Notes on several complex variables

183 Pages·1997·0.896 MB·English

by  Taylor J.L.

NOTES ON SEVERAL COMPLEX VARIABLES
J. L. Taylor
Department of Mathematics
University of Utah
July 27, 1994
Revised June 9, 1997
Notes from a 1993–94 graduate course
Revised for a 1996-97 graduate course
1
2 J. L. TAYLOR
TABLE OF CONTENTS
1. Holomorphic Functions p. 3
2. Local Theory p. 10
3. A Little Homological Algebra p. 15
4. Local Theory of Varieties p. 24
5. The Nullstellensatz p. 30
6. Dimension p. 46
7. Completion of Local Rings p. 54
8. Sheaves p. 63
9. Sheaf Cohomology p. 71
10. Coherent Algebraic Sheaves p. 87
11. Dolbeault Cohomology p. 99
12. Coherent Analytic Sheaves p. 106
13. Projective Varieties p. 116
14. Algebraic vs. Analytic Sheaves – Serre’s Theorems p. 124
15. Stein Spaces p. 136
16. Fr´echet Sheaves – Cartan’s Theorems p. 149
17. The Borel-Weil-Bott Theorem p. 166
NOTES ON SEVERAL COMPLEX VARIABLES 3
1. Holomorphic Functions
There are a number of possible ways to define what it means for a function defined on a
domain in Cn to be holomorphic. One could simply insist that the function be holomorphic
in each variable separately. Or one could insist the function be continuous (as a function
of several variables) in addition to being holomorphic in each variable separately. The a
priori strongest condition would be to insist that a holomorphic function have a convergent
expansion as a multi-variable power series in a neighborhood of each point of the domain.
The main object of this chapter is to show that these possible definitions are all equivalent.
In what follows, ∆(a,r) will denote the polydisc of radius r = (r ,r ,...,r ) about
1 2 n
a = (a ,a ,...,a ):
1 2 n
∆(a,r) = {z = (z ,z ,...,z ) : |z −a | < r j = 1,2,...,n}
1 2 n j j j
¯
and ∆(a,r) will denote the corresponding closed polydisc. Note that ∆(a,r) is just the
¯
Cartesian product of the open discs ∆(a ,r ) ⊂ C and ∆(a,r) is the Cartesian product of
i i
¯
the closed discs ∆(a ,r ).
i i
1.1 Proposition (Cauchy Integral Formula). If f is a function which is continuous
¯
in a neighborhood U of the closed polydisc ∆(a,r) and holomorphic in each variable z at
i
each point of U, then
(cid:181) (cid:182) (cid:90) (cid:90)
n
1 f(ζ ,...,ζ )dζ ...dζ
1 n 1 n
f(z) = ···
2πi (ζ −z )...(ζ −z )
|ζ −a |=r |ζ −a |=r 1 1 n n
n n n 1 1 1
for any z ∈ ∆(a,r).
ThisfollowsimmediatelyfromrepeatedapplicationoftheonevariableCauchyTheorem.
1.2 Proposition (Osgood’s Lemma). If a function is continuous in an open set U ⊂ Cn
and is holomorphic in each variable at each point of U then at each point a ∈ U there is
a power series of the form
(cid:88)∞
c (z −a )i1...(z −a )in
i ...i 1 1 n n
1 n
i ,...,i =0
1 n
which converges uniformly to f on every compact polydisc centered at a and contained in
U.
¯ ¯
Proof. Let ∆(a,s) be a compact polydisc centered at a and contained in U. Since ∆(a,s)
¯
is compact in U we may choose numbers r > s such that the compact polydisc ∆(a,r)
i i
with polyradius r = (r ,...,r ) is also contained in U. We substitute into the integrand
1 n
of 1.1 the series expansion:
1 (cid:88)∞ (z −a )i1...(z −a )in
1 1 n n
=
(ζ1 −z1)...(ζn −zn) (ζ1 −a1)i1+1...(ζn −an)in+1
i ,...,i =0
1 n
4 J. L. TAYLOR
This series converges uniformly in ζ and z for |z − a | < s < r = |ζ − a |. Thus, the
i i i i i i
series in the integrand of 1.1 can be integrated termwise and the resulting series in z is
¯
uniformly convergent for z ∈ ∆(a,s). The result is a series expansion of the form
(cid:88)∞
f(z) = c (z −a )i1...(z −a )in
i ...i 1 1 n n
1 n
i ,...,i =0
1 n
¯
uniformly convergent on ∆(a,s), with
(cid:181) (cid:182) (cid:90) (cid:90)
n
1 f(ζ ,...,ζ )dζ ...dζ
1 n 1 n
c = ···
i1...in 2πi |ζ −z |=r |ζ −z |=r (ζ1 −a1)i1+1...(ζn −an)in+1
n n n 1 1 1
This completes the proof.
