Counting Lines on Projective Hypersurfaces via Characteristic Classes

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DIPARTIMENTO DI MATEMATICA Corso di Laurea Magistrale in Matematica

Counting Lines on Projective Hypersurfaces

via Characteristic Classes

Relatore:

Chiar.ma Prof.ssa Rita Pardini

Controrelatore:

Chiar.mo Prof. Marco Franciosi

Candidato:

Filippo Fagioli

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Introduction

One of the most classical statements in enumerative geometry asserts that there are exactly 27 lines lying on a smooth cubic surface in P3

C; this result,

known as the 27 Lines Theorem, was proved by Cayley [8] in 1869. Over the reals the situation is dierent, since Schläi [29] showed that on a smooth cubic surface in P3

R there are either 27, 15, 7 or 3 lines.

It is natural to ask about the analog of these numerical results for a projective hypersurface of degree d in arbitrary dimension. We remark that in enumerative geometry one is naturally led to consider general objects (see Remark 3.4 for details about this notion). Hence, depending upon d, the general projective hypersurface either

• contains families of lines of positive dimension, if 0 < d < 2n − 3, • contains a nite number of lines, if d = 2n − 3,

• does not contain any line, if d > 2n − 3.

In this thesis we focus on the second case, by addressing the following enumerative geometry problem.

Problem. How many lines are contained in a general complex or real hy-persurface of degree 2n − 3 in the projective space Pn_?

We approach this problem via a topological method, by using character-istic classes of certain vector bundles, namely the Euler class and the top Chern class. Such method yields to an exact number of lines in the complex case, and to a non trivial lower bound in the real one; both results do not depend on the choice of the hypersurface. Our aim is to set and solve the problem in the dierentiable category, which is suitable for both the complex and the real case.

For the complex case we base our study on Harris' paper [19] of 1979, while for the real case we follow the article of 2013 by Finashin and Kharlamov [12] and that of 2014 by Okonek and Teleman [28].

What follows is a brief description of our study. Let V be a hypersurface of degree d in Pn

K, dened by a homogeneous

polynomial f with coecients in K, where K stands for the eld of real or complex numbers. We aim at counting the number of projective lines contained in V . The method is based on the following simple but key remark: a projective line l = P(W ), dened by a 2-plane W ⊂ Kn+1_{, is contained in}

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Denote by U the tautological bundle on the Grassmannian G2(Kn+1) of

2-planes in Kn+1_{. The symmetric d-linear form f induces a section s}

f of the

d-th symmetric power bundle Sd_(U∨₎_{of U}∨_{, obtained by restricting f to the}

linear 2-planes of Kn+1_{, which are the bers of U. The section s}

f is dened

in such a way that a line l = P(W ) lies on V if and only if sf(W ) = 0. Hence,

the problem of enumerating the lines on V reduces to the study of the zero locus Z(sf)of the section sf. This zero locus is said to be the Fano variety

of lines on V (see, e.g., [4]).

Given that we are considering only general hypersurfaces, for a general choice of f the section sf is transversal; we give a proof of this fundamental

result in Theorem 3.5. This implies that the Fano variety of lines is actually a submanifold of G2(Kn+1)of dimension depending on the degree d of f (see

Corollary 3.6). In particular, for d = 2n−3 the Fano variety is 0-dimensional, hence as already mentioned V contains a nite number of lines.

The method we use to count such lines is based on the following result from algebraic topology. We provide a proof specically tailored to our point of view in Theorem 1.50.

Theorem. The Euler class e(E) of an oriented vector bundle E → M dened over a compact and oriented manifold is Poincaré dual to the zero locus of any transversal section.

In particular, if the zero locus is a nite set of points, then the Euler num-ber he(E), [M]i of the bundle E equals the numnum-ber of these points counted with the sign ±1 induced by the orientation. Therefore, such number de-pends only on the oriented bundle and on the fundamental class of M, and not on the choice of the transversal section. Note that in the dierentiable category the Euler number of E equals the integral R_Me(E).

Returning to our problem, the Euler class of the bundle S2n−3(U∨_{) → G}

2(Kn+1)

is Poincaré dual to the Fano variety Z(sf), and to count how many points

Z(sf)contains it is necessary to make a distinction between the complex and

the real case.

In the complex case, which we examine in 3, all the objects involved are canonically oriented. Hence the Euler class of S2n−3_(U∨₎_{is well dened and}

equals the top Chern class. This leads to the following result.

Theorem. The number of lines contained in a general hypersurface of degree 2n − 3 in Pn C is exactly cn= Z G2(Cn+1) ctop(S2n−3(U∨)).

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There are several ways to compute cn numerically, as is proved in [18]

and [19]. In this thesis, at 3.4, we focus in particular on the formula

cn = [(1 − x) 2n−3

Y

a=0

(2n − 3 − a + ax)]xn−1

due to Zagier [18], where the notation [·]xn−1 stands for the coecient of xn−1.

In 3.3 we study the special case of n = 3, i.e. the case of cubic surfaces in P3

C. In this case, in fact, the number c3 equals 27 and, as we prove in

Proposition 3.14, the section sf is transversal not only for general but also

for smooth cubics. Both results conrm the 27 Lines Theorem.

In the real case, orientations on G2(Rn+1) and S2n−3(U∨) can be given

only when n is odd, as shown in 4.1, therefore we focus only on this case. For the non oriented case (n even) we refer to [28], in which the authors apply a general methodology valid for both even and odd cases.

In conclusion, the orientation of G2(Rn+1)induces a sign on each point of

the Fano variety of lines. This leads to a signed count of real lines, explicitly given by the Euler number

n =

Z

G2(Rn+1)

e(S2n−3_(U∨

)).

The sign of ndepends on the orientations, but its absolute value does not. To

calculate the above integral we base on the approach given in [12], obtaining the following nal result at 4.4.

Theorem. A general hypersurface of degree 2n−3 in Pn

R, for n odd, contains

at least

|n| = (2n − 3)!!

lines.

Note that for n = 3 we deduce that a general cubic surface of P3 R must

contain at least |3| = 3 lines, and this is compatible with the above

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List of Notations

pj: n Y s=1 Us → Uj j-th coordinate projection; p. 6

idM identity map of the topological space M; p. 7

GL_{(r, R)} real general linear group; p. 10 At transpose of the matrix A; p. 11

∧

k

V k-th exterior power of the vector space V ; p. 11 (M, S) pair consisting of a topological space M and a

sub-space S; p. 12

H∗(M, S; R) singular cohomology of the pair (M, S) with coe-cients in R; p. 12

f∗ pull-back in cohomology of the map f; p. 12

H∗(M, S; R) singular homology of the pair (M, S) with coecients

in R; p. 12 e

H∗(M ; R) reduced singular homology of M with coecients in

R; p. 12

H∗(M ; R) singular cohomology of M with coecients in R; p. 13 u ∪ v cup product in cohomology; p. 13

u × v cross product in cohomology; p. 14

f∗ push-forward in homology of the map f; p. 17

H∗(M ; R) singular homology of M with coecients in R; p. 18

u ∩ v cap product in homology; p. 18 TorR

1(A, B) the rst Tor functor; p. 18

Ext1R(A, B) the rst Ext functor; p. 18

(C∗(M ; R), ∂) singular chain complex of M with coecients in R;

p. 21

(A∗(M ), d) chain complex of dierential forms on M; p. 22 H_dR∗ (M ) de Rham cohomology of M; p. 22

In×n identity matrix of size n; p. 56

#A cardinality of the set A; p. 61 ∇f gradient of the function f; p. 62

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1 Real vector bundles and Euler class

This chapter introduces the Euler class of a vector bundle, which is the main object of our study. We start by recalling some basic notions about vector bundles. Then, we introduce the notion of orientability for a bundle, which is a key to dene the Euler class in the general setting of topological spaces. At the end of 1.2 some properties of the Euler class are listed and proved. Then, in order to dene the Euler number of a vector bundle, we recall some basic tools of algebraic topology, such as orientation of a manifold, Poincaré duality and singular cohomology. 1.4 interprets the Euler number as a result of a specic integral along a dierentiable manifold. Eventually, the last section considers Poincaré duality of submanifolds and transversality to provide the principal result of the chapter, namely Theorem 1.50.

This chapter follows mainly [1], [20], [27] and provides further biblio-graphic references when needed.

1.1 Preliminary notions

In this section we recall some basic denitions and results concerning vector bundles.

Let M be a topological space, which is called the base space.

Denition 1.1. A real vector bundle ξ of rank r (or r-plane bundle) over M is a continuous and surjective map

π: E → M

between a topological space E (called the total space of ξ) and M such that: 1. for each p ∈ M, Ep = π−1(p) is a real vector space of dimension r (with

zero vector 0p) called the ber over p;

2. for each p ∈ M there exists a neighborhood U of p in M and a home-omorphism

χ: π−1(U ) → U × Rr

called local trivialization of ξ on U, such that π = p1◦ χ;

3. for each local trivialization χ: π−1

(U ) → U × Rr the restriction

χ|Ep: Ep → {p} × R

r

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If it is possible to choose U equal to the base space M, then ξ is called a trivial bundle. A 1-plane bundle is called a line bundle.

Occasionally, we write E instead of ξ, identifying a vector bundle with its total space. If M and E are dierentiable manifolds and all the maps involved in Denition 1.1 are C∞_{, then ξ is called a smooth vector bundle.}

Notation. The symbol rk(ξ) (or rkR(ξ)) denotes the rank of the real vector

bundle ξ.

