [BBF]following ideas of Brasselet and Gonzalez-Sprinberg [BG].
Theorem 6.27 Let Y ⊂ PN , be a projective variety and ı : Y → KY the canonicalinclusion in the projective cone KY on Y with vertex s. Let us denote also by K :H∗(Y )→ H∗+2(KY ) the homological projective cone, one has
cj(KY ) = ı∗cj(Y ) +Kcj−1(Y ), (6.28)
where Kc−1(Y ) denotes the class [s] ∈ H0(KY ).
Let us consider an n-dimensional projective variety Y in PN and let us denote by Lthe restriction of the hyperplane bundle of PN to Y .
If H denotes an hyperplane in PN , then D = Y ∩H is a divisor in Y and we denoteby [D] ∈ H2n−2(Y ) the fundamental class. By Poincare duality, let ηD ∈ H2(Y ) be theassociated cohomology class. L is the bundle associated to D and one has c1(L) = ηD.
We denote by E the completed projective space of the total space of L, i.e. P(L⊕ 1Y )where 1Y is the trivial bundle of complex rank 1 on Y . The canonical projection p : E → Yadmits two sections, zero and infinite, with images Y(0) and Y(∞). The projective coneKY is obtained as a quotient of E by contraction of Y(∞) in a point s. It is the Thomspace associated to the bundle L, with basis Y .
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Let us consider p : E → Y as a sphere bundle with fibre S2, subbundle of a bundlep : E → Y with fibre the ball B3. We denote by θE ∈ H3(E, E) the associated Thomclass; One has a Gysin exact sequence
. . .→ Hj+1(Y )→ Hj−2(Y )γ→ Hj(E)
pj→ Hj(Y )→ . . . ;
in which the gysin map γ is the composition of
Hj−2(Y )(pj−2)−1
−→ Hj−2(E)(∩θE)−1
−→ Hj+1(E, E)∂→ Hj(E)
and can be explicited in the following way: If ζ is a cycle representing the class [ζ] ofHj−2(Y ), then γ([ζ]) is the class of the cycle p−1(ζ) in Hj(E).
Let π the canonical projection π : E → KY .
Definition 6.29 We call homological projective cone and we denote by κ the compositionκ = π∗γ : Hj−2(Y )→ Hj(KY ) for j ≥ 2. We let κ(0) := [s] ∈ H0(KY ) for 0 = H−2(Y ).
Let us remark that for j 6= 0, κ is an homomorphism.The theorem 6.27 is a direct consequence of the following proposition proved in [BBF].
Proposition 6.30 The Chern classes of E and Y are related by the formula
c∗(E) = (1 + η0 + η∞) ∩ γ(c∗(Y )), (6.31)
where ηj := c1(O(Y(j))
)∈ H2(E) for j = 0, ∞, and ∩ denotes the usual cap-product.
Proof of the Theorem 6.27 (from the Proposition 6.30). Let 1E the con-structible function which is the characteristic function of E, then one has
π∗(1E)(x) =
χ(Y ), if x = s
1, elsewhere,
i.e.
π∗(1E) = 1KY + (χ(Y )− 1)1s.
As one has
π∗c∗(1E) = c∗(π∗(1E))
one obtains
π∗c∗(E) = c∗(KY ) + (χ(Y )− 1) [s]. (6.32)
On another hand, from the formula (6.31) one obtains:
Let ι0 : Y → E and ι∞ : Y → E be the inclusions of Y as zero and infinite sectionsof E respectively. By definition of γ, for a cycle ζ in Y and for j = 0 or ∞, one has
where [s] is the class of the vertex s in H0(KY ). The comparison of the formulae (6.32)and (6.33) gives:
c∗(KY ) = ı∗c∗(Y ) + π∗γc∗−1(Y ) + [s],
and the Theorem 6.27.
