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classroom. Sister Mary Francis

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-
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


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
manifold at ¬nitely many points, then the resulting manifold will be
K¨hler as well.
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
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
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.

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

There is an orthogonal decomposition of H 2 (M, C) with respect to Q:
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
follows from the de¬nitions of Q and of primitive cohomology.
We will do some local computations aimed to show that Q is neg-
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
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
ω§ω = ’ dz1 §d¯1 §dz2 §d¯2 = o/2.
z z
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.
Finally, let [β] ∈ Hprim (M ). We can assume that β is a real har-
monic 2-form, which is written as
√ √
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:

(’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

(’1)p+q hp,q (M ).
Eu(M ) =

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.

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
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

¬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.


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