A function f defined on an open set U ⊂ Cn is called holomorphic at a if it has a power
series expansion as in 1.2, convergent to f in some open polydisc centered at a. If f is
holomorphicateachpointofU thenitiscalledholomorophiconU. Thus,Osgood’sLemma
says that a continuous function on U which is holomorphic in each variable separately at
each point of U is holomorphic on U. We will later prove that the continuity hypothesis
is redundant.
1.3 Proposition (Cauchy’s inequalities). If f is a holomorphic function in a neigh-
¯
borhood of the closed polydisc ∆(a,r) and |f| is bounded by M in this polydisc, then for
each multi-index (i ,...,i )
1 n
(cid:175) (cid:175)
(cid:175) ∂i1+...+in (cid:175)
(cid:175) f(a)(cid:175) ≤ M(i !)...(i !)r −i1...r −in
(cid:175)∂z1i1...∂znin (cid:175) 1 n 1 n
¯
Proof. If f is expressed as a power series convergent on ∆(a,r) as in 1.2, then repeated
differentiation yields
∂i1+...+in
f(a) = (i !)...(i !)c
∂z1i1...∂znin 1 n i1...in
where, according to the proof of 1.2,
(cid:181) (cid:182) (cid:90) (cid:90)
n
1 f(ζ ,...,ζ )dζ ...dζ
1 n 1 n
c = ···
i1...in 2πi |ζ −z |=r |ζ −z |=r (ζ1 −a1)i1+1...(ζn −an)in+1
n n n 1 1 1
From the obvious bounds on the integrand of this integral it follows that
|c | ≤ Mr −i1...r −in
i ...i 1 n
1 n
and the Cauchy inequalities follow from this.
In what follows, the notation |U| will be used to denote the volume of a set U ⊂ Cn.
NOTES ON SEVERAL COMPLEX VARIABLES 5
¯
1.4 Theorem (Jensen’s inequality). If f is holomorphic in a neighborhood of ∆ =
¯
∆(a,r) then
(cid:90)
1
log|f(a)| ≤ log|f(z)|dV(z)
|∆|
∆
Proof. We recall the single variable version of Jensen’s inequality:
(cid:90)
1 2π
log|g(a)| ≤ log|g(a+ρeiθ)|dθ
2π
0
and apply it to f considered as a function of only z with the other variables fixed at
1
a ,...,a . This yields:
2 n
(cid:90)
1 2π
log|f(a)| ≤ log|f(a +ρ eiθ1,a ,...,a )|dθ
1 1 2 n 1
2π
0
We now apply Jensen’s inequality to the integrand of this expression, where f is considered
as a function of z with z fixed at a +ρ eiθ1 and the remaining variables fixed at a ,...,a
2 1 a 1 3 n
to obtain:
(cid:181) (cid:182) (cid:90) (cid:90)
1 2 2π 2π
log|f(a)| ≤ log|f(a +ρ eiθ1,a +ρ eiθ2,a ,...,a )|dθ dθ
1 1 2 2 3 n 1 2
2π
0 0
Continuing in this way, we obtain:
(cid:181) (cid:182) (cid:90) (cid:90)
1 n 2π 2π
log|f(a)| ≤ ... log|f(a +ρ eiθ1,a +ρ eiθ2,...,a +ρ eiθn)|dθ ...dθ
1 1 2 2 n n 1 n
2π
0 0
Finally, we multiply both sides of this by ρ ...ρ and integrate with respect to dρ ...dρ
1 n 1 n
over the set {ρ ≤ r ; i = 1,...,n} to obtain the inequality of the theorem.
