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have a bijection between the critical points of fb and the п¬Ғbre ПҶвҲ’1 (b).
But we know that if b belongs to a dense subset of Rm , the п¬Ғber ПҶвҲ’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 п¬Ғber is actually п¬Ғnite.
Returning to our initial setup X вҠ‚ Cn we recall that X is algebraic,
hence the п¬Ғber ПҶвҲ’1 (b) is a real algebraic manifold. Therefore, if it is
discrete it must be п¬Ғnite.
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) вҲҲ ПҶвҲ’1 (b) the diп¬Җerential dy ПҶ is
bijective if and only if x вҲҲ X is an isolated non-degenerate critical point
of fb .
Proof. Let us deal with the situation when a real submanifold X вҠ‚ 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 вҲ‚g/вҲ‚xi = 0 for
1 вү¤ i вү¤ k (i.e. T0 X = Rk вҠ• 0). Both of these conditions can be met by
applying an isometry of Rk+1 .
Let ОІ : Rk+1 вҶ’ N be given by
вҲ‚g вҲ‚g
ОІ(x, О») = (x1 , ..., xk , g(x); вҲ’О» (x), ..., вҲ’О» (x), О»),
вҲ‚x1 вҲ‚xk
where x = (x1 , ..., xk ). Then we have
вҲ‚ 2g вҲ‚2g
вҲ‚ вҲ‚g
dОІ( ) = (0, ..., 0, 1, 0, ..., 0, ; вҲ’О» , ..., вҲ’О» , 0),
вҲ‚xj вҲ‚xj вҲ‚xj вҲ‚x1 вҲ‚xj вҲ‚xk
(1 is at the j-th position) and
вҲ‚ вҲ‚g вҲ‚g
dОІ( ) = (0, ..., 0; вҲ’ , ..., вҲ’ , 1).
вҲ‚О» вҲ‚x1 вҲ‚xk
4.11. LEFSCHETZ THEOREM 193

Let us denote О± := ПҶ в—¦ ОІ:
ОІ
Rk+1 вҶ’ X Г— Rk+1
вҶ’ N
О± вҶ“ПҶ +
Rk+1

What is left for us to do is to compute dО±(x, О») and check if it is
invertible. We have
вҲ‚ 2g вҲ‚ 2g
вҲ‚ вҲ‚g
dОІ(0,...,0,О») ( ) = (0, ..., 0, 1, 0, ..., 0, ; вҲ’О» , ..., вҲ’О» , 0),
вҲ‚xj вҲ‚xj вҲ‚xj вҲ‚x1 вҲ‚xj вҲ‚xk
вҲ‚
dОІ(0,...,0,О») (
) = (0, ...0, ..., 1).
вҲ‚О»
Then we have dО± = dПҶ в—¦ dОІ, but since the map ПҶ is given simply by
the addition, we have
вҲ‚
dО±(0,...,0,О») ( )=
вҲ‚xj
вҲ‚ 2g вҲ‚ 2g вҲ‚ 2g
вҲ‚g
(0, ..., 0, 1, 0, ..., 0, ; вҲ’О» , ..., вҲ’О» + Оҙij , .., вҲ’О» , 0),
вҲ‚xj вҲ‚xj вҲ‚x1 вҲ‚xj вҲ‚xi вҲ‚xj вҲ‚xk
вҲ‚
dО±(0,...,0,О») ( ) = (0, ..., 0, ..., 1).
вҲ‚О»
So we have
Id вҲ’ О»H0 (g) 0
dО±(0,...,0,О») = ,
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 вҲ’ О»H0 (g)) = 0.
Since x is a critical point of fb we have b = (0, ..., 0, О») for some О».
Locally we identify X with Rk and thus

f (x) = ||x||2 + (g(x) вҲ’ О»)2

viewed as a function on Rk . We have
вҲ‚ 2f вҲ‚ 2g
= sОҙij вҲ’ 2О» ,
вҲ‚xi вҲ‚xj вҲ‚xi вҲ‚xj
вҲ‚2f
= 2(Id вҲ’ О»H0 (g)).
вҲ‚xi вҲ‚xj
194 CHAPTER 4. COMPLEX ALGEBRAIC VARIETIES

Therefore we proved the assertion of the lemma that non-degeneracy of
вҲ‚2f
x (det вҲ‚xi вҲ‚xj = 0) is equivalent to 0 being not an eigenvalue of H0 (g)
(det(Id вҲ’ О»H0 (g)) = 0) which in turn is equivalent to the bijectivity of
dy ПҶ at any y вҲҲ ПҶвҲ’1 (b).

Now we pass to step 2). Let fb (x) be a Morse function as above.
We shall see that О»(x) вү¤ d for each critical point x of fb . We recall
the notion of harmonic function g on C. By deп¬Ғnition, a real valued
function g : C вҶ’ R is harmonic if
вҲҡ
вҲ‚ 2g вҲ‚ 2g
вҲҶg = вҲ’ 2 вҲ’ 2 = 0, z = x + вҲ’1y.
вҲ‚x вҲ‚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 вҲҲ C then
g(z)ds
вҲ‚D
g(a) = ,
ds
вҲ‚D

where s parametrizes the circle вҲ‚D. A function g(z) : C вҶ’ R is called
subharmonic if for any point a in the domain of the deп¬Ғnition we have
the following modiп¬Ғcation of the above equality:
g(z)ds
вҲ‚D
g(a) вү¤ .
ds
вҲ‚D

