But we know that if b belongs to a dense subset of Rm , the ¬ber φ’1 (b)

is a discrete set and all the critical points of fb are isolated. What is

left to do is to show that in such a situation the ¬ber is actually ¬nite.

Returning to our initial setup X ‚ Cn we recall that X is algebraic,

hence the ¬ber φ’1 (b) is a real algebraic manifold. Therefore, if it is

discrete it must be ¬nite.

Next, we will make sure that the point b is such that fb (x) is a

Morse function.

LEMMA 4.11.2 For any y = (x, v) ∈ φ’1 (b) the di¬erential dy φ is

bijective if and only if x ∈ X is an isolated non-degenerate critical point

of fb .

Proof. Let us deal with the situation when a real submanifold X ‚ Rk+1

is of dimension k (co-dimension 1). In general, the proof is very similar.

Thus, at least locally X is given by

X = {(x1 , ..., xk , g(x1 , ..., xk ))}.

Without any loss of generality we can assume that g(0, ..., 0) = 0 (i.e.

X passes through the origin of Rk+1 ) as well as that ‚g/‚xi = 0 for

1 ¤ i ¤ k (i.e. T0 X = Rk • 0). Both of these conditions can be met by

applying an isometry of Rk+1 .

Let β : Rk+1 ’ N be given by

‚g ‚g

β(x, ») = (x1 , ..., xk , g(x); ’» (x), ..., ’» (x), »),

‚x1 ‚xk

where x = (x1 , ..., xk ). Then we have

‚ 2g ‚2g

‚ ‚g

dβ( ) = (0, ..., 0, 1, 0, ..., 0, ; ’» , ..., ’» , 0),

‚xj ‚xj ‚xj ‚x1 ‚xj ‚xk

(1 is at the j-th position) and

‚ ‚g ‚g

dβ( ) = (0, ..., 0; ’ , ..., ’ , 1).

‚» ‚x1 ‚xk

4.11. LEFSCHETZ THEOREM 193

Let us denote ± := φ —¦ β:

β

Rk+1 ’ X — Rk+1

’ N

± “φ +

Rk+1

What is left for us to do is to compute d±(x, ») and check if it is

invertible. We have

‚ 2g ‚ 2g

‚ ‚g

dβ(0,...,0,») ( ) = (0, ..., 0, 1, 0, ..., 0, ; ’» , ..., ’» , 0),

‚xj ‚xj ‚xj ‚x1 ‚xj ‚xk

‚

dβ(0,...,0,») (

) = (0, ...0, ..., 1).

‚»

Then we have d± = dφ —¦ dβ, but since the map φ is given simply by

the addition, we have

‚

d±(0,...,0,») ( )=

‚xj

‚ 2g ‚ 2g ‚ 2g

‚g

(0, ..., 0, 1, 0, ..., 0, ; ’» , ..., ’» + δij , .., ’» , 0),

‚xj ‚xj ‚x1 ‚xj ‚xi ‚xj ‚xk

‚

d±(0,...,0,») ( ) = (0, ..., 0, ..., 1).

‚»

So we have

Id ’ »H0 (g) 0

d±(0,...,0,») = ,

0 1

where H0 (g) is the Hessian of the function g at (0, ..., 0). And this

matrix is invertible if and only if det(Id ’ »H0 (g)) = 0.

Since x is a critical point of fb we have b = (0, ..., 0, ») for some ».

Locally we identify X with Rk and thus

f (x) = ||x||2 + (g(x) ’ »)2

viewed as a function on Rk . We have

‚ 2f ‚ 2g

= sδij ’ 2» ,

‚xi ‚xj ‚xi ‚xj

‚2f

= 2(Id ’ »H0 (g)).

‚xi ‚xj

194 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

Therefore we proved the assertion of the lemma that non-degeneracy of

‚2f

x (det ‚xi ‚xj = 0) is equivalent to 0 being not an eigenvalue of H0 (g)

(det(Id ’ »H0 (g)) = 0) which in turn is equivalent to the bijectivity of

dy φ at any y ∈ φ’1 (b).

Now we pass to step 2). Let fb (x) be a Morse function as above.

