4.1 Blow-up

The notion of blow-up is a corner stone of classical algebraic geometry.

There are many constructions of spectacular, intriguing and perplexing

algebraic varieties which essentially use the blow-up procedure. We will

consider a number of examples of blown-up manifolds, compute their

cohomology and other important invariants. We start with de¬nition

of the blow-up at a point of a complex vector space.

DEFINITION 4.1.1 The blow-up at the origin of V = Cn is the

˜

smooth subvariety V ‚ V — CPn’1 de¬ned by the following set of equa-

tions:

zi Xj = zj Xi , 1 ¤ i, j ¤ n,

where z1 , ..., zn are the coordinates in V = Cn and X1 , ..., Xn are homo-

geneous coordinates in CPn’1 .

˜

The natural projection onto the ¬rst factor de¬nes the map π : V ’ Cn .

The pre-image of any point except for the origin of Cn under π is just

137

138 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

˜

one point in V , though the pre-image of the origin is isomorphic to

CPn’1 . Of course, using local charts, this procedure can be repeated

for any complex analytic manifold M of dimension n. Intuitively, the

˜

blow-up M of M at a point is replacing this point by the projective

˜

space CPn’1 . There is always a birational morphism π : M ’ M with

the described above properties. It is true that if one blows-up a K¨hler

a

manifold at ¬nitely many points, then the resulting manifold will be

K¨hler as well.

a

We shall mostly consider the compact complex manifold Xm , which

by de¬nition is the blow-up of CP2 at m di¬erent points. Of course as

we vary the location of these points we get di¬erent complex manifolds

in general. The cohomology groups are easy to compute:

H 0 (Xm , Z) = H 4 (Xm , Z) = Z,

H 1 (Xm , Z) = H 3 (Xm , Z) = 0, H 2 (Xm , Z) = Zm+1 .

We notice that all the classes in H 2 (Xm , Z) are generated by complex

submanifolds; one can take as generators the complete pre-images un-

der the natural map π : Xm ’ CP2 of a complex line in CP2 and those m

points we have blown-up at. It means that all the classes in H 2 (Xm , C)

are of type (1, 1). Let us call them l0 , ..., lm respectively. The intersec-

tion pairing obviously gives li · lj = 0, i = j, and l0 · l0 = 1. A somewhat

less trivial is the fact that li · li = ’1, 1 ¤ i, which deserves to be

explained.

Let us consider the case of X1 which is the blow-up of CP2 at a point

a. Let π : X1 ’ CP2 is the natural map such that π ’1 (a) = l1 CP1 .

Let l0 be any line matching the above description. We take ∆ ∈ CP2 -

a line that passes through a. It follows that π ’1 (∆) = l 0 ∪ l1 , and ∆

is homologous to π(l0 ). Pulling back the cohomology class of ∆ we get

π — ([∆]) = [l1 ] + [l 0 ].

On the other hand we have [l0 ] = π — [∆] because both classes are pull-

backs of the cohomology class of a line in CP2 . It remains to notice

that

0 = [l1 ][l0 ] = [l1 ][l1 ] + [l1 ][l 0 ] = [l1 ][l1 ] + 1.

So we get the desired [l1 ][l1 ] = ’1.

4.2. SIGNATURE 139

The notion of blow-up helps one to relate CP2 and CP1 — CP1 . Let us

be given a smooth quadric surface Q in CP3 , which can be represented

by the equation X0 X1 = X2 X3 in homogeneous coordinates [X0 : X1 :

X2 : X3 ] on CP3 . There are two families of rulings on Q. The ¬rst family

of lines is given by the system »X2 = µX0 , µX3 = »X1 ; the second

family is given by similar formulae: »X2 = µX1 , µX3 = »X0 . In both

cases (», µ) stands for a point in CP1 . We will try to map birationally

this quadric to CP2 using a stereographic-type projection p from a point

a ∈ Q to a plane Π CP2 in CP3 . In order for this map to be de¬ned

˜

everywhere, we blow-up Q at the point a to get the smooth variety Q.