Denition 1.2. Let

ξ1 = π1: E1 → M1 and ξ2 = π2: E2 → M2

be two vector bundles. A morphism between ξ1 and ξ2 is a pair of continuous

(or, in the dierentiable case, smooth) maps

G: E1 → E2 and g : M1 → M2 such that 1. the diagram E1 G −−−→ E2 π1   y   yπ2 M1 g −−−→ M2 commutes,

2. G|(E1)p: (E1)p → (E2)g(p) is a linear homomorphism for all p ∈ M.

In this case, we say that G covers g. If (G, g) is an invertible morphism (i.e., Gand g are homeomorphisms or, in the dierentiable case, dieomorphisms) we call (G, g) an isomorphism and we say that ξ1 and ξ2 are isomorphic,

writing ξ1 ∼= ξ2. When the base spaces M1 and M2 are equal to the same

space M, we always assume that g = idM, writing G instead of (G, idM).

In order to follow the terminology used in [27], we call bundle map a morphism of vector bundles ξ1 → ξ2 such that the induced linear map on

each ber is an isomorphism of vector spaces.

Lemma 1.3 ([27], Lemma 2.3). Let ξ and ξ0 _{be vector bundles over M, with}

total spaces E and E0 _{respectively. Let G: ξ → ξ}0 _{be a bundle map. Then G}

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Remark 1.4. The above Lemma 1.3 is, in general, false if the base spaces of ξ and ξ0 are dierent. See p. 15 for a counterexample.

Denition 1.5. Let g : M0 _{→ M} _{be a continuous map of topological spaces.}

Consider an r-plane bundle ξ = π : E → M. One can construct the pullback bundle g∗_ξ _{over M}0 _{as follows. The total space of g}∗_ξ _{is the subspace}

M0×M E = {(b, v) ∈ M0 × E | g(b) = π(v)}

of M0_{× E}_{. The projection map π}0_{: M}0_×

M E → M0 is dened by

π0(b, v) = b. Thus, there is a commutative diagram

M0 ×M E G0 −−−→ E π0   y   yπ M0 −−−→ Mg

where G0_{(b, v) = v}_{. The real vector space structure over (M}0 _×

M E)b =

(π0₎−1_(b) _{is dened by}

α1(b, v) + α2(b, w) = (b, α1v+ α2w).

In this way, G0 _{restricts to an isomorphism between (M}0_×

M E)b and Eg(b).

The local trivializations of g∗_ξ _{are constructed as follows. If}

h_{: U × R}r→ π−1(U )

is the inverse of a local trivialization of ξ in U, then the map h0: g−1(U ) × Rr _{→ (π}0

)−1(g−1(U ))

dened by h0_{(b, x) = (b, h(g(b), x))} _{is the inverse of a local trivialization of}

g∗ξ in g−1(U ). Hence, g∗ξ has an r-plane bundle structure over M0.

Lemma 1.6 ([27], Lemma 3.1). If (G, g) is a bundle map from ξ0 _{= π}0_{: E}0 _→

M0 to ξ = π : E → M, then ξ0 ∼= g∗ξ.

It is possible to construct new vector bundles out of old ones, for instance the pullback bundle of Denition 1.5. Hereafter, we give another example.

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Denition 1.7. Given two vector bundles

ξ1 = π1: E1 → M1 and ξ2 = π2: E2 → M2

such that rkR(ξ1) = r1 and rkR(ξ2) = r2, the Cartesian product ξ1 × ξ2 is

dened to be the (r1+ r2)-plane bundle with projection map

π1× π2: E1× E2 → M1× M2

where the ber over the point (b1, b2) is

(E1× E2)(b1,b2) = (E1)b1 × (E2)b2.

Notation. For the Cartesian product bundle π1× π2: E1× E2 → M1× M2,

we use the notation

P1: E1× E2 → E1 and P2: E1× E2 → E2

for the projections on the total spaces, while

p1: M1× M2 → M1 and p2: M1× M2 → M2

denote the projections on the base spaces.

Let V be the category of nite dimensional real vector spaces. Every continuous functor T : V × · · · × V → V of k ≥ 1 variables (see [27], p. 32 for the continuity assumption, which is satised in all of our cases) can be used to dene new vector bundles in the following way. We refer to [27], Theorem 3.6 for a proof.

Theorem 1.8. Let T : V ×· · ·×V → V be a continuous functor of k variables and let

ξ1 = π1: E1 → M, . . . , ξk= πk: Ek → M

be vector bundles of ranks r1, . . . , rk respectively. A new vector bundle over

M is constructed as follows. For every p ∈ M dene Ep = T ((E1)p, . . . ,(Ek)p).

Consider the disjoint union

E = a

p∈M

Ep

and dene the projection

π: E → M

as π(Ep) = p. Then, there exists a canonical topology for E such that E

is the total space of a vector bundle of rank r = dimRT(R

r1_{, . . . , R}rk) with

projection π and bers Ep, for each p ∈ M. This bundle is denoted by

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For example, starting with the direct sum functor ⊕ one obtains the direct sum bundle ξ1⊕ ξ2 of rank r1+ r2. Starting with the tensor product functor

⊗ one obtains the tensor product bundle ξ1⊗ ξ2 of rank r1· r2. Starting with

the duality functor ·∨ _{one obtains the dual bundle ξ}∨

1 of rank r1.

Therefore, when we will dene a bundle specifying only the bers (in terms of a certain functor) and the projection, such denition will be well given thanks to Theorem 1.8.

If ξ1, . . . , ξk are smooth vector bundles, then T (ξ1, . . . , ξk) can be given

the structure of a smooth vector bundle (see [27], p. 34). Furthermore, in the smooth case, T (ξ1, . . . , ξk) can be dened also by using the transition

functions of the bundles involved. See [1], Section 3.1 for details.

Denition 1.9. Let ξ = π : E → M be a smooth r-plane bundle on a dierentiable manifold M. We choose the atlas A = {(Uα, ϕα)}α of M such

that for every chart (Uα, ϕα), there is a local trivialization χα of ξ dened on

π−1(Uα). The maps

χα◦ χ−1_β : (Uα∩ Uβ) × Rr → (Uα∩ Uβ) × Rr

must induce on Uα∩ Uβ smooth maps

gαβ: Uα∩ Uβ → GL(r, R),

such that χα◦ χ−1_β (p, ·) = (p, gαβ(p)·) is a linear isomorphism of {p} × Rr.

The maps {gαβ} are called the transition functions of ξ associated to A.

Now we recall some of the bundle operations that we are going to use. The following denition is well given thanks to [1], Proposizione 3.1.8. Denition 1.10. Let

ξ= π : E → M and ˜ξ = ˜π : ˜E → M

be two smooth vector bundles of rank r and ˜r with transition functions {gαβ}

and {˜gαβ} respectively. Dene the following vector bundles over M.

• The direct sum of ξ and ˜ξ, denoted by ξ ⊕ ˜ξ, is given by transition functions

g_αβ(p) 0 0 g˜αβ(p)

∈ GL(r + ˜r, R), and the ber over p ∈ M is

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• The dual of ξ, denoted by ξ∨_{, is given by transition functions}

(gαβ(p)−1)t∈ GL(r, R),

and the ber over p ∈ M is (E∨₎

p = (Ep)∨.

• The tensor product of ξ and ˜ξ, denoted by ξ ⊗ ˜ξ, is given by transition functions

gαβ(p) ⊗ ˜gαβ(p) ∈ GL(r · ˜r, R),

where gαβ(p) ⊗ ˜gαβ(p) stands for the Kronecker product of the matrices

gαβ(p) and ˜gαβ(p) (see [1], Denizione 1.E.6). The ber over p ∈ M is

(E ⊗ ˜E)p = Ep⊗ ˜Ep.

• The determinant bundle of ξ, denoted by det(ξ) or

∧

rξ, is given by transition functions

det(gαβ(p)) ∈ R \ {0},

and the ber over p ∈ M is

det(E)p =

∧

r

(Ep) ∼= R.

Denition 1.11. A section of a vector bundle ξ = π : E → M is a continuous map s: U → E, where U is an open set of M, such that π ◦ s = idU, i.e.,

s(p) ∈ Ep for each p ∈ U. If U = M, then s is called a global section of

ξ. The section 0ξ: M → E such that 0ξ(p) = 0p is called the zero section

of ξ. If the vector bundle ξ is smooth, we require its sections to be smooth maps. The vector space of global sections of ξ is denoted by Γ(ξ)(M), or by C∞(ξ)(M )in the dierentiable case.

Remark 1.12. By denition, every global section s ∈ Γ(ξ)(M) is a homeo-morphism (or dieohomeo-morphism in the dierentiable case) between M and its image s(M) in E. Moreover, since the zero section 0ξ always exists for every

vector bundle, we can consider the base space embedded canonically in the total space. Therefore, M can be seen as a deformation retract of E with retraction mapping 0ξ◦ π.

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1.2 Orientation of a vector bundle and Euler class

In this section we introduce the concept of orientation for a vector bundle and use it to dene a cohomology class of the bundle called the Euler class. Such class will be the central object of our work.

In this section ξ = π : E → M is a rank r ≥ 1 real vector bundle.