We shall see that »(x) ¤ d for each critical point x of fb . We recall

the notion of harmonic function g on C. By de¬nition, a real valued

function g : C ’ R is harmonic if

√

‚ 2g ‚ 2g

∆g = ’ 2 ’ 2 = 0, z = x + ’1y.

‚x ‚y

Locally, such a function g can be represented as the real part of a holo-

morphic function. The average value theorem for harmonic functions

tells us that if D is a small disk around a ∈ C then

g(z)ds

‚D

g(a) = ,

ds

‚D

where s parametrizes the circle ‚D. A function g(z) : C ’ R is called

subharmonic if for any point a in the domain of the de¬nition we have

the following modi¬cation of the above equality:

g(z)ds

‚D

g(a) ¤ .

ds

‚D

It can be shown that for a smooth function g this property is equivalent

to ∆g ¤ 0. As an example of a subharmonic function g(z) one can take

g(z) = |z ’ b|2 for an arbitrary ¬xed b ∈ C. In this situation one has

∆g(z) = ’4.

In higher dimensions we have similar notions. Let X be a complex

manifold and let g : X ’ R be a smooth function. We say that g is

pluriharmonic if and only if for any Riemann surface Σ and a holomor-

phic map ψ : Σ ’ Cn the composed map f —¦ ψ : Σ ’ R is harmonic.

Similarly one de¬nes the notion of plurisubharmonic function. We shall

see e.g. that the function fb (z) = ||z ’ b||2 is plurisubharmonic. The

property to be (pluri)(sub)harmonic is intrinsic and does not depend

upon a choice of holomorphic coordinates. For example, let w = h(z)

4.11. LEFSCHETZ THEOREM 195

be another holomorphic coordinate on C (i.e. h(z) is a holomorphic

function). Then w = h(z) does not depend upon z and we have

¯

‚ 2f ‚ 2 f ‚w ‚ w ‚w 2 ‚ 2 f

¯

= =| | .

‚z‚ z¯ ‚w‚ w ‚z ‚ z

¯ ¯ ‚z ‚w‚ w ¯

LEMMA 4.11.3 Let · : Y ’ X be a holomorphic map and let f be a

pluri(sub)harmonic function on X. Then f —¦ · is a pluri(sub)harmonic

function on Y .

Proof. Let Σ be a Riemann surface and let φ : Σ ’ Y be a holomorphic

map. Then · —¦ φ : Σ ’ X is a holomorphic map too and thus by

assumption f —¦ · —¦ φ is a (sub)harmonic function.

The plurisubharmonicity condition is a strong condition and thus it

is very important to know that there are non-trivial examples of such

functions.

LEMMA 4.11.4 Let h : X ’ C be a holomorphic function on a

complex manifold X. Then the function f (x) = ||h(x)||2 is plurisub-

harmonic.

Proof. We can assume that X is a one-dimensional manifold with a

local coordinate z. Then f (z) = |h(z)|2 = h(z)h(z). Therefore,

‚f ‚h

= h(z) and

‚z ‚z

‚ 2f ‚h ‚h ‚h

= | |2 ≥ 0.

=

‚z‚ z¯ ‚z ‚z ‚z

As an immediate corollary we see that if t1 , ..., tn are non-negative

real numbers and h1 , ..., hn are holomorphic functions on X then the

function n ti ||hi (x)||2 is a plurisubharmonic function on X.

i=1

The above discussion allows us to conclude that the function f (z) =

||z ’ b||2 is plurisubharmonic on X ‚ Cn , where z = (z1 , ..., zn ) and

b ∈ Cn .

Let us have a real analytic function f (z) on a neighbourhood U of

the origin in Cn . Then we can expand f as a Taylor power series:

f (z) = f (0) + f1 (z) + f2 (z) + · · · .

196 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

LEMMA 4.11.5 Let 0 be a critical point of f (z) and assume that f (z)

is plurisubharmonic in U then f2 (z) is plurisubharmonic as well.