i i
In the next lemma we will write a point z ∈ Cn as z = (z(cid:48),z ) with z(cid:48) ∈ Cn−1. Similarly
n
we will write a polyradius r as r = (r(cid:48),r ) so that a polydisc ∆(a,r) can be written as
n
∆(a(cid:48),r(cid:48))×∆(a ,r ).
n n
1.5 Lemma (Hartogs’ lemma). Let f be holomorphic in ∆(0,r) and let
(cid:88)
f(z) = f (z(cid:48))zk
k n
k
be the power series expansion of f in the variable z , where the f are holomorphic in
n k
∆(0,r(cid:48)). If there is a number c > r such that this series converges in ∆¯(0,c) for each
n
z(cid:48) ∈ ∆(0,r(cid:48)) then it converges uniformly on any compact subset of ∆(0,r(cid:48))×∆(0,c). Thus,
f extends to be holomorphic in this larger polydisc.
Proof.. Choose any point a(cid:48) ∈ ∆(0,r(cid:48)), choose a closed polydisc ∆¯(a(cid:48),s(cid:48)) ⊂ ∆(0,r(cid:48)) and
choose some positive b < r . Then we may choose an upper bound M > 1 for f on the
n
6 J. L. TAYLOR
polydisc ∆¯(a(cid:48),s(cid:48))×∆¯(0,b). It follows from Cauchy’s inequalities that |f (z(cid:48))| ≤ Mb−k for
k
all z(cid:48) ∈ ∆¯(a(cid:48),s(cid:48)). Hence,
k−1log|f (z(cid:48))| ≤ k−1logM −logb ≤ logM −logb = M
k 0
for all z(cid:48) ∈ ∆¯(a(cid:48),s(cid:48)) and all k. By the convergence of the above series at (z(cid:48),c) we also
have that |f (z(cid:48))|ck converges to zero for each z(cid:48) ∈ ∆¯(a(cid:48),s(cid:48)) and hence:
k
limsupk−1log|f (z(cid:48))| ≤ −logc
k
k
for all z(cid:48) ∈ ∆¯(a(cid:48),s(cid:48)).
We have that the functions k−1log|f (z(cid:48))| are uniformly bounded and measurable in
k
∆¯(a(cid:48),s(cid:48)) and by Fatou’s lemma
(cid:90) (cid:90)
limsup k−1log|f (z(cid:48))|dV(z(cid:48)) ≤ limsupk−1log|f (z(cid:48))|dV(z(cid:48))
k k
k ∆(a(cid:48),s(cid:48)) ∆(a(cid:48),s(cid:48)) k
≤ −|∆(a(cid:48),s(cid:48))|logc
By replacing c by a slightly smaller number if neccessary we may conclude that there is a
k0 such that (cid:90)
k−1log|f (z(cid:48))|dV(z(cid:48)) ≤ −|∆(a(cid:48),s(cid:48))|logc
k
∆(a(cid:48),s(cid:48))
for all k ≥ k . Now by shrinking s(cid:48) to a slightly smaller multiradius t(cid:48) and choosing (cid:178) small
0
enough we can arrange that
∆(a(cid:48),t(cid:48)) ⊂ ∆(w(cid:48),t(cid:48) +(cid:178)) ⊂ ∆(a(cid:48),s(cid:48))
for all w(cid:48) ∈ ∆(a(cid:48),(cid:178)), from which we conclude that
(cid:90)
k−1log|f (z(cid:48))|dV(z(cid:48)) ≤ −|∆(a(cid:48),t(cid:48))|logc+M |∆(w(cid:48),t(cid:48) +(cid:178))−∆(w(cid:48),t(cid:48))|
k 0
∆(w(cid:48),t(cid:48)+(cid:178))
Now if we choose a c slightly smaller than c we may choose (cid:178) small enough that the right
0
hand side of this inequality is less than −|∆(w(cid:48),t(cid:48) +(cid:178))|logc . This yields
0
(cid:90)
k−1log|f (z(cid:48))|dV(z(cid:48)) ≤ −|∆(w(cid:48),t(cid:48) +(cid:178))|logc
k 0
∆(w(cid:48),t(cid:48)+(cid:178))
and this, in turn, through Jensen’s inequality, yields
k−1log|f (w(cid:48))| ≤ −logc
k 0
or
|f (w(cid:48))|ck ≤ 1
k 0
for all k ≥ k and w(cid:48) ∈ ∆(a(cid:48),(cid:178)). This implies that our series is uniformly convergent
0
in ∆(a(cid:48),(cid:178)) × ∆(0,c ) and since a(cid:48) is arbitrary in ∆(0,r(cid:48)) and c is an arbitrary positive
0 0
number less than c, we have that our power series serves to extend f to a function which
is continuous on ∆(0,r(cid:48))×∆(0,c) and certainly holomorphic in each variable. It follows
from Osgood’s lemma that this extension of f is, in fact, holomorphic.