Notation. Let E0 be the space E \0ξ(M )of nonzero vectors in E. For every

p ∈ M we denote by

ιp: (Ep, Ep\ {0p}) ,→ (E, E0)

the inclusion of pairs, and by

ι∗_p: H∗(E, E0; Z) → H∗(Ep, Ep\ {0p}; Z)

the restriction morphism in singular cohomology induced by pull-back of cochains, where H∗

(X, Y ; Z) denotes the singular cohomology of the pair (X, Y ) with coecients in Z. In a similar fashion if S is a subspace of E then S0 is the subspace S \ 0ξ(M ) of E0. If there is an inclusion of pairs

(Ep, Ep\ {0p}) ,→ (S, S0), we again denote it with the symbol ιp.

Remark 1.13. For each real vector space V of dimension r, recall that Hi(V, V \ {0}; Z) is nonzero only for i = r. In fact, one has that

Hi(V, V \ {0}; Z) ∼= Hi(Rr, Rr\ {0}; Z)

∼

= eHi−1(Rr\ {0}; Z)

∼

= eHi−1(Sr−1; Z),

where He denotes the reduced singular homology. Moreover, from the Uni-versal Coecient Theorem on p. 18 we have

Hr_{(V, V \ {0}; Z) ∼}= HomZ(Hr(V, V \ {0}; Z), Z) ∼= Z.

Thanks to Remark 1.13, we can give the next

Denition 1.14. An orientation for ξ is the choice of a generator up ∈ Hr(Ep, Ep\ {0p}; Z)

for each ber Ep of ξ such that: for all p ∈ M there exists a neighborhood

U of p in M and a cohomology class

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so that for every q ∈ U,

ι∗_q(uU) = uU|(Eq,Eq\{0q}) = uq.

We say that ξ is orientable if an orientation for ξ exists. A pair (ξ, u) where uis an orientation for ξ, is called an oriented bundle. In order to simplify the notation, an oriented bundle (ξ, u) is simply denoted by ξ if the orientation u is understood.

The next theorem is the central result that allows us to dene the Euler class of a vector bundle. This is proved in [27], Theorem 10.4.

Theorem 1.15. Let (ξ, u) be an oriented vector bundle of rank r. Then there exists one and only one cohomology class

t(ξ) ∈ Hr_{(E, E} 0; Z),

called the Thom class of ξ, such that for every p ∈ M, the restriction ι∗_p(t(ξ)) = t(ξ)|(Ep,Ep\{0p}) equals up. Moreover, for all k ∈ Z, the map

Tk: Hk(E; Z) → Hk+r(E, E0; Z)

dened by

y 7→ y ∪ t(ξ),

where ∪ denotes the cup product, is an isomorphism.

Remark 1.16. The projection π : E → M induces an isomorphism π∗: H∗(M ; Z) → H∗(E; Z),

since, by Remark 1.12, M is a deformation retract of E. Denition 1.17. The Thom isomorphism is the map

φk: Hk

(M ; Z) → Hk+r_{(E, E}

0; Z) (1.1)

dened by the formula

φk(x) = (Tk_{◦ π}∗

)(x) = (π∗x) ∪ t(ξ). (1.2) Remark 1.18. Consider the inclusion j : (E, ∅) ,→ (E, E0). The latter gives

rise to a restriction homomorphism

j∗: H∗(E, E0; Z) → H∗(E; Z)

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Now we are ready to give the following

Denition 1.19. The Euler class of an oriented real vector bundle ξ of rank r is the class

e(ξ) ∈ Hr(M ; Z) such that π∗_{(e(ξ)) = t(ξ)|}

E. Often, we will write e(E) instead of e(ξ).

Here we collect some of the most important properties of the Euler class. Proposition 1.20. Let

ξ = π : E → M and ξ0 = π0: E0 → M0

be two oriented vector bundles of ranks r and r0 _{respectively. The following}

properties hold:

1. If r = r0 _{and (G, g): ξ}0 _{→ ξ} _{is an orientation preserving bundle map}

(i.e., (G, g) preserves the chosen orientation for each bre), then e(ξ0_{) =}

g∗e(ξ),

2. If ξ is a trivial bundle, then e(ξ) = 0,

3. If −ξ is the bundle ξ with the opposite orientation, then e(−ξ) = −e(ξ), 4. If r is odd, then e(ξ) + e(ξ) = 0,

5. e(ξ × ξ0_{) = e(ξ) × e(ξ}0₎_,

6. If M = M0_{, then e(ξ ⊕ ξ}0_{) = e(ξ) ∪ e(ξ}0₎_,

7. If s ∈ Γ(ξ)(M) is a nowhere vanishing section, then e(ξ) = 0. Proof.

1. If (G, g) is a bundle map then, thanks to Lemma 1.6, we can suppose that ξ0 ∼= g∗_ξ_{. So the bundle map (G, g) becomes}

g∗E −−−→ E∼= 0 _{−−−→ E}G   yπ˜   yπ 0   yπ M0 M0 −−−→ Mg

By the uniqueness property of Theorem 1.15, we note that G∗_t(ξ)_{is equal to}

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(see [22], p. 255). Now, 1. follows from the commutative diagram H∗(E, E0; Z) ·|E −−−→ H∗ (E; Z) ←−−− Hπ∗ ∗ (M ; Z)   yG ∗   yG ∗   yg ∗ H∗(g∗_E,_(g∗_E) 0; Z) ·|g∗E −−−→ H∗_(g∗_E ; Z) ←−−− Hπ˜∗ ∗_(M0 ; Z)

2. If {p} × Rr_{→ {p}}_{is an r-plane bundle over a point, then e({p} × R}r_{) = 0}

because Hr_{({p}; Z) ∼}_{= (0)} for r ≥ 1. Now let M × Rr _{→ M} be a trivial

r-plane bundle over M. Consider the constant map cp: M → {p}. We obtain

an induced bundle map (Cp, cp)

M × Rr Cp −−−→ {p} × Rr p1   y   y p1 M −−−→cp {p} Again, by Lemma 1.6 we have that M × Rr ∼_{= c}∗

p({p} × Rr). Now 2. follows

from the previous point 1., since e_{(M × R}r) = e(c∗ p({p} × R r_{)) = c}∗ pe({p} × R r_{) = c}∗ p(0) = 0.

3. If ξ is oriented by the choice of the generator up ∈ Hr(Ep, Ep\ {0p}; Z)

for each p ∈ M and t(ξ) is the Thom class of ξ, it follows immediately that t(−ξ)|(Ep,Ep\{0p}) = −up = −t(ξ)|(Ep,Ep\{0p}).

Thus, t(−ξ) = −t(ξ) implies that e(−ξ) = −e(ξ).

4. Apply the Thom isomorphism φr _{to the Euler class e(ξ),}

φr(e(ξ)) = (π∗e(ξ)) ∪ t(ξ) = (t(ξ)|E) ∪ t(ξ) = t(ξ) ∪ t(ξ).

Hence, the following relation holds e(ξ) = (φr₎−1

(t(ξ) ∪ t(ξ)).

By the anticommutativity of cup product (see [20], Theorem 3.11), one has that

t(ξ) ∪ t(ξ) = (−1)r2

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Thus e(ξ) = −e(ξ).

5. We note that the direct sum (E ⊕ E0₎

p = Ep⊕ E0p is oriented by taking

an orientation (i.e., an ordered basis) for Ep followed by an orientation for

E0p. It is possible to show (see [27], p. 92) that

t(ξ × ξ0) = (−1)rr0t(ξ) × t(ξ0), (1.3) where × denotes the cross product in cohomology. Consider the commutative diagrams E × E0 π×π 0 −−−→ M × M0 M × M0 π×π 0 ←−−− E × E0 P1   y p1   y   yp2   yP2 E −−−→π M M0 π 0 ←−−− E0 .

By functoriality, we obtain the commutative diagrams

H∗(E × E0₎ _{←−−−− H}(π×π0)∗ ∗_{(M × M}0₎ _H∗_{(M × M}0₎ _{−−−−→ H}(π×π0)∗ ∗_{(E × E}0₎ P1∗ x   p1 ∗ x   x  p2 ∗ x  P2 ∗ H∗(E) ←−−−π∗ H∗(M ) H∗(M0₎ _−−−→π0∗ _H∗_(E0₎

where we have omitted the coecients group Z in the notation. Applying the restriction homomorphism to both sides of (1.3), one has that

(π × π0)∗(e(ξ × ξ0)) = t(ξ × ξ0)|E×E0 = (−1)rr0_t(ξ)| E× t(ξ0)|E0 = (−1)rr0_π∗ e(ξ) × π0∗e(ξ0) = (−1)rr0 P1∗π∗e(ξ) ∪ P2∗π0∗e(ξ0) = (−1)rr0(π × π0)∗p1∗e(ξ) ∪ (π × π0)∗p2∗e(ξ0) = (−1)rr0(π × π0)∗(p1∗e(ξ) ∪ p2∗e(ξ0)) = (−1)rr0(π × π0)∗(e(ξ) × e(ξ0)).

Thus, being (π × π0₎∗_{, π}∗_{, π}0∗ _{all isomorphisms, we get the relation}

e(ξ × ξ0_{) = (−1)}rr0

e(ξ) × e(ξ0_), _(1.4)

where the sign can be ignored because, by point 4., the right side of (1.4) is an element of order two if r or r0 _{is odd.}

6. Consider the diagonal embedding ∆: M → M × M. Then, one has that (e(ξ ⊕ ξ0)) = ∆∗(e(ξ × ξ0))

= ∆∗(e(ξ) × e(ξ0)) = e(ξ) ∪ e(ξ0).