6.9 Schwartz-MacPherson classes of Thom spaces associated toembeddings
The previous construction associates canonically a Thom space to the embedding of asmooth variety Y in PN . As examples, let us consider the image of the Segre embeddingP1 × P1 → P3, defined in homogeneous coordinates by
That is an embedding whose bidegree is (1, 1). Image ϕS(P1 × P1) is a non degeneratequadric Q provided with two families of generatices. Euler class of the bundle E inH2(P1
x × P1y) = H2(P1
x) ⊕ H2(P1y) = Z ⊕ Z is c1(E) = (ηx, ηy) where ηx is Euler class of
the hyperplane bundle of P1x, i.e. such that ηx ∩ [P1
x] = 1.The image of the Veronese embedding P2 → P5 defined by
ϕV : (x0 : x1 : x2) 7→ (x20 : x0x1 : x0x2 : x2
1 : x1x2 : x22).
That is an embedding whose degree is 2. Image ϕV (P2) is smooth and has degree 4. It iscalled Veronese surface. Euler class of the bundle E in H2(P2) is c1(E) = 2ηK where Hbeing hyperplane in P5, one has H ∩ V is a divisor in P2 ∼= 2K, K being hyperplane inP2. Then ηK = c1(EK) is generator of H2(P2), one has ηK ∩ [K] = 1.
With the previous construction, KY is the Thom space associated to the fibre bundleL, of complex rank 1 and restriction to Y of the hyperplane bundle of PN . Chern classesand intersection homology of these exemples have been computed in [BG]. In the case
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of the Segre embedding, let d1 and d2 two fixed lines belonging each to a system ofgeneratrices of the quadric Y = P1 × P1. Let us denote by ω the canonical generator ofH2(P1), one has c∗(P1) = 1 + 2ω and c∗(P1) = [P1] + 2??. Then
where a is a point in Y and where ∗ denotes the intersection of cycles or homology classes.One has
κ(c∗(Y )) = [KY ] + 2([Kd1] + [Kd2]) + 4[Ka].
let us denote by ∼ the homology relation of cycles. In KY , one has [BG], §3:
Y ∼ Kd1 +Kd2, d1 ∼ d2 ∼ Ka, a ∼ s,
and, with 6.27
c∗(KY ) = [KY ]︸ ︷︷ ︸H6(KY )
+ 3([Kd1] + [Kd2])︸ ︷︷ ︸H4(KY )
+ 8[Ka]︸ ︷︷ ︸H2(KY )
+ 5[s]︸︷︷︸H0(KY )
,
which is the result of [BG].In the case of the Veronese embedding, let d be a projective line in Y := P2, one has:
c∗(P2) = 1 + 3ω + 3ω2 where ω is the canonical generator of H2(P2), and is dual, byPoincare isomorphism of the class [d] ∈ H2(P2). One has, by Poincare duality
c∗(Y ) = [Y ] + 3[d] + 3[a]
where a is a point in Y . One has
K(c∗(Y )) = [KY ] + 3[Kd] + 3[Ka]
such that, in KY , [BG], §3.b, Y ∼ 2Kd, d ∼ 2Ka and a ∼ s. One has
c∗(KY ) = [KY ]︸ ︷︷ ︸H6(KY )
+ 5[Kd]︸ ︷︷ ︸H4(KY )
+ 9[Ka]︸ ︷︷ ︸H2(KY )
+ 4[s].︸︷︷︸H0(KY )
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7 Other classes and comparisons
7.1 Fulton classes
The Fulton classes ([Fu] exemple 4.2.6 (a)) and Fulton-Johnson classes ([FJ], [Fu] exemple4.2.6 (c)) have been defined in a general framework. In the case of Local CompleteIntersection, these classes coincide and can be defined in the following (simpler) way:
If X is a local complete intersection, then the normal bundle of Xreg in M extendscanonically to X as a vector bundle NXM and the Fulton class is equal to
cF (X) = c(TM |X)c(NXM)−1 ∩ [X] = c(τX) ∩ [X].
Here τX = TM |X − NXM denotes the virtual tangent bundle on X, defined in theGrothendieck group of vector bundles on X.
If X is a manifold, then cF (X) is the usual Chern class c∗(X).
Let M be a non-singular compact complex analytic variety of pure dimension n + 1and let L be a holomorphic line bundle on M . Take f ∈ H0(M,L), a holomorphic sectionof L, such that the variety X of zeroes of f is a (nowhere dense) hypersurface in M . Then,the Fulton class of X is
cF (X) = c(TM |X − L|X) ∩ [X].