NOTES ON SEVERAL COMPLEX VARIABLES 7
1.6 Theorem (Hartog’s theorem). If a complex-valued function is holomorphic in each
variable separately in a domain U ⊂ Cn then it is holomorphic in U.
Proof. We prove this by induction on the dimension n. The theorem is trivial in one
dimension. Thus, we suppose that n > 1 and that the theorem is true for dimension n−1.
¯
Let a be any point in U and let ∆(a,r) be a closed polydisc contained in U. As in the
previous lemma, we write z = (z(cid:48),z ) for points of Cn and write ∆¯(a,r) = ∆¯(a(cid:48),r(cid:48)) ×
n
¯
∆(a ,r ).
n n
Consider the sets
X = {z ∈ ∆¯(a ,r /2) : |f(z(cid:48),z )| ≤ k ∀z(cid:48) ∈ ∆¯(a(cid:48),r(cid:48))}
k n n n n
For each k this set is closed since f(z(cid:48),z ) is continuous in z for each fixed z(cid:48). On the
n n
other hand, since f(z(cid:48),z ) is also continuous in z(cid:48) and, hence bounded on ∆¯(a(cid:48),r(cid:48)) for each
n (cid:83)
¯
z , we have that ∆(a ,r /2) ⊂ X . It follows from the Baire category theorem that
n n n k k
for some k the set X contains a neighborhood ∆(b ,δ) of some point b ∈ ∆(a ,r /2).
k n n n n
We now know that f is separately holomorphic and uniformly bounded in the polydisc
∆(a(cid:48),r(cid:48))×∆¯(b ,δ). It follows from Cauchy’s inequalities in each variable separately that
n
the first order complex partial derivatives are also uniformly bounded on compact subsets
of this polydisc. This, together with the Cauchy-Riemann equations implies that all first
order partial derivatives are bounded on compacta in this polydisc and this implies uniform
continuity on compacta by the mean value theorem. We conclude from Osgood’s theorem
that f is holomorphic on ∆(a(cid:48),r(cid:48)) × ∆(b ,δ) and, in fact, has a power series expansion
n
about (a(cid:48),b ) which converges uniformly on compact subsets of this polydisc.
n
Now choose s > r /2 so that ∆(b ,s ) ⊂ ∆(a ,r ). Then f(z(cid:48),z ) is holomorphic in
n n n n n n n
z on ∆(b ,s ) for each z(cid:48) ∈ ∆(a(cid:48),r(cid:48)) and, hence, its power series expansion about (a(cid:48),b )
n n n n
converges as a power series in z on ∆(b ,s ) for each fixed point z(cid:48) ∈ ∆(a(cid:48),r(cid:48)). It follows
n n n
from Hartog’s lemma that f is actually holomorphic on all of ∆(a(cid:48),r(cid:48))×∆(b ,s ). Since
n n
a is in this set and a was an arbitrary element of U, the proof is complete.