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7. The composition

M −→ Es 0 ⊂ E π

− → M

is, by denition, the map idM. Then the induced composition in cohomology

Hr_{(M ; Z)}←s− H∗ r_(E 0; Z) ·|_E0 ←−− Hr (E; Z)←− Hπ∗ r (M ; Z) becomes the map idHr_{(M ;Z)}. Thus

e(ξ) = s∗((π∗e(ξ))|E0) = s

∗

((t(ξ)|E)|E0).

But the composition

Hr(E, E0; Z) ·|E

−→ Hr

(E; Z)−−→ H·|E0 r_(E

0; Z) (1.5)

is zero since (1.5) is a part of the long exact sequence of the pair (E, E0)

(see [20], p. 199). Hence, e(ξ) = s∗_{(0) = 0}_.

1.3 Orientation of a manifold and Euler number

In this section, M is a connected topological manifold of dimension n without boundary. For ease of use, we call it an n-manifold.

We begin with the following

Remark 1.21. For each p ∈ M recall that (see for example [20], p. 231)

Hi(M, M \ {p}; Z) ∼=

(

Z, if i = n (0), if i 6= n.

Denition 1.22. A topological orientation for the manifold M is a function which assigns to each p ∈ M a generator µp ∈ Hi(M, M \ {p}; Z) such that

for all p ∈ M there exists a compact neighborhood U of p in M with a homology class

µU ∈ Hn(M, M \ U ; Z)

so that (iq)∗(µU) = µq for each q ∈ U, where

iq: (M, M \ U ) ,→ (M, M \ {q})

is the inclusion. The manifold M is called C0_{-orientable if a topological}

orientation exists for M. Consequently, M is C0_{-oriented if a topological}

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In particular [20], Theorem 3.26 says that Hn(M ; Z) ∼= Z if M is C0

-oriented and compact. A homology class µ ∈ Hn(M ; Z) whose image µp ∈

Hn(M, M \ {p}; Z) is a generator for each p ∈ M is called a fundamental

class for M and is denoted by [M]. Now we state a fundamental theorem of algebraic topology, the proof of which can be found in [20], p. 247.

Theorem 1.23 (Topological Poincaré Duality). If M is a C0_{-oriented and}

compact n-manifold, then the map

DM: Hk(M ; Z) → Hn−k(M ; Z)

dened by

DM(α) = [M ] ∩ α

is an isomorphism for all k ≥ 0.

Remark 1.24. We consider the cap product operation ∩ : Hk(M ; Z) × Hl(M ; Z) → Hk−l(M ; Z),

dened for example in [20], p. 239, in the following way: let α ∈ Hk(M ; Z)

and φ ∈ Hl_{(M ; Z). Then α ∩ φ is the only (k − l) homology class such that}

for every ψ ∈ Hk−l

(M ; Z)

ψ(α ∩ φ) = φ ∪ ψ(α). (1.6)

Another result used hereafter is the Universal Coecient Theorem, which is given in two versions, one for homology and another for cohomology. De-tails can be found in [20], Theorem 3.2 and Theorem 3A.3 or in [33], p. 222 and p. 243. These are stated in the following result and in a less general case.

Theorem 1.25 (Universal Coecient Theorem). Let M be an n-manifold. If R is a principal ideal domain and G is an R-module, then we have

• U.C.T for Singular Homology:

Hk(M ; G) ∼= (Hk(M ; R) ⊗RG) ⊕ TorR1(Hk−1(M ; R), G).

• U.C.T for Singular Cohomology:

Hk(M ; G) ∼= HomR(Hk(M ; R), G) ⊕ Ext1R(Hk−1(M ; R), G).

Remark 1.26. Both the isomorphisms appearing in Theorem 1.25 are not natural, as pointed out in [20], p. 196. For the signicance of the functors TorR

1 and Ext 1

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Corollary 1.27. If M is an n-manifold then

Hk_{(M ; R) ∼}= HomZ(Hk(M ; Z), R) ∼= Hk(M ; R) ∨

. Moreover, if M is C0-orientable and compact then

Hn_{(M ; Z) ∼}= HomZ(Hn(M ; Z), Z) ∼= Z.

Proof. Using Theorem 1.25 with R = Z and G = R we obtain:

Hk(M ; R) ∼= Hk(M ; Z) ⊗ZR. (1.7)

Since R is a torsion-free Z-module, the direct summand TorZ

1(Hk−1(M ; Z), R)

equals 0 (see [20], Proposition 3A.5). Again, by Theorem 1.25 with R = G = R we have:

Hk_{(M ; R) ∼}= HomR(Hk(M ; R), R) = Hk(M ; R) ∨

. The term Ext1

R(Hk−1(M ; R), R) = 0, since R is a eld. In fact Hk−1(M ; R)

is a R-vector space, i.e., it is a free R-module (see [20], p. 195). If we apply another time Theorem 1.25 with R = Z and G = R we have:

Hk_{(M ; R) ∼}= HomZ(Hk(M ; Z), R).

The reason why Ext1

Z(Hk−1(M ; Z), R) = 0 is that R is a divisible abelian

group (see [20], p. 320). For the second part, we can use Theorem 1.23 to obtain

Hn_{(M ; Z) ∼}= H0(M ; Z) ∼= Z.

Consequently, it holds that Z∼= Hn(M ; Z) ∼ = HomZ(Hn(M ; Z), Z) ⊕ Ext 1 Z(Hn−1(M ; Z), Z) by Theorem 1.25 ∼ = HomZ(Z, Z) ⊕ Ext 1 Z(Hn−1(M ; Z), Z) since Hn(M ; Z) ∼= Z ∼

= Z ⊕ Ext1_Z(Hn−1(M ; Z), Z) since HomZ(Z, Z) ∼= Z

and it follows that Ext1

Z(Hn−1(M ; Z), Z) = 0, so we can conclude.

Corollary 1.27 says that every class in Hn

(M ; Z) can be seen as a homo-morphism from the top homology group of M with value in Z. In particular, it is possible to give the next

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Denition 1.28. Let M be a C0_{-oriented and compact n-manifold. If ξ =}

π: E → M is an oriented vector bundle with rk_R(E) = dim(M ) = n, let e(ξ) ∈ Hn

(M ; Z) ∼= HomZ(Hn(M ; Z), Z)

be its Euler class. The Euler number of ξ is the integer ξ = e(ξ)([M ]).

We use the notation

he(ξ), [M ]i for the Euler number ξ.

1.4 Euler number for dierentiable manifolds

Here we suppose that M is a dierentiable n-manifold. Our aim is to calculate the Euler number through integration along M.

Denition 1.29. A smooth n-manifold M is C∞_{-orientable if it admits an}

atlas A = {(Uα, ϕα)}αsuch that for every couple of charts (Uα, ϕα), (Uβ, ϕβ) ∈

A one has that

det(d(ϕα◦ ϕ−1β )x) > 0

for all x ∈ ϕβ(Uα ∩ Uβ). The maximal atlas with this property is called

a smooth orientation for M. The manifold M is C∞_{-oriented if a smooth}

orientation is assign to M.

Thus, for a C∞_{manifold, we have two notions of orientability: one}

dier-entiable and the other topological. The next theorem shows that these two notions are equivalent (see [7], Theorem 7.15 at p. 347 for a proof).

Theorem 1.30. Let M be a smooth n-manifold. M is C∞_{-orientable if and}

only if is C0_-orientable.

Thanks to Theorem 1.30 we can simply call orientable a dierentiable manifold admitting a topological or a smooth orientation.

The next step is to give an interpretation of the Euler class as a de Rham cohomology class. This is possible thanks to a version of the de Rham Theorem, made for singular cohomology. All the details of what follows can be found in [26], Chapter 16. We state the fundamental steps.

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Denition 1.31. Let ∆k = ( _k X i=0 tiei | 0 ≤ ti ≤ 1and k X i=0 ti = 1 ) ⊂ Rk

be the standard k-simplex. Let M be a dierentiable n-manifold. A smooth k-simplex is a C∞ map σ : ∆k → M. The subgroup of Ck(M ; Z) generated

by smooth simplices is denoted by C∞

k (M ; Z) and called the smooth k-chain

group. It is possible to restrict the boundary operator ∂k: Ck(M ; Z) →

Ck−1(M ; Z) to the subgroup Ck∞(M ; Z) and, as for the singular homology

theory, one has that

∂k(Ck∞(M ; Z)) ⊆ C ∞

k−1(M ; Z) and ∂k◦ ∂k+1 = 0.

Thus, dene

• the group of smooth k-cycles as

Z_k∞_{(M ; Z) = Ker(∂}k),

• the group of smooth k-boundaries as

B_k∞_{(M ; Z) = Im(∂}k+1),

• the k-th smooth singular homology group of M as H_k∞_{(M ; Z) =} Z ∞ k (M ; Z) B∞ k (M ; Z) .

Remark 1.32. The inclusion ι: C∞

k (M ; Z) ,→ Ck(M ; Z) commutes with the

boundary ∂, so induces a map

ι∗: H_k∞(M ; Z) → Hk(M ; Z)

dened by ι∗([c]) = [ι(c)].

Theorem 1.33. For a smooth n-manifold M the map ι∗: Hk∞(M ; Z) → Hk(M ; Z)

is an isomorphism for every k ≥ 0.

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Remark 1.34. Theorem 1.33 tells us that for every [c] ∈ Hk(M ; Z) it is

pos-sible to nd a smooth k-cycle ˜c ∈ Z∞

k (M ; Z) such that ˜c represents the class

[c]. Thus, from now, we can suppose that for every class [c] ∈ Hk(M ; Z), the

representative c is a smooth k-cycle.