˜ ˜

If π : Q ’ Q is the usual map, we require now that p : Q ’ Π sends

π ’1 (a) to the line ”at in¬nity” of the projective plane Π. Actually, the

map p is almost an isomorphism except for it sends two lines l1 and l2

from the ¬rst and the second rulings (these lines meet at a on Q) to

two points c1 , and c2 in Π respectively. We correct this abnormality by

blowing-up Π at the points c1 and c2 . Our conclusion is that CP2 being

blown-up at two points is isomorphic to CP1 — CP1 which is blown-up

at one point.

4.2 Signature

Here we shall discuss the signature, which is an integer number as-

signed to every closed oriented real manifold of dimension divisible by

4. We pursue this topic by illuminating the special case of 4-manifolds,

though it can be easily generalized. Let M be a compact, oriented

manifold (with no boundary) of dimension 4. The cup-product de¬nes

a symmetric non-degenerate pairing

H 2 (M, R) — H 2 (M, R) ’ R

and we let I(M ) be the signature of this bilinear form (the di¬erence

between the numbers of positive and negative eigenvalues). In a suitable

basis, this pairing is given by a b2 (M ) — b2 (M ) matrix

1i—i 0j—i

.

0i—j 1j—j

We have

I(M ) = i ’ j.

140 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

PROPOSITION 4.2.1 Let M be a compact K¨hler manifold of (com-

a

plex) dimension 2. Then

I(M ) = 2 ’ h1,1 (M ) + 2h0,2 (M ),

where by de¬nition hi,j (M ) = dim Hi,j (M ).

Proof. The bilinear form de¬ning the signature on H 2 (M, R) gives rise

to a hermitian form Q on H 2 (M, C) by the formula

¯

Q([±], [β]) = ±§β.

M

There is an orthogonal decomposition of H 2 (M, C) with respect to Q:

1,1

H 2 (M, C) = C · [ω] • Hprim (M ) • H 2,0 (M ) • H 0,2 (M ),

where C · [ω] is one-dimensional linear subspace spanned by the class of

the K¨hler form ω. The fact that this decomposition is really orthogonal

a

follows from the de¬nitions of Q and of primitive cohomology.

We will do some local computations aimed to show that Q is neg-

1,1

atively de¬nite on Hprim (M ) and positively de¬nite on all the other

terms. It is certainly enough to √ prove the result. The statement is

’1

punctual, so we may take ω = ’ 2 (dz1 §d¯1 + dz2 §d¯2 ) in local co-

z z

ordinates z1 , z2 . The orientation (as a √local 4-form) is taken to be

o = dx1 §dy1 §dx2 §dy2 , where zk = xk + ’1yk , k = 1, 2.

First, Q is positively de¬ned on C · [ω], since

1

ω§ω = ’ dz1 §d¯1 §dz2 §d¯2 = o/2.

z z

2

Second, we show that Q is positively de¬ned on H 0,2 (M ) and on

H 2,0 (M ). Since one follows from the other, we will only consider

H 2,0 (M ). Let ± be a holomorphic 2-form on M ; locally ± = f dz1 §dz2 .

We have

¯ ¯

±§¯ = f f = ’f f dz1 §d¯1 §dz2 §d¯2 ,

± z z

which is a positive multiple of o.

1,1

Finally, let [β] ∈ Hprim (M ). We can assume that β is a real har-

monic 2-form, which is written as

√ √

¯z

β = ’1f dz1 §d¯1 + ’1gdz2 §d¯2 + hdz1 §d¯2 + hd¯1 §dz2 ,

z z z

4.2. SIGNATURE 141

∞

where f, g ∈ CR (M ). The condition that β de¬nes a primitive coho-

mology class is clearly equivalent to β§ω = 0 on the level of di¬erential

forms. It is a consequence of the fact that ω § β is harmonic as we

proved in Lemma 3.6.9. But

β§ω = (f + g)(dz1 §d¯1 §dz2 §d¯2 )/2.

z z

Therefore, g = ’f and now we easily obtain that Q is negative de¬nite

on H 1,1 (M ), because

¯ ¯

β§β = 2(hh ’ f g)(dz1 §d¯1 §dz2 §d¯2 ) = ’2(hh + f 2 )o.

z z

COROLLARY 4.2.2 (Hodge index theorem.) Let M be a com-

pact K¨hler manifold with the K¨hler form ω of (complex) dimension

a a

2. Then the intersection pairing is negative de¬nite on the orthogonal

complement of [ω] in H 1,1 (M ).

