Introduction to Several Complex Variables T. M. Wolniewicz February 5, 1998

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Contents 1 Preliminaries

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2 Complex derivatives 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rules of complex differentiation . . . . . . . . . . . . . . . . .

3 3 3

3 Differential forms 3.1 Real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Operators ∂ and ∂¯ . . . . . . . . . . . . . . . . . . . . . . . .

6 6 7 9

4 Holomorphic functions 4.1 One variable . . . . . . . . . . . 4.2 Several variables . . . . . . . . 4.3 Power series . . . . . . . . . . . 4.4 Cauchy formula in the polydisk 4.5 Holomorphic mappings . . . . .

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5 Harmonic and subharmonic functions 5.1 Harmonic functions . . . . . . . . . . . . . . . 5.2 Subharmonic functions . . . . . . . . . . . . . 5.3 Pluriharmonic and plurisubharmonic functions 5.4 Smooth approximation . . . . . . . . . . . . .

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25 25 26 33 38

6 Some applications of subharmonicity to SCV 41 6.1 The Bergman space . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Hartogs theorem on separate analyticity . . . . . . . . . . . . 42 7 Pseudoconvexity and domains of 7.1 Introduction . . . . . . . . . . . 7.2 Smoothly bounded domains . . 7.3 Geometric convexity . . . . . . 7.4 Pseudoconvexity . . . . . . . . 7.5 Domains of holomorphy . . . . i

holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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46 46 46 49 55 66

8 The 8.1 8.2 8.3

∂¯ problem The problem and its consequences . . . . . . . . . . . . . . . . Solution for compactly supported forms . . . . . . . . . . . . . Elements of H¨ormander’s solution for smooth forms . . . . . .

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72 72 77 80

1

1 Preliminaries

1

Preliminaries

First we introduce some notational conventions. If w, z ∈ Cn , w = (w1 , . . . , wn ), z = (z1 , . . . , zn ) then hw|zi = |z| =

q

X

wj z¯j ,

X

hz|zi = (

|zj |2 )

1/2

.

A multi-index is any n-tuple α = (α1 , . . . , αn ) with αj ∈ N (we include 0 in N). If α is a multi-index then α! = α1 ! . . . αn !, |α| = α1 + · · · + αn , z α = z1α1 . . . znαn . Exercise. (z + w)α =

X

α! β γ z w β+γ=α β!γ!

where β, γ are multi-indices and the sum β +γ is taken in the ordinary vector sense. Exercise. n k X X k! aj = aα α! j=1 |α|=k where a = (a1 , . . . , an ). If r = (r1 , . . . , rn ), rj > 0 then the polydisk centered at w with polyradius r is D n (w, r) = {z ∈ Cn : |zj − wj | < rj , j = 1, . . . , n}. A ball centered at w with radius ρ is B(w, ρ) = {z ∈ Cn : |z − w| < ρ}. If z = (z1 , . . . , zn ) then z ′ = (z1 , . . . , zn−1 ) so that z = (z ′ , zn ). Whenever it is convenient we will identify Cn with R2n . Thus a vector w = (w1 , . . . , wn ) = (a1 + ib1 , . . . , an + ibn ) is the same as (a1 , . . . , an , b1 , . . . , bn )R . To get used to this identification let’s realize how multiplication by i works on this R2n . We have i(a1 , . . . , an , b1 , . . . , bn )R = iw = (iw1 , . . . , iwn ) = (−b1 + ia1 , . . . , −bn + ian ) = (−b1 , . . . , −bn , a1 , . . . , an )R .

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1 PRELIMINARIES

If D, Ω are subsets of Rm then we will write D ⊂⊂ Ω if D ⊂ int Ω and D is compact. Thus for instance D ⊂⊂ Rm means simply that D is bounded. Whenever we say that Ω is a domain we mean that it is open and connected. By ν we will always denote the Lebesgue measure of the underlying space, σ will be reserved for surface measures. Uniform convergence on compact subsets of a given set plays a very important role in function theory of several complex variables and it is usually referred to as normal convergence. We will always use this term in this meaning.