Denition 1.35. Suppose M is a smooth n-manifold, σ : ∆k → M is a

smooth k-simplex and ω ∈ Ak(M ). The integral of ω over σ is

Z σ ω = Z ∆k σ∗ω. If c = Ps

j=1cjσj is a smooth k-chain the integral of ω over c is

Z c ω = s X j=1 cj Z σj ω = s X j=1 cj Z ∆k σ_j∗ω.

The next statement is an alternative version of the Stokes' Theorem (see [26], Theorem 16.10 for the proof). This is the last result that we need to connect the singular cohomology theory with the de Rham cohomology. Theorem 1.36 (Stokes' Theorem for Chains). Let c be a smooth k-chain in a smooth n-manifold M. Consider ω ∈ Ak−1_{(M )}_{. Then}

Z ∂c ω = Z c dω. For all k ≥ 0, consider the linear map

Jk: HdRk (M ) → H k (M ; R) (1.8) dened by hJk([ω]), [c]i = Z c ω,

for all [c] ∈ Hk(M ; Z). The map (1.8) is well dened since

• if [ω] ∈ Hk

dR(M )(in particular dω = 0) and c, c

0 _{are smooth k-cycle}

rep-resenting the same homology class [c] ∈ Hk(M ; Z), then Theorem 1.33

says that there exists b ∈ C∞

k+1(M ; Z) such that c − c 0 _{= ∂b}_{. Thus, by} Theorem 1.36, we have Z c ω − Z c0 ω= Z c−c0 ω= Z ∂b ω = Z b dω= 0.

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• If [c] ∈ Hk(M ; Z) (in particular ∂c = 0) and ω = dη is an exact k-form,

then by Theorem 1.36 one has that Z c ω = Z c dη= Z ∂c η= 0.

Moreover, by Denition 1.35 and by linearity of the integral, one has that hJk([ω]), r[c] + s[c0]i = rhJk([ω]), [c]i + shJk([ω]), [c0]i

and

hJk(α[ω] + β[η]), [c]i = αhJk([ω]), [c]i + βhJk([η]), [c]i

for all [c], [c0_{] ∈ H}

k(M ; Z), r, s ∈ Z, [ω], [η] ∈ HdRk (M ) and α, β ∈ R. Thus,

for each k ≥ 0, Jk([ω]) is a well dened element of HomZ(Hk(M ; Z), R) ∼=

Hk_{(M ; R) and} Jk ∈ HomR(H k dR(M ), H k (M ; R)).

Theorem 1.37 (Singular de Rham Theorem). Let M be a smooth n-manifold. For every k ≥ 0 the homomorphism

Jk: HdRk (M ) → Hk(M ; R)

is an isomorphism called de Rham isomorphism.

A proof of the above result can be found in [26], Theorem 16.12.

If M is orientable and compact, then we can express the pairing with the fundamental class [M] as integration along M in the following way.

Proposition 1.38. For all ω ∈ An_{(M )} _{one has that}

hJn([ω]), [M ]i = Z [M ] ω = Z M ω.

Proof. Choose ω ∈ An(M ), clearly ω is automatically closed. Theorem 1.37

says that

hJn([ω]), [M ]i =

Z

[M ]

ω.

Every compact and oriented dierentiable n-manifold admits a smooth tri-angulation (see [34]), i.e., a smooth cycle

c=X

j

σj ∈ Zn∞(M ; Z)

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1. each σj: ∆n→ M is a C∞ orientation preserving embedding,

2. if j 6= s then σj(Int ∆n) ∩ σs(Int ∆n) = ∅,

3. M = Sjσj(∆n).

The triangulation c has the property that Z c ω = Z M ω

see [26], Problem 16-2 for a reference. Moreover, (see [20], p. 238 or [14], pp. 157-158), the triangulation c represents the fundamental class [M] of M. Thus the integral of a form in the sense of Denition 1.35 equals the integral along M in the usual sense of dierential forms.

Remark 1.39. Let ξ be a smooth oriented vector bundle over a compact and oriented n-manifold M, with rkR(ξ) = n. We can see the Euler class

e(ξ) ∈ Hn_{(M ; Z) as an element of H}n_{(M ; R) considering the composition}

Hn(M ; Z) → Z ⊂ R.

Thanks to Proposition 1.38, the Euler number he(ξ), [M]i of a smooth vector bundle ξ over M can be interpreted as the integral R_Mν, where ν ∈ An_{(M )} _{is a representative for the unique class [ν]}

ξ ∈ HdRn (M ) such that

Jn([ν]ξ) = e(ξ). From now on, we identify the class [ν]ξ with the Euler class

e(ξ), without specifying the isomorphism Jn.

We conclude this part stating a result that explains the names Euler class and Euler number for e(ξ) and ξ respectively.

Let M be a smooth manifold and denote by T M its tangent bundle. Lemma 1.40. M is an orientable smooth manifold if and only if T M is orientable in the sense of Denition 1.14. In particular, if M is oriented then the Euler class e(T M) is well dened.

A proof can be found in [27], Lemma 11.6.

Denition 1.41. Let M be a compact n-manifold. The Euler characteristic of M is χ(M ) = n X k=0 (−1)k_dim RH k (M ; R).

The characteristic χ(M) is well dened. Indeed, since M is compact, by [1], Proposizione 5.8.3 and Theorem 1.37

dim_RH_dRk (M ) = dim_RHk_{(M ; R) < ∞} for all k ≥ 0.

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Proposition 1.42. If M is a compact and oriented smooth n-manifold, then χ(M ) = he(T M ), [M ]i,

i.e., the Euler characteristic of M coincides with the Euler number of T M. See [27], Corollary 11.12 for a proof.

1.5 Poincaré dual of submanifolds and transversality of

sections

In this part, we recall the Poincaré duality for the de Rham cohomology, and introduce the notion of Poincaré dual for a submanifold. The latter is subsequently related to Poincaré duality in the sense of Denition 1.45. Consequently, we introduce the concept of transversality for sections of vector bundles. At the end, we prove a result that allows to connect the Euler class of a bundle with the zero locus of a transversal section.

As before, M denotes a connected smooth manifold of dimension n. Let us now recall the C∞ _{version of the Poincaré Duality, the proof of which can}

be found in [1], Teorema 5.6.6.

Theorem 1.43 (Smooth Poincaré Duality). If M is oriented and compact then, for all k ≥ 0, the map

Z M : HdRk (M ) × H n−k dR (M ) → R given by ([ω], [η]) 7→ Z M ω ∧ η

is a well dened and non-degenerate bilinear map. In particular, we denote with the same symbol the induced isomorphism

Z M : HdRk (M ) → H n−k dR (M ) ∨ , which is given by Z M [ω], [η] = Z M ω ∧ η, for each [ω] ∈ Hk dR(M )and [η] ∈ H n−k dR (M ).

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Corollary 1.44. Let M be an oriented and compact n-manifold and let ι: S ,→ M be an oriented and closed submanifold of dimension k. Then, there exists a unique class [ηS] ∈ HdRn−k(M )such that

Z S ι∗ω= Z M ηS∧ ω

for all closed ω ∈ Ak_{(M ).}

Proof. Consider the linear functional λ: Ak_{(M ) → R dened by}

hλ, ωi = Z

S

ι∗ω.

If ω, ν ∈ Ak_{(M )} _{are two closed forms such that ν = ω + dφ for some φ ∈}

Ak−1_{(M )}_{, then} hλ, νi = Z S ι∗ν = Z S ι∗(ω + dφ) = Z S ι∗ω+ Z S d(ι∗φ) = hλ, ωi + Z ∂S ι∗φ where the last equality follows from Stokes' Theorem ([1], Teorema 4.5.12) and R_∂Sι∗φ = 0 since ∂S = ∅. Thus, λ is well dened as a linear functional on Hk

dR(M ). By Theorem 1.43 there is a unique class [ηS]in HdRn−k(M )such

that

λ= Z

M

[ηS]

Thus, for each [ω] ∈ Hk

dR(M ), we have Z S ι∗ω= hλ, [ω]i = Z M [ηS], [ω] = Z M ηS ∧ ω.

Denition 1.45. The class [ηS]dened in Corollary 1.44 is called the Poincaré

dual of the submanifold S ,→ M. Sometimes, we will call Poincaré dual even a representative of the class [ηS].

The next step is to connect the Poincaré duality of Theorem 1.23 with the duality in the sense of Denition 1.45. To do that, consider a compact and orientable n-manifold M with an oriented and closed embedding f : S → M of dimension k.

Denition 1.46. The embedding f induces a map f∗: H∗(S; Z) → H∗(M ; Z)

in homology. Consider the fundamental class [S] ∈ Hk(S; Z). Note that if

S is disconnected, a fundamental class is the direct sum of the fundamental classes for each connected component. We call the class

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fundamental class of S in M through f. If f is the inclusion ι: S ,→ M then we call [ι(S)]M = ι∗([S]) fundamental class of S in M and denote it by [S]M.

Clearly, if f is the identity idM: M → M we get the fundamental class

[M ]M = idM ∗([M ]) = [M ].

Poincaré duality of Theorem 1.23 holds not only in Z, but also in R coecients (see [20], Theorem 3.30). Furthermore, if [S] ∈ Hk(S; Z) is a

fundamental class for a manifold S, we can consider it as an element of Hk(S; R) using, for instance, the Universal Coecient Theorem 1.25. Thus,

we can rewrite Denition 1.46 using R coecients.

Proposition 1.47. Let ι: S ,→ M be as in Corollary 1.44. The fundamental class [S]M ∈ Hk(M ; R) is the Poincaré dual (in the sense of Theorem 1.23

with R coecients) of [ηS] ∈ HdRn−k(M ) ∼= Hn−k(M ; R).