In general one has the following formula for computing the signature

of a compact K¨hler manifold M of an arbitrary dimension:

a

(’1)p hp,q (M ).

I(M ) =

p+q=0(mod 2)

Due to the fact that hp,q = hq,p one can omit the condition that p + q

is an even number and take the above sum for all p, q.

We will sometimes refer to the Euler characteristic of M , which for

any manifold is de¬ned by Eu(M ) = i (’1)i dimR H i (M, R). When

M is compact K¨hler , it is the same as

a

(’1)p+q hp,q (M ).

Eu(M ) =

p,q

EXAMPLES. Now we consider a couple of examples illustrating the

above Proposition. We start with Xm de¬ned as CP2 blown-up at m

points. The right hand side of the formula

I(Xm ) = 2 ’ h1,1 (Xm ) + 2h0,2 (Xm )

is easy to compute. According to our previous knowledge, it is equal to

2 ’ (m + 1) + 2 · 0 = 1 ’ m.

142 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

Now we would like to compute the left hand side by applying another

fact that we did establish before. Namely we recall that the intersection

pairing on H 2 (Xm , Z) is given by the matrix

1 0m—1

.

01—m ’1m—m

Thus the left hand side is equal to 1 ’ m too.

Next, we consider a ¬‚at compact K¨hler manifold M = C2 /Λ, where

a

Λ is a complete lattice. We have already seen that here h0,2 = 1 and

h1,1 = 4, so Proposition gives

I(M ) = 2 ’ 4 + 2 · 1 = 0.

On the other hand side, it follows from the ¬‚atness of M that the

signature is 0. Indeed, there is a well-known formula of Atiyah-Singer

for I(M ) which involves the curvature of M . For a ¬‚at manifold, this

formula gives I(M ) = 0.

4.3 Examples in low dimensions and the

Siegel upper-half space

To start with, let us consider the so-called Kummer surface. Let Λ be

a cocompact lattice in C2 , and let A = C2 /Λ be the quotient, which is

a torus. We also assume that A is a projective manifold, which is not

always the case and depends upon Λ. We consider the involution ι of

A de¬ned by ι(x) = ’x. We notice that ι is a holomorphic map. The

quotient X = A/ι is not a manifold, since it has singularities at ¬xed

points of ι. It is easy to see that there are exactly 16 isolated singular

points corresponding to points of order 2 in A, which is isomorphic to

(R/Z)4 considered as real Lie group. We need to understand what sort

of singularities do we have. For this we shall construct a local model of

ι at a ¬xed point x.

LEMMA 4.3.1 The di¬erential dx ι : Tx A ’ Tx A is equal to ’Id.

Proof. Of course, the di¬erential map can be computed directly, but

here we will give a proof which will work for any involution with isolated

4.3. EXAMPLES AND SIEGEL SPACE 143

¬xed points. We always have

(Tx A)ι = Tx (Aι ).

It is true just because A is smooth and ι is a di¬eomorphism of ¬nite

order. Due to the fact that Tx (Aι ) = 0, we see that the (+1) eigenspace

of dz ι of Tx A is zero. There are only two eigenvalues: (+1) and (’1),

because dx ι2 = Id. It follows that dx ι = ’Id.

Let us pick local holomorphic coordinates (z1 , z2 ) at x such that

ι(z1 ) = ’z1 and ι(z2 ) = ’z2 . We reduced our problem to the under-

standing of the quotient C2 /ι. The ι-invariants in the ring of polynomial

functions in z1 , z2 is

C[z1 , z2 ]ι = aij z i z j .

i+j even

Thus the algebra of polynomial functions on C2 /ι has three generators

2 2

u = z1 , v = z2 , and w = z1 z2 . The only relation between them is

uv = w2 . Thus we got a local model for X = A/ι. It is a cone in C3

de¬ned in coordinates (u, v, w) by uv = w2 with the vertex 0. It shows

that X is really singular at x.