It can be shown that for a smooth function g this property is equivalent
to вҲҶg вү¤ 0. As an example of a subharmonic function g(z) one can take
g(z) = |z вҲ’ b|2 for an arbitrary п¬Ғxed b вҲҲ C. In this situation one has
вҲҶg(z) = вҲ’4.
In higher dimensions we have similar notions. Let X be a complex
manifold and let g : X вҶ’ R be a smooth function. We say that g is
pluriharmonic if and only if for any Riemann surface ОЈ and a holomor-
phic map ПҲ : ОЈ вҶ’ Cn the composed map f в—¦ ПҲ : ОЈ вҶ’ R is harmonic.
Similarly one deп¬Ғnes the notion of plurisubharmonic function. We shall
see e.g. that the function fb (z) = ||z вҲ’ 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
ВҜ
вҲ‚ 2f вҲ‚ 2 f вҲ‚w вҲ‚ w вҲ‚w 2 вҲ‚ 2 f
ВҜ
= =| | .
вҲ‚zвҲ‚ zВҜ вҲ‚wвҲ‚ w вҲ‚z вҲ‚ z
ВҜ ВҜ вҲ‚z вҲ‚wвҲ‚ w ВҜ
LEMMA 4.11.3 Let О· : Y вҶ’ X be a holomorphic map and let f be a
pluri(sub)harmonic function on X. Then f в—¦ О· is a pluri(sub)harmonic
function on Y .
Proof. Let ОЈ be a Riemann surface and let ПҶ : ОЈ вҶ’ Y be a holomorphic
map. Then О· в—¦ ПҶ : ОЈ вҶ’ X is a holomorphic map too and thus by
assumption f в—¦ О· в—¦ ПҶ 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 вҶ’ 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,
вҲ‚f вҲ‚h
= h(z) and
вҲ‚z вҲ‚z
вҲ‚ 2f вҲ‚h вҲ‚h вҲ‚h
= | |2 вүҘ 0.
=
вҲ‚zвҲ‚ zВҜ вҲ‚z вҲ‚z вҲ‚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 вҲ’ b||2 is plurisubharmonic on X вҠ‚ Cn , where z = (z1 , ..., zn ) and
b вҲҲ 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) + В· В· В· .
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) + В· В· В·. Moreover,
вҲ‚ 2f вҲ‚ 2 f2
0вү¤ (0) = (0).
вҲ‚zвҲ‚ zВҜ вҲ‚zвҲ‚ zВҜ

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 вҲҡ real-valued
вҲҡ
bilinear form on Cd . There exists v вҲҲ Cd such that B( вҲ’1x, вҲ’1v) =
В±B(x, v) for any x вҲҲ Cd .
Let us п¬ҒxвҲҡ вҲҲ Cd . There exists w вҲҲ Cd such that for any y вҲҲ Cd we
u
have вҲ’B( вҲ’1y, u) вҲҡ B(y, w). Let us deп¬Ғne an invertible R-linear map
=
T : Cd вҶ’ Cd by T ( вҲ’1u) = w. Then we will have for any x, v вҲҲ Cd
вҲҡ вҲҡ
that B( вҲ’1x, T ( вҲ’1v)) = B(x, v). Moreover,
вҲҡ вҲҡ вҲҡ вҲҡ
2
B(x, T (x)) = B( вҲ’1(вҲ’ вҲ’1x), T (T ( вҲ’1(вҲ’ вҲ’1v)))) =
вҲҡ вҲҡ
B(вҲ’ вҲ’1x, T (вҲ’ вҲ’1v)) = B(вҲ’x, вҲ’v) = B(x, v).
Therefore, T 2 = Id has eigenvalues 1 and/or вҲ’1 and there exists w вҲҲ Cd
вҲҡ
such that T w = В±w. Let w = В± вҲ’1v, then
вҲҡ вҲҡ вҲҡ
B(x, v) = B( вҲ’1x, T ( вҲ’1v)) = В±B( вҲ’1x, T (w)) =
вҲҡ вҲҡ вҲҡ
В±B( вҲ’1x, w) = В±B( вҲ’1x, вҲ’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 Г— 2 real matrix:
вҲҡ
ab x
Q(z) = ( x y ) , z = x + вҲ’1y.
cd y

Then we have that the plurisubharmonicity condition

вҲ‚ 2f вҲ‚ 2f вҲ‚ 2f
4 = + 2 вүҘ0
ВҜ вҲ‚x2
вҲ‚zвҲ‚ z вҲ‚y

ab
implies that a + d вүҘ 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) вҲ’ Q(u) вҲ’ Q(w)).
2

Let us have a non-zero v вҲҲ Cd from the above Lemma and consider
the complex line l = C.v spanned by v. Let also lвҠҘ be the orthogonal
вҲҡ
complement to l with respect to B. One easily sees that вҲ’1lвҠҘ =
lвҠҘ .вҲҡ Indeed, if x вҲҲ lвҠҘ and B(x, v) = вҲҡ then by the above Lemma
0
вҲҡ вҲҡ
B( вҲ’1x, вҲ’1v) = 0 as well and hence вҲ’1x вҲҲ lвҠҘ because вҲ’1v вҲҲ l.
Therefore we have the following direct sum with respect to B of complex
vector spaces: Cd = l вҠ• lвҠҘ . By the inductive assumption the index of
Q on l is at most 1 and the index of Q on lвҠҘ is at most d вҲ’ 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 :
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 О»(x) attached for every critical point x вҲҲ X.

Therefore the proof of the Lefschetz theorem is now complete.
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