In [BLSS1], the authors show the following result:
Theorem 7.1 Let us assume that X ⊂ M is a hypersurface, defined by X = f−1(0),where f : M → D is a holomorphic function into an open disc around 0 in C. Foreach point a ∈ X, let Fa denote a local Milnor fiber, and let χ(Fa) be its Euler-Poincarecharacteristic. Then the Fulton-Johnson class cFJr−1(X) of X of degree (r−1) is representedin H2(r−1)(X) by the cycle ∑
σα⊂X, dimσα=2(r−1)
χ(Fσα) I(vr, σα) · σα (7.2)
7.2 Milnor classes
The comparison between the Schwartz-MacPherson classes and the Fulton-Johnson clas-ses can be viewed in two ways, which coincide in some classical situations.
In the case of isolated singularities, the difference between Schwartz-MacPherson classesand Fulton-Johnson classes is given by the Milnor numbers at the singular points:
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Theorem 7.3 [SS] If X is compact and the singularities of X are isolated points xiwhere X is a local complete intersection. Then
c∗(X)− cF (X) = (−1)n+1
q∑i=1
µxi [xi] ∈ H0(X).
This motivates the following definition given by various authors:
Definition 7.4 ([A1],[BLSS1],[PP2], [Yo]) The difference class
µ∗(X) = (−1)n(cF (X)− c∗(X))
is called the Milnor class of X.
7.2.1 Description in terms of constructible functions
The following description comes from [PP2].Let us come back to the hypersurface situation: M is a non-singular compact complex
analytic variety of pure dimension n + 1 and L is a holomorphic line bundle on M . Thehypersurface X in M is the set of zeroes of a holomorphic section of L.
Consider the function χ : X → Z defined by χ(x) := χ(Fx), where Fx denotes theMilnor fibre at x and χ(Fx) its Euler characteristic. Define also the function µ : X → Zby µ = (−1)n−1(χ− 1X).
Fix any stratification S of X such that µ is constant on the strata of S, for instancea Whitney stratification of X. The topological type of the Milnor fibre is constant alongthe strata of the Whitney stratification of Z. Let us denote by µS the value of µ on thestratum S.
Let
α(S) = µS −∑
S′ 6=S,S⊂S′
α(S ′)
be the numbers defined inductively on descending dimensions of S.
Theorem 7.5 [PP2] We have
µ∗(X) =∑S∈S
α(S)c(L|X)−1 ∩ (iS,X)∗c∗(S) = c(L|X)−1 ∩ c∗(µ),
where iS,X : S → X denotes the natural inclusion.
The formula was conjectured in [Yo] when X is projective. Under this last assumption,[PP1] proved earlier that∫
X
µ∗(X) =∑S∈S
α(S)
∫S
c(L|S)−1 ∩ c∗(S)
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7.3 Motivic Chern classes: Hirzebruch theory for singular vari-eties
In the same way that the MacPherson Chern functor generalises the Chern class, the Toddclass and the Thom-Hirzebruch class have been generalised as natural transformationsrespectively by Baum-Fulton-MacPherson and by Cappell-Shaneson. We show that themotivic theory allows to unify the three generalisations in the case of singular varieties.
In the case of singular varieties, the three classes (Chern Todd and L-class) have beengeneralized as natural transformations of functors, in the following way:
Definition 7.6 [Chern Transformation (MacPherson)] (see [MP])Theorem 6.13 tells us that there is an unique natural transformation
c∗ : F(X)→ H∗(X)
from the group of constructible functions F(X) to homology, satisfying the suitable prop-erties. In particular for the constructible function 1X , one defines c∗(X) := c∗(1X) :Schwartz-MacPherson class of X.
Definition 7.7 [Todd Transformation (Baum-Fulton-MacPherson)] [BFM]There is an unique natural transformation
td∗ : G0(X)→ H∗(X)⊗Q
from the Grothendieck group of coherent sheaves on X, satisfying suitable axioms, inparticular, for the structure sheaf OX on a smooth variety, td∗(OX) is the Todd class ofX. In general, one defines td∗(X) := td∗([OX ]) as being the Baum-Fulton-MacPhersonTodd class of X.