Proof. For all ω ∈ Ak_{(M )}_{closed form, the following chain of equalities holds:}

Z [M ]∩[ηS] ω = [ω]([M ] ∩ [ηS]) by Theorem 1.37 = [ηS] ∪ [ω]([M ]) by (1.6) = Z M ηS∧ ω by Proposition 1.38 = Z S ι∗ω by Corollary 1.44 = Z [S] ι∗ω by Proposition 1.38 = Z ι∗[S] ω by denition of ι∗ and ι∗.

This means that R_{[M ]∩[η}_S_] and R_ι_∗_[S] are the same functional in Hk dR(M )

∨_,

hence

[M ] ∩ [ηS] = ι∗[S]. (1.9)

The third equality also follows from the fact that the cup product in singular cohomology corresponds to wedge product in de Rham cohomology. More precisely, (H∗

dR(M ), ∧) ∼= (H ∗

(M ; R), ∪) is an algebra isomorphism. We assume this fact referring to [6], Theorem 14.28 combined with p. 192, where a rigorous proof (that uses the advanced tool of Spectral Sequences) can be found. Instead, in [26], Problem 15-3 the cup product is dened directly by

[ω] ∪ [η] = [ω ∧ η], thus the third equality follows immediately.

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We can now consider transversality of vector bundles' sections.

Let ξ = π : E → M a smooth r-plane bundle over an n-dimensional smooth manifold M.

Denition 1.48. Take s and ˜s two sections in C∞_{(ξ)(M ). We say that s}

and ˜s are transverse if for every p ∈ M with s(p) = ˜s(p) = v we have

TvE = dsp(Tp(M )) + d˜sp(Tp(M )) (1.10)

or, equivalently, that

TvE = Tvs(M ) + Tvs(M ).˜

In this case we write s t ˜s. If ˜s is equal to the zero section 0ξ and s t 0ξ,

we call s a transversal section of ξ.

Notation. If s ∈ C∞_{(ξ)(M )}_{is a section, then}

Z(s) = s−1(0ξ(M )) = {p ∈ M | s(p) = 0p}

denotes the zero locus of s.

Proposition 1.49. If s ∈ C∞_{(ξ)(M )} _{is a transversal section, then Z(s) is}

a submanifold in M of dimension

dim(Z(s)) = dim(M ) − rk(ξ) = n − r.

Proof. Suppose n ≥ r and take p ∈ Z(s). Since s: M → E is an injective immersion, we can choose a chart (U, ϕ) of M centered at p and a chart (π−1_{(U ), ψ = (ϕ, id}

Rr) ◦ χ) of E centered in v = s(p), such that s(U) ⊆

π−1(U ),

ψ ◦ s ◦ ϕ−1(x1, . . . , xn) = (x1, . . . , xn,0, . . . , 0)

and

ψ(s(U )) = ψ(π−1(U )) ∩ (Rn_{× {0} Rr}).

In other words, we may suppose that E = Rn+r_{, v = 0 and M = 0}

ξ(M ) =

Rn ⊂ Rn+r. Now we consider the projection q : Rn+r → Rr onto the last r coordinates. Around p we have that Z(s) = (q ◦ s)−1_({0

Rr}) and (q ◦ s) is

a submersion at p by transversality. Thus Z(s) is a submanifold in M of codimension r. Finally, we note that if n < r, condition (1.10) is satised only when s(M) ∩ 0ξ(M ) = ∅. The only possibility is that Z(s) = ∅.

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Theorem 1.50. Let π : E → M be an oriented smooth vector bundle over an oriented and compact manifold M. Let s ∈ C∞_{(E)(M )} _{be transversal to}

the zero section. Then e(E) is the Poincaré dual of Z(s).

Prior to proving Theorem 1.50 we need some results. As before, ξ = π: E → M denotes a smooth vector bundle.

Denition 1.51. A Riemannian structure on ξ is a section g ∈ C∞(ξ∨ ⊗ ξ∨)(M )

that is

• symmetric, i.e., gp(v, w) = gp(w, v),

• positive denite, i.e., gp(v, v) ≥ 0 and gp(v, v) = 0 ⇔ v = 0p,

for all p ∈ M and v, w ∈ Ep.

Example 1.52. If ξ = T M is the tangent bundle of M, then a Riemannian structure on T M is exactly a Riemannian metric on M.

Lemma 1.53. Any smooth vector bundle can be endowed with a Riemannian structure.

We refer to [6], p. 55 for a proof.

Denition 1.54. Let (ξ, g) a smooth r-plane bundle endowed with a Rie-mannian structure. We give two denitions.

• The unit disk bundle associated to (ξ, g) is the bre bundle Dξ = π|D(E): D(E) → M

with total space

D(E) = {v ∈ E | gp(v, v) ≤ 1, where p = π(v)}.

Hence, the bre D(E)p = D(Ep) over p is the unitary disk in Ep.

• The unit sphere bundle associated to (ξ, g) is the bre bundle Sξ= π|S(E): S(E) → M

with total space

S(E) = {v ∈ E | gp(v, v) = 1, where p = π(v)}.

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Remark 1.55. By excision and homotopy we have two canonical isomorphisms H∗(E, E0; Z) ∼= H∗(D(E), D(E) \ 0ξ(M ); Z) ∼= H∗(D(E), S(E); Z).

In this way, we can regard the Thom class t(ξ) of the bundle ξ as an element of Hr

(D(E), S(E); Z).

Remark 1.56. If X is a compact, orientable, n-manifold with boundary ∂X, there is a relative fundamental class [X] ∈ Hn(X, ∂X; Z) and a Poincaré

duality isomorphism

DX: Hk(X, ∂X; Z) → Hn−k(X; Z)

dened by

DX(α) = [X] ∩ α.

See [20], Theorem 3.43 for details.

Lemma 1.57. Let ξ be a smooth, oriented, r-plane bundle over a compact and oriented n-manifold M. Then

(0ξ)∗[M ] = [D(E)] ∩ t(ξ) ∈ Hn(D(E); Z)

or, in other words, the Thom class of ξ is just the Poincaré dual of the submanifold M ∼= 0ξ(M ) ,→ D(E).

Proof. Recall that every manifold is assumed to be connected. By denition, ∂D(E) = S(E). In order to simplify the notation, denote by (D, S) the couple (D(E), S(E)) and by t the Thom class t(ξ). Consider the following chain of isomorphisms Z∼= H0(M ; Z) π∗(·)∪t −−−−→ Hr_{(D, S; Z)} _{−−−→ H}[D]∩ n(D; Z) π∗ −−−→ Hn(M ; Z) ∼= Z that map 1 7−→ π∗(1) ∪ t = t 7−→ [D] ∩ t 7−→ π∗([D] ∩ t).

The class π∗([D] ∩ t)must be a generator of Hn(M ; Z). Thus,

π∗([D] ∩ t) = ±[M ]

and, since π∗ is an isomorphism with inverse (0ξ)∗, we obtain

[D] ∩ t = ±(0ξ)∗[M ].

To eliminate the sign in the last equality we would need of an explicit deni-tion of the cap product. We avoid this technical passage referring the reader to [6], p. 67, where the proof of this lemma is given completely in terms of de Rham cohomology and dierential forms. In this latter case the sign is understood in terms of integration on oriented manifolds.

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Now it is possible to conclude this section. Proof of Theorem 1.50. Endow ξ = E π

−

→ M with a Riemannian structure g and consider the unit disk bundle D π

−

→ M associated to (ξ, g). Denote by S0

the image 0ξ(M )of the zero section in D. Since M is compact, assume that

S = s(M ) is entirely contained in Int D. Now, the section s is homotopic to the zero section 0ξ (see [6], Remark 12.4.2), thus

[M ]D = (0ξ)∗[M ] = s∗[M ]

is the Poincaré dual of the Thom class t = t(ξ) of ξ, by the previous lemma. It follows that t∪t is the Poincaré dual of the transversal intersection S ∩S0

ι

,→ D. Indeed, using the formula

[ηS∩S0] = [ηS] ∪ [ηS0]

valid for transversal intersections (see [21], Proposition 5.4.12 for a proof in terms of cohomology, or [6], p. 69 for another one in terms of dierential forms), we have that

[S ∩ S0]D = ι∗[S ∩ S0]

= [D] ∩ [ηS∩S0]

= [D] ∩ ([ηS] ∪ [ηS0])

= [D] ∩ (t ∪ t). Note that the cap product

∩ : Hn(D; Z) × Hr(D, S; Z) → Hn−r(D; Z)

is dened, by composition, in the following way Hn(D) × Hr(D, S) id ×j∗ −−−→ Hn(D) × Hr(D) ∩ − → Hn−r(D) (1.11)

where j : (D, ∅) ,→ (D, S) is the pair inclusion and the second arrow is the usual cap product (see [21], p. 152). As Z(s) ,→ Mi and s: M → D, one has that

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Thus,

[Z(s)]M = π∗([S ∩ S0]D) since π ◦ s = idM

= π∗([D] ∩ (t ∪ t))

= π∗(([D] ∩ t) ∩ t) by [20], p. 259

= π∗([M ]D ∩ t) by the previous lemma

= π∗([M ]D ∩ j∗t) by (1.11)

= π∗([M ]D ∩ π∗(e(ξ))) by denition of the Euler class

= π∗([M ]D) ∩ e(ξ) by [20], p. 241

= [M ] ∩ e(ξ).

