(VII) b1 = 1 and pg = 0 (unde¬ned type)

It is clear that the number of algebraically independent meromor-

phic functions on S carries an important information about the struc-

ture of S. If there exist two such functions on S then S is an algebraic

surface. In fact, if there exists only one such function then S is an

elliptic surface.

We already gave a thorough investigation to K3 surfaces. Another

good point in this study is

186 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

THEOREM 4.9.7 (A. Weil) Any K3 surface is a deformation of a

quartic surface in CP3 .

Another deep result which uses the above classi¬cation is

THEOREM 4.9.8 (Kodaira) A surface is a deformation of an al-

gebraic surface if and only if its ¬rst Betti number is even.

4.10 Cohomology of a smooth projective

quadric

A very interesting type of complex algebraic varieties is given by the

varieties paved by a¬ne spaces. One of the important features of such

varieties is the simple structure of their cohomology which is underlied

by an explicit geometric structure.

DEFINITION 4.10.1 A complex variety Z of dimension d is paved

by a¬ne spaces if there exists a ¬ltration

Z0 ‚ Z1 ‚ · · · ‚ Zd = Z,

where Zj is a closed complex algebraic subvariety (maybe reducible) of

dimension j and Zj \ Zj’1 is a disjoint union of kj copies of Cj .

Right now we shall establish the basic result about the cohomology

of paved varieties.

PROPOSITION 4.10.2 Let us assume that Z is a compact complex

algebraic variety paved as above. Then H 2j (Z, Z) = Zkj and the odd

degree cohomology groups of Z vanish.

Proof. We shall conduct the proof by induction on j. When d = 0 then

the statement follows automatically. Therefore, we shall assume that

the cohomology groups of Zj’1 satisfy the conditions of the proposition.

Consider the exact sequence in cohomology:

· · · ’ Hc (Zj \ Zj’1 , Z) ’ H i (Zj , Z) ’ H i (Zj’1 , Z) ’ · · · .

i

4.10. COHOMOLOGY OF A QUADRIC 187

i

The cohomology group with compact supports Hc (Zj \ Zj’1 , Z) vanish

unless i = 2j due to our assumption that the space Zj \ Zj’1 is a

disjoint union of a¬ne spaces of complex dimension j. It also means

that Hc (Zj \ Zj’1 , Z) = Zkj . From the exact sequence we then deduce

2j

that the cohomology groups of Zj and Zj’1 coincide up to the degree

2j ’ 2. For the rest we invoke the following piece of the same exact

sequence:

· · · ’ Hc (Zj \ Zj’1 , Z) ’ H 2j’1 (Zj , Z) ’ H 2j’1 (Zj’1 , Z) ’

2j’1

Hc (Zj \ Zj’1 , Z) ’ H 2j (Zj , Z) ’ H 2j (Zj’1 , Z) ’ · · · .

2j

Since H 2j’1 (Zj’1 , Z) = H 2j (Zj’1 , Z) = 0 by the dimension considera-

tions, Hc (Zj \ Zj’1 , Z) = 0, and Hc (Zj \ Zj’1 , Z) = Zkj , we conclude

2j’1 2j

that H 2j’1 (Zj , Z) = 0 and H 2j (Zj , Z) = Zkj .

Our main objective in this setion is to compute the cohomology of

a smooth projective quadric X ‚ CPn+1 given as the zero locus of a

homogeneous polynomial Q(z) of degree 2, where z = [z0 : · · · : zn+1 ]

as usual parametrizes the points of CPn+1 . Such a quadric is obviously

of complex dimension n. Let A = (aij ) be the matrix corresponding to

the quadratic form Q(z) meaning that

Q(z) = aij zi zj .

i,j

It is fairly easy to see that the quadric X is smooth if and only if the

determinant of the matrix A is not equal to zero. In such a situation

one is able to make a change of coordinates in such a way that the

equation of the quadric Q will be given by

Q(z) = z0 zn+1 + z1 zn + · · · + zm zm+1 , n = 2m,

2

Q(z) = z0 zn+1 + z1 zn + · · · + zm+1 , n = 2m + 1.

We notice that the quadratic form vanishes identically on the totally

isotropic subspace CPm ‚ CPn+1 given by zm+1 = zm+2 · · · = zn+1 = 0.

The main result about the cohomology of X is then as follows.