We introduce the first order partial differential operators
(cid:181) (cid:182) (cid:181) (cid:182)
∂ 1 ∂ ∂ ∂ 1 ∂ ∂
= −i = +i
∂z 2 ∂x ∂y ∂z¯ 2 ∂x ∂y
j j j j j j
Clearly, the Cauchy-Riemann equations mean that a function is holomorphic in each
variable separately in a domain U if and only if its first order partial derivatives exist in U
and ∂ f = 0 in U for each j. We may write these equations in a more succinct form by
∂z¯
j
using differential forms. We choose as a basis for the complex one forms in Cn the forms
dz = dx +idy , dz¯ = dx −idy , j = 1,...,n
j j j j j j
¯
then the differential df of a function f decomposes as ∂f +∂f, where
(cid:88) (cid:88)
∂f ∂f
¯
∂f = dz , ∂f = dz¯
j j
∂z ∂z¯
j j
j j
8 J. L. TAYLOR
¯
and the Cauchy-Riemann equations become simply ∂f = 0. Thus, a function defined
in a domain U is holomorphic in U provided its first order partial derivatives exist and
∂¯f = 0 in U. A much stronger resultis true: A distribution ψ defined in a domain in Cn
¯
which satisfies the Cauchy-Riemann equations ∂ψ = 0 in the distribution sense is actually
a holomorphic function. We will not prove this here, but it follows from appropriate
regularity theorems for elliptic PDE’s.
If U is an open subset of Cn then the space of all holomorphic functions on U will be
denoted by H(U). This is obviously a complex algebra under the operations of pointwise
addition and multiplication of functions. It also has a natural topology - the topology of
uniform convergence on compact subsets of U. We end this section by proving a couple of
important theorems about this topology.
The topology of uniform convergence on compacta is defined as follows. For each com-
pact set K ⊂ U we define a seminorm ||·|| on H(U) by
K
||f|| = sup|f(z)|
K
z∈K
Then a neighborhood basis at g ∈ H(U) consists of all sets of the form
{f ∈ H(U) : ||f −g|| < (cid:178)}
K
for K a compact subset of U and (cid:178) > 0. If {K } is an increasing sequence of compact
n
subsets of U with the property that each compact subset of U is contained in some K ,
n
then a basis for this topology is also obtained using just the sets of the above form with
K one of the K(cid:48)s and (cid:178) one of the numbers m−1 for m a positive interger. Thus, there
n
is a countable basis for the topology at each point and, in fact, H(U) is a metric space
in this topology. Clearly, a sequence converges in this topology if and only if it converges
uniformly on each compact subset of U. A nice application of Osgoods’s Lemma is the
proof that H(U) is complete in this topology:
1.7 Theorem. If {f } is a sequence of holomorphic functions on U which is uniformly
n
convergent on each compact subset of U, then the limit is also holomorphic on U.
Proof. For holomorphic functions of one variable this is a standard result. Its proof is a
simple application of Morera’s Theorem. In the several variable case we simply apply this
result in each variable separately (with the other variables fixed) to conclude that the limit
of a sequence of holomorphic functions, converging uniformly on compacta, is holomorphic
in each variable separately. Such a limit is also obviously continuous and so is holomorphic
by Osgood’s Lemma (we could appeal to Hartog’s Theorem but we really only need the
much weaker Osgoods Lemma).
A topological vector space with a topology defined by a sequence of seminorms, as
above, and which is complete in this topology is called a Frechet space. Thus, H(U) is a
Frechet space. It is, in fact, a Montel space. The content of this statement is that it is a
Frechet space with the following additional property:
NOTES ON SEVERAL COMPLEX VARIABLES 9
1.8 Theorem. Every closed bounded subset of H(U) is compact, where A ⊂ H(U) is
bounded if {||f|| ;f ∈ A} is bounded for each compact subset K of U.
K
Proof. Since H(U) is a metric space we need only prove that every bounded sequence in
H(U) has a convergent subsequence. Thus, let {f } be a bounded sequence in H(U). By
n
theprevioustheorem,weneedonlyshowthatithasasubsequencethatconvergesuniformly
on compacta to a continuous function - the limit will then automatically be holomorphic
as well. By the Ascoli-Arzela Theorem, a bounded sequence of continuous functions on a
compact set has a uniformly convergent subsequence if it is equicontinuous. It follows from
theCauchyestimatesthat{f }hasuniformlyboundedpartialderivativesoneachcompact
n
set. This, together with the mean value theorem implies that {f } is equicontinuous on
n
each compact set and, hence, has a uniformly convergent subsequence on each compact
set. We choose a sequence {K } of compact subsets of U with the property that each
m
compact subset of U is contained in some K . We then choose inductively a sequence
m
of subsequences {f } of {f } with the property that {f } is a subsequence of
n i n n i
m,i m+1,i
{f } and {f } is uniformly convergent on K for each m. The diagonal of the
n i n i m
m,i m,i
resulting array of functions converges uniformly on each K and, hence, on each compact
m
subset of U. This completes the proof.