Proof. Since the origin is a critical point, we have f1 (z) = 0 and

f2 (z) = H0 (f )(z), where H0 (f ) is the Hessian of f . The Hessian H0 (f )

is independent upon a choice of local holomorphic coordinates because

f1 = 0. Thus we can reduce the problem to the one-dimensional case.

Let f (z) be a subharmonic function on a neighbourhood of zero

in C and let z be a local holomorphic coordinate. We have f (z) =

f (0) + f2 (z) + · · ·. Moreover,

‚ 2f ‚ 2 f2

0¤ (0) = (0).

‚z‚ z¯ ‚z‚ z¯

This lemma reduces our problem to questions of linear algebra. In

fact, what is left to prove for the step 2) is to show that the main

assertion of this step is valid for a plurisubharmonic quadratic form

on a vector space Cd . (One has to think of Cd as Tx X and apply our

results.) This will be shown in the next Proposition, but before that

we need

LEMMA 4.11.6 Let B(u, v) be a non-degenerate symmetric √ real-valued

√

bilinear form on Cd . There exists v ∈ Cd such that B( ’1x, ’1v) =

±B(x, v) for any x ∈ Cd .

Let us ¬x√ ∈ Cd . There exists w ∈ Cd such that for any y ∈ Cd we

u

have ’B( ’1y, u) √ B(y, w). Let us de¬ne an invertible R-linear map

=

T : Cd ’ Cd by T ( ’1u) = w. Then we will have for any x, v ∈ Cd

√ √

that B( ’1x, T ( ’1v)) = B(x, v). Moreover,

√ √ √ √

2

B(x, T (x)) = B( ’1(’ ’1x), T (T ( ’1(’ ’1v)))) =

√ √

B(’ ’1x, T (’ ’1v)) = B(’x, ’v) = B(x, v).

Therefore, T 2 = Id has eigenvalues 1 and/or ’1 and there exists w ∈ Cd

√

such that T w = ±w. Let w = ± ’1v, then

√ √ √

B(x, v) = B( ’1x, T ( ’1v)) = ±B( ’1x, T (w)) =

√ √ √

±B( ’1x, w) = ±B( ’1x, ’1v).

4.11. LEFSCHETZ THEOREM 197

PROPOSITION 4.11.7 Let Q(x) be a non-degenerate real-valued

plurisubharmonic quadratic form on Cd , then the index of Q is at most

d.

Proof. We shall do the proof by induction on d. When d = 1 the form

Q is given by a 2 — 2 real matrix:

√

ab x

Q(z) = ( x y ) , z = x + ’1y.

cd y

Then we have that the plurisubharmonicity condition

‚ 2f ‚ 2f ‚ 2f

4 = + 2 ≥0

¯ ‚x2

‚z‚ z ‚y

ab

implies that a + d ≥ 0. Thus, at most one eigenvalue of Q =

cd

is negative.

Now we shall make the inductive step. Consider the symmetric

bilinear form B(u, w) corresponding to Q:

1

B(u, w) = (Q(u + w) ’ Q(u) ’ Q(w)).

2

Let us have a non-zero v ∈ Cd from the above Lemma and consider

the complex line l = C.v spanned by v. Let also l⊥ be the orthogonal

√

complement to l with respect to B. One easily sees that ’1l⊥ =

l⊥ .√ Indeed, if x ∈ l⊥ and B(x, v) = √ then by the above Lemma

0

√ √

B( ’1x, ’1v) = 0 as well and hence ’1x ∈ l⊥ because ’1v ∈ l.

Therefore we have the following direct sum with respect to B of complex

vector spaces: Cd = l • l⊥ . By the inductive assumption the index of

Q on l is at most 1 and the index of Q on l⊥ is at most d ’ 1. We

conclude that the index of Q on Cd is at most d.

Thus we have completed the proof of step 2) in which we clearly saw

the interaction of complex analytic properties and topology of complex

algebraic manifolds. Now for step 3) of our program we quote a theorem

from [42]:

198 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

THEOREM 4.11.8 (Morse, Smale.) Let f be a Morse function on

a manifold X. Then X is homotopy equivalent to a CW complex with

one cell of dimension »(x) attached for every critical point x ∈ X.

Therefore the proof of the Lefschetz theorem is now complete.