188 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

THEOREM 4.10.3 (i) If n = 2m + 1 then

H 2i (X, Z) = Z, 0 ¤ i ¤ n

and all other cohomology groups vanish.

(ii) If n = 2m then

H 2i (X, Z) = Z, 0 ¤ i ¤ n, i = m;

H 2m (X, Z) = Z • Z

and all other cohomology groups vanish.

Proof. Let us denote by Yi the linear subspace CPi ‚ CPn+1 de¬ned

by the vanishing of the coordinates zi+1 , ..., zn+1 . Then we obtain so-

called complete ¬‚ag Y1 ‚ Y2 ‚ · · · ‚ Yn+1 = CPn+1 . We claim that the

quadric X is paved by a¬ne spaces. More precisely, let Zi := X © Yi ;

then we shall see that Zi+1 \Zi is (isomorphic to) either Cai or a disjoint

union of two copies of Cai . Since we have noticed that Ym is a totally

isotropic subspace, then it follows that for i ¤ m we have Zi = Yi = CPi

and thus Zi+1 \ Zi Ci .

Next, let us deal with the case n = 2m + 1. We see right from the

equation for X that

Zm+1 = {[z0 : · · · : zm+1 : 0 : · · · : 0], zm+1 = 0} = Zm .

Therefore, Zm+1 \ Zm is an empty set. Next,

2

Zm+2 = {[z0 : · · · : zm+2 : 0 : · · · : 0], zm+1 + zm zm+2 = 0}.

Since zm+2 = 0 in Zm+2 \ Zm+1 we can let zm+2 = 1 and then the

2

equation of Zm+2 reads as zm = ’zm+1 . Therefore we have m + 1 unre-

lated coordinates on Zm+2 \ Zm+1 which makes it isomorphic to Cm+1 .

Similarly one continues for j ≥ m + 2 to establish the isomorphisms

Cj’1 . We conclude that for an odd n the manifold X is

Zj \ Zj’1

paved by a¬ne spaces Cj , 0 ¤ j ¤ n, one in each dimension, and thus

the theorem follows.

Now we we deal with the case n = 2m. The only di¬erence with

the previous case appears when we consider

Zm+1 = {[z0 : z1 : · · · : zm+1 : 0 : · · · : 0], zm zm+1 = 0}.

4.10. COHOMOLOGY OF A QUADRIC 189

This implies that either zm or zm+1 vanishes. The latter situation cor-

responds to the points of Zm . In the former case zm = 0 we obtain

Zm+1 \ Zm Cm but it is the second time we get the a¬ne space of the

dimension m in the paving. Therefore we have H 2m (X, Z) = Z • Z and

the proof of the theorem is now complete.

The above result is a perfect illustration of the following general

fact.

PROPOSITION 4.10.4 Let (X, ω) be a compact symplectic manifold

(for example, a projective manifold). Then H 2i (X, R) = 0 for 0 ¤ i ¤

n, where 2n stands for the real dimension of X.

Proof. Non-degeneracy of the form ω implies that §n ω is a volume

form and thus

§n ω = vol(X) = 0.

X

We shall show that the class [§i ω] in H 2i (X, R) is non-zero for 0 ¤ i ¤ n.

The form §i ω is clearly closed and thus the class [§i ω] is well de¬ned.

Now, assume that the form §i ω is exact, i.e. §i ω = d± for some smooth

di¬erential (2i ’ 1)-form ± on X. Then the form §n ω would be exact

as well and equal to d(± § (§n’i ω)). But then by the Stokes™ theorem

one has

§n ω = d(± § (§n’i ω)) = ± § (§n’i ω) = 0,

X X ‚X

since X has no boundary. Thus our assumption on the exactness of

§i ω led to a contradiction.

The cohomology class [§i ω] is of type (i, i) and when X is a pro-

jective manifold, it is represented by the (smooth) cycle given by the

intersection of X (embedded in CPN ) with dim(X) ’ i generic hyper-

planes CPN ’1 ‚ CPN .

Returning to our quadric X ‚ CPn+1 given by a homogeneous

quadratic equation Q(z) = 0, it is not hard to ¬nd the Hodge num-

bers of X. When n = 2m + 1 then by the Dirichlet™s principle we must

have hp,p = 1 and hp,q = 0 for 0 ¤ p ¤ n and p = q.