This means that e(ξ) is Poincaré dual to the class [Z(s)]M. Passing to real

co-ecients, recall, by Proposition 1.47, that [ηZ(s)]is Poincaré dual to [Z(s)]M.

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2 Complex vector bundles and Chern classes

This chapter deals with the Chern classes of complex vector bundles. In 2.1 we introduce some notions (such as complex structures and complexication of vector bundles) which allow to pass from real bundles to complex ones and vice versa. In particular, it is proved that all complex vector bundles are canonically oriented. Such property is used in 2.2 to dene by induc-tion the Chern classes of a complex bundle, starting from the Euler class of the underlying real bundle. This approach follows the one given in [27], 14. At the end of 2.2 some useful properties of Chern classes are stated. Subsequently, we introduce the symmetric power bundles which are closely related to the problem of counting lines on generic hypersurfaces in Pn_{. We}

conclude this chapter by talking about holomorphic vector bundles on com-plex manifolds and by recalling, in particular, some basic facts concerning the line bundles OPn(d) on projective spaces.

2.1 Constructions on complex bundles

In order to x the notation, we rewrite Denition 1.1 in the complex case. Denition 2.1. A complex vector bundle ζ of rank r (or complex r-plane bundle) is a vector bundle

π: E → M such that:

1. for each p ∈ M, the ber Ep = π−1(p) is a complex vector space of

dimension r;

2. for each p ∈ M there exists a neighborhood U of p in M such that the corresponding local trivialization has the form

χ: π−1(U ) → U × Cr

and the restriction

χ|Ep: Ep → {p} × C

r

is a complex-linear vector space isomorphism.

As for the real case, if it is possible to choose U equal to the base space M, then ζ is called a trivial bundle, while a complex 1-plane bundle is called a line bundle.

Sometimes we will identify the vector bundle ζ with its total space E. The complex rank of ζ is denoted by rkC(ζ), or rk(ζ) if the eld C is understood.

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Remark 2.2. All the notions given in the previous chapter for real vector bundles can be transported to the complex case by substituting the words real with complex and, when needed, dierentiable with holomorphic. Clearly the base eld is intended to be C instead of R. In this way all the concepts like: bundle maps, sections, operations between vector bundles and transversality, can be dened in a similar way.

Notation. If ζ = π : E → M is a complex vector bundle, then the vector space of global sections of ζ is denoted by Γ(ζ)(M), as in the real case. If M is a complex manifold we write H0_{(M, O(ζ))} (or O(ζ)(M)) for the space of

holomorphic global sections.

Now we describe methods for passing from real vector bundles to complex ones and vice versa. It is possible to construct bundles of complex rank r starting from bundles of real rank 2r, as follows.

Denition 2.3. A complex structure over a real vector bundle ξ of rkR(ξ) =

2r is a continuous map

J: E → E where E is the total space of ξ, such that:

1. for each p ∈ M, J|Ep: Ep → Ep is a R-linear homomorphism,

2. for each v ∈ E, J(J(v)) = −v.

Proposition 2.4. Let (ξ, J) be a real 2r-plane bundle with a complex struc-ture. Then ξ is a complex vector bundle of complex rank r.

Proof. The map J allows to dene a complex vector space structure on each ber Ep ∼= R2r by setting

(x + iy)v = xv + J(yv)

for every x + iy ∈ C. Every ber Ep becomes a complex vector space of

dimension r.

Now we want to show that for every p ∈ M there is a neighborhood with a real local frame of the form

s1, J s1, . . . , sr, J sr.

In this way, s1, . . . , sr form a complex local frame. A local trivialization

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around p gives rise to a continuous local frame σ1, . . . , σ2r. Set s1 = σ1.

At p we have two linearly independent vectors s1(p), Js1(p) and so there is

some j such that s1(p), Js1(p), σj(p) are linearly independent (if not, then

s1(p), Js1(p) form a basis for Ep, i.e., r = 1 and we stop). Set s2 = σj. At p

we have linearly independent vectors s1(p), Js1(p), s2(p), Js2(p). Continuing

in this way, we will have a set of continuous local sections s1, J s1, . . . , sr, J sr

which form a basis s1(p), Js1(p), . . . , sr(p), Jsr(p)of Ep. Thus, the continuous

function det(s1, J s1, . . . , sr, J sr)is non zero at p, hence it must be non zero in

a neighborhood W of p such that p ∈ W ⊆ U. Therefore, s1, J s1, . . . , sr, J sr

is the required local frame in W .

Conversely, by taking a complex bundle ζ of rank r it is possible to obtain a real 2r-plane bundle, forgetting about the complex structure over ζ. In this way every ber Ep ∼= Cr of ζ becomes a real vector space of dimension 2r.

The bundle obtained in this way is called the underlying real vector bundle of ζ and is denoted by ζR.

Lemma 2.5. If ζ is a complex vector bundle, then the real vector bundle ζR

is canonically oriented.

Proof. Let V be a nite dimensional complex vector space. If v1, . . . , vr is a

complex basis for V , then the ordered vectors

v1, iv1, . . . , vr, ivr (2.12)

form a real basis for V seen as a real vector space. We claim that by applying this process to every ber Ep of ζ, then the required orientation for ζR is

obtained. To do this, we show that the orientation induced by basis (2.12) does not depend on the choice of any complex basis. Let

v = {v1, . . . vr} and u = {u1, . . . ur}

be two complex bases for the space V , with change of basis matrix in GL(r, C). There is an embedding GL(r, C) ,→ GLı _{(2r, R), performed componentwise in} this way z= x + iy 7→ x −y y x =Re(z) − Im(z) Im(z) Re(z) . It corresponds to writing a matrix A ∈ GL(r, C) in the basis

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where e1, . . . , eris the canonical basis of Cr. The following diagram commutes GL_{(r, C)} _{−−−→ GL(2r, R)}ı detR −−−→ R∗   ydetC C∗ −−−→ GL(2, R)ı −−−→ RdetR ∗ i.e., the formula

det_Rı(A) = det_Rı(det_CA) = |det_CA|2 (2.13) is valid for every A ∈ GL(r, C). In [32], Theorem 1, there is a proof of this fact for which (2.13) is a particular case. Instead, we show (2.13) in the following way. Let

D=    z1 · · · 0 ... ... ... 0 · · · zr   ∈ GL(r, C)

be a diagonal matrix, where zs = xs+ iys for s = 1, . . . , r. Formula (2.13)

holds for the matrix D because

   z1 · · · 0 ... ... ... 0 · · · zr    ı −−−→        x1 −y1 · · · 0 0 y1 x1 · · · 0 0 ... ... ... ... ... 0 0 · · · xr −yr 0 0 · · · yr xr        det_R −−−→ |z1|2· · · |zr|2   ydetC z = z1· · · zr ı −−−→ Re(z) − Im(z) Im(z) Re(z) det_R −−−→ |z|2

Formula (2.13) holds also for diagonalizable matrices. Indeed, if A ∈ GL(r, C) is such that D = P−1_AP _{for some P ∈ GL(r, C), by Binet Theorem we have}

det_CA= det_C(P−1AP) = det_CD.

Since diagonalizable matrices are dense in GL(r, C), Formula (2.13) follows by continuity.

Thus, it is sucient to show the claim of the lemma for V of complex dimension 1. Take {v} and {u} two bases of V of coordinates a1+ ia2 and

b1+ ib2 respectively. They are related by

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for α = (α1+ iα2) ∈ C∗. By (2.14), the matrix ı(α) =

α₁ −α2

α2 α1

takes v into u, when V is seen as a real vector space of dimension 2. Hence,

det_Rı(α) = det_Rα1 −α2 α2 α1

= (α1)2+ (α2)2 >0

and the orientation induced on V is determined uniquely for every choice of v and u.

Corollary 2.6. For every complex bundle ζ of rank r over the base space M, the Euler class

e(ζR) ∈ H 2r

(M ; Z) is well dened.

Now we introduce an operation that can be performed starting from a complex vector bundle.

Denition 2.7. Let ζ = E → M a complex r-plane bundle. The conjugate bundle ¯ζ = ¯E → M of ζ is the complex vector bundle with underlying real vector bundle

¯ ζ_R= ζR

(in particular ¯E = E) and with complex structure ¯J(v) = −J(v) = −iv, for every v ∈ E. Here J(v) = iv denotes the complex structure of ζ. Equiva-lently, ¯ E = a p∈M ¯ Ep

and a local trivialization of ¯ζ is given by ¯

χ: ¯E|U → U × Cr

which is the composition ¯ E|U χ − → U × ¯_Cr idU×conj −−−−−→ U × Cr;

where ¯Ep is the conjugate vector space of the ber Ep, χ is a local

trivializa-tion of ζ and conj stands for the conjugatrivializa-tion operatrivializa-tion in Cr.

Remark 2.8. Despite ζ and ¯ζ are equal as real vector bundles, they are not isomorphic in general as complex vector bundles. This is a consequence, for example, of Lemma 2.22 on p. 43.

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Example 2.9. The identity map id: E → ¯E is conjugate linear, id(zv) = ¯zid(v)

for every z ∈ C and every v ∈ E. In particular, id(iv) = −i id(v). We denote by ¯v the vector id(v) in ¯E.

We have seen in Proposition 2.4 that starting from a real 2r-plane bundle equipped with a complex structure, it is possible to obtain a complex r-plane bundle. Now we start from a real bundle without a complex structure, introducing a way to obtain a complex vector bundle of the same (complex) rank.