If U is a domain in Cn and F : U → Cm is a map, then F is called holomorphic if each
of its coordinate functions is holomorphic.
1. Problems
1. Prove that the composition of two holomorphic mappings is holomorphic.
2. Formulate and prove the identity theorem for holomorphic functions of several variables.
3. Formulate and prove the maximum modulus theorem for holomorphic functions of sev-
eral variables.
4. Formulate and prove Schwarz’s lemma for holomorphic functions of several variables.
5. Provethatafunctionwhichisholomorphicinaconnectedneighborhoodoftheboundary
of a polydisc in Cn, n > 1, has a unique holomorphic extension through the interior
of the polydisc.
6. Prove that if a holomorphic function on a domain (a connected open set) is not iden-
tically zero, then the set where it vanishes has 2n-dimensional Lebesgue measure zero
(hint: use Jensen’s inequality).
10 J. L. TAYLOR
2. Local theory
Let X be a topological space and x a point of X. If f and g are functions defined
in neighborgoods U and V of x and if f(y) = g(y) for all y in some third neighborhood
W ⊂ U ∩ V of x, then we say that f and g are equivalent at x. The equivalence class
consisting of all functions equivalent to f at x is called the germ of f at x.
The set of germs of complex valued functions at x is clearly an algebra over the complex
field with the algebra operations defined in the obvious way. In fact, this algebra can be
described as the inductive limit limF(U) where F(U) is the algebra of complex valued
→
functions on U and the limit is taken over the directed set consisting of neighborhoods of
x. The germs of continuous functions at x obviously form a subalgebra of the germs of all
complex valued functions at x and, in the case where X = Cn, the germs of holomorphic
functions at x form a subalgebra of the germs of C∞ functions at x which, in turn, form a
subalgebra of the germs of continuous functions at X.
We shall denote the algebra of holomorphic functions in a neighborhood U ⊂ Cn by
H(U) and the algebra of germs of holomorphic functions at z ∈ Cn by H or by H in case
z n z
it is important to stress the dimension n. We have that H = limH(U) where the limit is
z
→
taken over a system of neighborhoods of z. We also have as an immediate consequence of
the definition of holomorphic function that:
2.1 Proposition. The algebra H may be described as the algebra C{z ,...,z } of con-
n 0 1 n
vergent power series in n variables.
Another important algebra of germs of functions is the algebra O of germs of regular
n z
functions at z ∈ Cn. Here we give Cn the Zariski topology. This is the topology in which
the closed sets are exactly the algebraic subvarieties of Cn. By an algebraic subvariety of
Cn we mean a subset which is the set of common zeroes of some set of complex polynomials
in C[z ,...,z ]. A regular function on a Zariski open set U is a rational function with a
1 n
denominator which does not vanish on U. The algebra of regular functions on a Zariski
open set U will be denoted O(U). The algebra of germs of regular functions at z ∈ Cn is
then O = limO(U). One easily sees that:
n z
→
2.2 Proposition. The algebra O is just the ring of fractions of the algebra C[z ,...,z ]
n z 1 n
with respect to the multiplicative set consisting of polynomials which do not vanish at z.
Byalocalringwewillmeanaringwithauniquemaximalideal. Itisatrivialobservation
that:
2.3 Proposition. The algebras O and H are local rings and in each case the maximal
n z n z
ideal consists of the elements which vanish at z.
The algebras O and H are, in fact, Noetherian rings (every ideal is finitely gener-
n z n z
ated). For O this is a well known and elementary fact from commutative algebra. We
n z
will give the proof here because the main ingredient (Hilbert’s basis theorem) will also be
needed in the proof that H is Noetherian.
n z
We will use the elementary fact that if M is a finitely generated module over a Noe-
therian ring A then every submodule and every quotient module of M is also finitely
generated.