190 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

When n = 2m, then for p = m we have exactly the same con-

clusions. Because of the equalities hm,m ≥ 1, hm’j,m+j = hm+j,m’j ,

and m m’j,m+j

= 2 we are forced to conclude that hm,m = 2

j=’m h

and hm’j,m+j = 0 for j = 0. It is also clear that in this case the

group Hn (X, Z) is generated by the following two algebraic cycles in X:

zm+1 = zm+2 = · · · = zn+1 = 0 and zm = zm+2 = zm+3 = · · · = zn+1 =

0. Thus, the Hodge conjecture is trivial in this situation.

4.11 Lefschetz theorem

In this section we shall prove a basic result about the homology of a¬ne

algebraic manifolds. Let X ‚ Cn be an a¬ne algebraic manifold of

(complex) dimension d. By de¬nition, X is given as the zero locus of a

¬nite number of holomorphic polynomials and at every point x ∈ X the

dimension of the (Zariski) tangent space to X is equal to d (smoothness

condition). It turns out that such a manifold X can have non-trivial

homology groups only up to degree d, i.e. Hi (X, Z) = 0 for i > d. In

fact, we shall see that the following stronger statement is true.

THEOREM 4.11.1 (Lefschetz). If X is an a¬ne algebraic man-

ifold of dimension d then X is homotopy equivalent to a ¬nite CW

complex with all cells having (real) dimension at most d.

This theorem will also imply that the groups Hi (X, Z) are ¬nitely gen-

erated.

It is easy to see that in the case when X = C \ {p1 , p2 , ..., pk }

is the complement to a ¬nite number of points there exists a graph

“ (a continuous map [0, 1] ’ C) which is a deformation retract of

X. Moreover, if pi ™s are distinct then H1 (X, Z) = H1 (“, Z) = Zk and

H2 (X, Z) = H2 (“, Z) = 0. The manifold X is actually a¬ne and alge-

braic since it can be realized as an a¬ne submanifold of C2 given by

one polynomial equation:

(z1 ’ p1 ) · · · (z1 ’ pk )z2 ’ 1 = 0.

We shall conduct our proof of Lefschetz theorem in three major

steps:

4.11. LEFSCHETZ THEOREM 191

1). We show that for b ∈ Cn general enough the function f (x) = ||x’b||2

is a Morse function on X.

2). We establish that for a critical point x ∈ X of f the index »(x) of

x does not exceed d.

3). We use the fact that X is homotopically equivalent to a CW complex

with the number of attached k-cells less or equal to the number of

critical points of a Morse function f of index k.

We shall underline the major results of the Morse theory. An ex-

cellent reference for this subject is a book of Milnor [42]. Let X be

a manifold of (real) dimension n. A Morse function f : X ’ R is a

proper function (the pre-image f ’1 (A) of any compact subset A ∈ R

is compact) which is bounded from below, has isolated critical points

and for each critical point x ∈ X and a (real) local coordinate system

(x1 , ..., xn ) the Hessian Hx (f ) de¬ned by

‚2f

Hx (f ) = ( )|x

‚xi ‚xj

is non-degenerate on Tx X. This property as well as the index »(x) of

a critical point x given by

1

»(x) := (rkH ’ signH)

2

does not depend upon a choice of a local coordinate system. The index

»(x) is the same as the number of the negative eigenvalues of the matrix

Hx (f ).

To prove part 1) we shall forget about the complex structure and

do some real geometry. Let X ‚ Rm be a submanifold and let f (x) =

||x ’ b||2 , where b ∈ Rm . Let us see when the point x is a critical point

of the function f . For u ∈ Tx X we have dx f (u) = 2u · (x ’ b), where

the dot product has sense if we identify Tx Rm with Rm . Therefore dfx

vanishes on Tx X if and only if Tx X ⊥ (x ’ b) or (x ’ b) is a normal

vector at x. Let N ’ X be the normal bundle to X. The dimension

of the total space of N is m and we shall consider N as a submanifold

of X — Rm . Let us introduce a mapping φ : N ’ Rm given by

φ(x, v) = x + v ∈ Rm , x ∈ X, v ∈ Nx .

192 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

The map φ is called the developing map. Now we see that x ∈ X is a

critical point of the function fb (x) = ||x ’ b||2 if and only if we have

(x, v) ∈ N ‚ X — Rm , where v := b ’ x. Given a point b ∈ Rm we