Definition 7.8 [L-Transformation (Cappell-Shaneson)] [CS1, CS2]There is an unique natural transformation
L∗ : Ω(X)→ H2∗(X;Q)
from the group of constructible self-dual sheaves on X, satisfying suitable axioms, inparticular, for the intersection sheaf ICX on a smooth variety, L∗([ICX ]) is the L-class ofX. In general, one defines L∗(X) := L∗([ICX ]) as being the Cappell-Shaneson L-class ofX.
In short, one has the following table:
X manifold X singular variety
number cohomology classes homology classes
χ(X) Chern Schwartz-MacPherson
— — —
χa(X) Todd Baum-Fulton-MacPherson
— — —
sign(X) Thom-Hirzebruch Cappell-Shaneson
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The problem is that the three transformations are defined on different spaces:
F(X), G0(X) and Ω(X)
and one asks for the possibility of unifying in the same way thatn the Hirzebruch theory inthe smooth case. The problem has been solved (see [BSY]) using the motivic framework.Let us recall some ingredients which will be useful:
Definition 7.9 Let X be an algebraic variety, the Grothendieck relative group of algebraicvarieties over X denoted by
K0(var/X)
is the quotient of the free abelian group of isomorphy classes of algebraic maps Y −→ X,modulo the “additivity relation”:
[Y −→ X] = [Z −→ Y −→ X] + [Y \ Z −→ Y −→ X]
for closed algebraic sub-spaces Z in Y .
In [BSY], the authors prove the following 4 theorems:
Theorem 7.10 The map
e : K0(var/X) −→ F(X) defined by e([f : Y → X]) := f!1Y
is the unique group morphism which commutes with direct images for proper maps andsuch that
e([idX ]) = 1X for X smooth and pure dimensional.
Theorem 7.11 There is an unique group morphism
mC : K0(var/X) −→ G0(X)
which commutes with direct images for proper maps and such that
mC([idX ]) = [OX ] for X smooth and pure dimensional.
Theorem 7.12 The morphism
sd : K0(var/X) −→ Ω(X) defined by sd([f : Y → X]) := [Rf∗QY [dimC(Y )+dimC(X)]]
is the unique group morphism which commutes with direct images for proper maps andsuch that
sd([idX ]) = [QX [2 dimC(X)]] = [ICX ] for X smooth and pure dimensional.
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Theorem 7.13 There is an unique group morphism
Ty : K0(var/X) −→ H∗(X)⊗Q[y]
which commutes with direct images for proper maps and such that
Ty([idX ]) = td(y)(TX) ∩ [X] for X smooth and pure dimensional.
In particular, one has: T−1([idX ]) = c∗(X).
Remark 7.14 If a complex algebraic variety X has only rational singularities (for exam-ple if X is a toric variety), then:
mC([idX ]) = [OX ] ∈ G0(X) and in this case T0([idX ]) = td∗(X).
That is not true in general !
The main result is the following:
Theorem 7.15 One has a commutative “tripode” diagramme:
F(X)e←− K0(var/X)
mC−→ G0(X)
sd c∗ ↓ Ty ↓ Ω(X) ↓ td∗
H∗(X)⊗Q y=−1←− H∗(X)⊗Q[y] L∗ ↓y=0−→ H∗(X)⊗Q.
y=1 H∗(X)⊗Q
7.4 Verdier Riemann-Roch Formula
Theorem 7.16 Let f : X ′ → X be a smooth map (or a map with constant relativedimension), then one has
td(y)(Tf ) ∩ f ∗Ty([Z −→ X]) = Tyf∗([Z −→ X]).
Here Tf is the bundle over X ′ of tangent spaces to fibres of f .
Proposition 7.17 (Factorisation of Ty) Let us define
td(1+y)([F ]) :=∑<∞
i=0 tdi([F ]) · (1 + y)−i. Then one has:
Ty = td(1+y) mC : K0(var/X) −→ H∗(X)⊗Q[y].
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