Denition 2.10. Let ξ = E → M be a real r-plane bundle. The complexi-cation of ξ is the complex r-plane bundle

ξ ⊗ C = E ⊗ C → M,

such that the ber over p ∈ M is the complex vector space (E ⊗ C)p = Ep⊗RC.

Remark 2.11. Every element in the complex vector space Ep ⊗RC can be

written uniquely as

v+ iw = v ⊗_R1 + w ⊗_Ri,

where v, w ∈ Ep. With the multiplication by a complex number (x + iy) ∈ C

given by

(x + iy)(v + iw) = (xv − yw) + i(xw + yv), (2.15) we have a linear isomorphism

Ep⊗RC∼= {v ⊗R1 | v ∈ Ep} ⊕ {v ⊗Ri | v ∈ Ep} ∼= Ep⊕ iEp.

It follows that there is a vector bundle isomorphism

(ξ ⊗ C)R∼= ξ ⊕ ξ. (2.16)

By (2.15) and the previous isomorphism (2.16), ξ ⊕ξ has a complex structure J given by

J(v, w) = i(v + iw) = −w + iv = (−w, v). In this way, we have an isomorphism of complex vector bundles

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Lemma 2.12. For any real r-plane bundle ξ = E → M, there is an isomor-phism of complex vector bundles

ξ ⊗ C ∼= ξ ⊗ C. Proof. The map G: E ⊗ C → E ⊗ C dened by

G(v + iw) = v − iw

is a homeomorphism of E ⊗ C onto itself, that is conjugate linear in each ber since

G(i(v + iw)) = G(−w + iv) = −w − iv = −i(v − iw) = −iG(v + iw). Thus, we can perform the composition

E ⊗ C−_{→ E ⊗ C}G −_{→ E ⊗ C}id

obtaining an isomorphism that is complex linear in each ber.

Lemma 2.13. For any complex r-plane bundle ζ = E → M, there is an isomorphism of complex vector bundles

ζ_R_{⊗ C ∼}= ζ ⊕ ¯ζ.

Proof. Denote by F any ber Ep of the bundle ζ, for some p ∈ M. By

Remark 2.11, there is an isomorphism

F_R⊗_R_C∼= (FR⊕ FR, J)

where J is the complex structure J(v, w) = (−w, v).

The map g : F → FR⊕ FR given by g(v) = (v, −iv) is complex linear,

since

g(iv) = (iv, v) = J(v, −iv) = J(g(v)).

Similarly, the map ¯g: F → FR ⊕ FR given by g(v) = (v, iv) is conjugate

linear, since

¯

g(iv) = (iv, −v) = −J(v, iv) = −J(¯g(v)).

In this way, every point (v, w) ∈ (FR⊕ FR, J)can be written uniquely as the

sum g v + iw 2 + ¯g v − iw 2

of an element in g(F ) ∼= F and an element in ¯g(F ) ∼= ¯F. Thus, ([27], p. 177), there are isomorphisms

F_R⊗_R_C∼= (FR⊕ FR, J) ∼= g(F ) ⊕ ¯g(F ) ∼= F ⊕ ¯F .

This holds for each ber of ζ. So, combining these isomorphisms berwise, we get the lemma.

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Lemma 2.14. For any real oriented r-plane bundle ξ = E → M, the iso-morphism (2.16) of Remark 2.11 either preserves or reverses the orientation, according to whether r(r − 1)/2 is even or odd.

Proof. Let F be any ber of the bundle ξ. If v1, . . . , vr is an ordered basis

for F then, according to Lemma 2.5, the vectors

v1, iv1, . . . , vr, ivr (2.17)

give the preferred orientation for F ⊗RC as an R-vector space. By (2.16) we

have an R-linear isomorphism

F ⊗_R_C∼= F ⊕ iF ∼= F ⊕ F.

Thus, the basis v1, . . . , vr for F followed by the basis iv1, . . . , ivr for iF gives

a dierent ordered basis

v1, . . . , vr, iv1, . . . , ivr (2.18)

for F ⊗RC. The permutation that takes (2.18) to (2.17) has sign

(−1)(r−1)+(r−2)+···+1_{= (−1)}r(r−1)₂ _.

2.2 Chern classes

In this section we dene the Chern classes of a complex vector bundle, stating some properties about them. We give an inductive denition of such classes as in [27], 14.

Let ζ = π : E → M be a complex vector bundle of rank r. Suppose that M is a paracompact space, for instance a (Hausdor, second-countable) manifold. We dene (see [27], p. 157) a complex vector bundle

ζ0 = E → E0

of rank r − 1 in this way:

• the base space of ζ0 is E0 = E \ 0ζ(M ),

• the total space E of ζ0 consists of bers

Ev = Ep

Cv for every v ∈ E0 such that π(v) = p ∈ M.

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Lemma 2.15. To any oriented r-plane (real) bundle ξ = E π

−

→ B there is an associated exact sequence

· · · → Hk (M ; Z)−−−→ H∪e(ξ) k+r (M ; Z) π ∗ 0 −→ Hk+r_(E 0; Z) → Hk+1(M ; Z) → · · ·

called the Gysin sequence of ξ with coecients in Z, where the map π0 is

equal to the restriction π|E0.

Proof. We omit the Z coecients and write e, t in place of e(ξ), t(ξ) respec-tively. We start with the cohomology exact sequence

· · · → Hk_{(E, E} 0) j∗ −→ Hk_(E)₋_{→ H}i∗ k_(E 0) δ − → Hk+1_{(E, E} 0) → · · · associated to (E0, ∅) i

,→(E, ∅),→j (E, E0), with connecting morphism δ. We

use the isomorphism

∪t : Hk−r(E) → Hk(E, E0)

of Theorem 1.15, by substituting Hk−r_(E)_{in place of H}k_{(E, E}

0)in the above

sequence. In this way, the latter becomes · · · → Hk−r_(E)₋_{→ H}g k_(E)₋_{→ H}i∗ k_(E 0) δ − → Hk−r+1_{(E) → · · ·} where g(y) = j∗(y ∪ t) = (y ∪ t)|E = y ∪ t|E.

Again, by Remark 1.16, we can substitute Hk_{(M )}_{in place of H}k_(E)_{via the}

isomorphism π∗_{. Recalling that the Euler class e corresponds to t|}

E through

π∗, we obtain the required sequence · · · → Hk−r(M )−→ H∪e k(M ) π ∗ 0 −→ Hk(E0) δ − → Hk−r+1(M ) → · · · where π∗

0 is obtained as the composition i

∗_{◦ π}∗ _{= (π ◦ i)}∗ _{= (π|} E0)

∗ _{= π}∗ 0.

Take the Gysin sequence of the vector bundle ζR dened before

· · · → Hk−2r_{(M )}_{−−−→ H}∪e(ζR) k_{(M )} π∗0

−→ Hk_(E

0) → Hk−2r+1(M ) → · · ·

with integer coecients. If k < 2r − 1 then Hk−2r_{(M )} _{and H}k−2r+1_{(M )} _are

zero. In this case the map π∗

0 is an isomorphism.

Denition 2.16. The Chern classes of the complex vector bundle ζ are cs(ζ) ∈ H2s(M ; Z),

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• Using Corollary 2.6, the top Chern class cr(ζ) (or ctop(ζ)) is dened to

be the Euler class e(ζR).

• For 0 < s < r, dene cs(ζ) = (π0∗) −1_c

s(ζ0).

• For s = 0, set c0(ζ) = 1.

• For s > r, set cs(ζ) = 0.

The total Chern class of ζ is the formal sum

c(ζ) = c0(ζ) + c1(ζ) + · · · + cr(ζ).

Remark 2.17. Since ctop(ζ) = e(ζR)all the properties of the Euler class listed

in Proposition 1.20 also hold for the top Chern class.

Here we list some of the most important properties of the total Chern class with references for the proofs. In what follows, let

ζ = π : E → M and ζ0 = π0: E0 → M0

be two complex vector bundles of (complex) ranks r and r0 _{respectively.}

Lemma 2.18 ([27], Lemma 14.2). If r = r0 _{and (G, g): ζ}0 _{→ ζ} _{is a bundle}

map, then c(ζ0_{) = c(g}∗_{ζ) = g}∗_c(ζ)_.

Remark 2.19. If M = M0_{, then the underlying real vector bundle (ζ ⊕ ζ}0₎ R

of the direct sum ζ ⊕ ζ0 _{is isomorphic (as an oriented bundle) to ζ} R⊕ ζ

0 R.

Indeed, if v1, . . . , vr is a basis for the ber Ep of ζ and w1, . . . , wr0 is a basis

for the ber Ep0 of ζ0, then the canonical orientation induced by

v1, iv1, . . . , vr, ivr

on Ep(as a 2r-real vector space) followed by the canonical orientation induced

by

w1, iw1, . . . , wr0, iw_r0

on Ep0 (as a 2r0-real vector space), yields the canonical orientation

v1, iv1, . . . , vr, ivr, w1, iw1, . . . , wr0, iw_r0

for Ep⊕ Ep0 as a (2r + 2r0)-real vector space. In particular, one has that

ctop(ζ ⊕ ζ0) = e((ζ ⊕ ζ0)R) = e(ζR⊕ ζ 0 R) = e(ζR) ∪ e(ζ 0 R) = ctop(ζ) ∪ ctop(ζ 0 ). Lemma 2.20 ([27], Lemma 14.3). If M = M0 _{and ζ}0 _{is a trivial complex}

Counting Lines on Projective Hypersurfaces via Characteristic Classes