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2 2.1

Complex derivatives Definitions

Let Ω ⊂ Cn be open and f : Ω → C be differentiable in the real sense at a point P ∈ Ω. We define complex derivatives of f at P as 



∂f 1 ∂f ∂f (P ) = (P ) − i (P ) ∂zj 2 ∂xj ∂yj   ∂f 1 ∂f ∂f (P ) = (P ) + i (P ) . ∂ z¯j 2 ∂xj ∂yj

(2.1.1) (2.1.2)

These definitions are extremely useful but they are purely formal and do not carry any particular geometrical meaning. As a consequence of (2.1.1) and (2.1.2) we get ∂f ∂f ∂f (P ) = (P ) + (P ) ∂xj ∂zj ∂ z¯j   ∂f ∂f ∂f (P ) = i (P ) − (P ) . ∂yj ∂zj ∂ z¯j

2.2

Rules of complex differentiation X ∂f ∂gk X ∂f ∂¯ ∂ gk (f ◦ g) = + ∂zj ¯k ∂zj k ∂zk ∂zj k ∂z

and similarly for

∂ . ∂ z¯j

We also have ∂f ∂zj

!

=

∂ f¯ . ∂ z¯j

We can express the action of the real global derivative of a function f on a vector w ∈ Cn as DR f (P )(w) =

n X

n X ∂f ∂f (P )wj + (P )wj ¯j j=1 ∂zj j=1 ∂ z

(2.2.1)

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2 COMPLEX DERIVATIVES

To prove this let w = (aj + ibj )j . Then X

X ∂f ∂f (P )aj + (P )bj ∂xj ∂yj   X ∂f X  ∂f ∂f ∂f = aj + i bj + − ∂zj ∂ z¯j ∂zj ∂ z¯j X ∂f X ∂f = (aj + ibj ) + (aj − ibj ) ∂zj ∂ z¯j X ∂f X ∂f wj = wj + ∂zj ∂ z¯j

DR f (P )(w) =

We define higher order complex derivatives in the obvious way. Of course these derivatives do not depend on the order of differentiation provided that f is smooth enough. Sometimes we will use shorthand notation like ∂ |α|+|β| f ∂ α1 ∂ αn ∂ β1 ∂ βn = · · · f. · · · α ∂z α ∂ z¯β ∂z1 1 ∂znαn ∂ z¯1β1 ∂ z¯nβn Observe also that 1 ∂ ∂ = ∂zj ∂ z¯j 4

∂ ∂ −i ∂xj ∂yj

hence we get that

!

∂ ∂ +i ∂xj ∂yj

!

1 = 4

∂2 ∂2 + ∂x2j ∂yj2

n X

1 ∂2f = ∆f. ¯j 4 j=1 ∂zj ∂ z

!

(2.2.2)

To get a bit more feeling about complex derivatives let us derive the complex form of the Taylor expansion. If Ω ⊂ Rm and f : Ω → Rp is of class C k in Ω then f (x0 + h) = =

k X

1 j DR f (x0 )(hj ) + o(khkk ) j! j=0 k X

1 X |α|! ∂ |α| f (x0 )hα + o(khkk ) α j! α! ∂x j=0 |α|=j

The reason for the factor |α!|/α! is that to each ∂ |α| /∂xα correspond |α!|/α! derivatives which have a ‘different ordering’ of variables.

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2.2 Rules of complex differentiation If we consider the differential operator m X l=1

∂ (x0 )hl ∂xl

(2.2.3)

corresponding to the global derivative evaluated on the vector h then the composition of j of such operators gives m X l=1

∂ (x0 )hl ∂xl

!j

=

|α|! ∂ |α| (x0 )hα α α! ∂x |α|=j X

which is (modulo 1/j!) exactly the j-th term of the Taylor expansion. Now let us pass to Cn ≈ R2n substituting P for x0 and w for h. It directly follows from (2.2.1) that in this case operator (2.2.3) can be written as n X

n X ∂ ∂ (P )wl + (P )wl ∂zl ∂ z¯l l=1 l=1

hence the j-th term of the Taylor expansion will be: n n X ∂ ∂ 1 X (P )wl + (P )wl j! l=1 ∂zl ∂ z ¯ l l=1

!j

=

1 X (|α| + |β|)! ∂ |α|+|β| α β w w . j! |α|+|β|=j α!β! ∂z α ∂ z¯β

Thus we get Taylor’s formula f (P + w) =

1 ∂ |α|+|β| f (P )w αwβ + o(|w|k ). α∂z β α!β! ∂z ¯ |α|+|β|=j X

(2.2.4)

Now let us use Tylor’s formula to derive another useful identity n X

∂2f ∂2f ∂2f (P )wiw j = + ¯j ∂R w 2 ∂R (iw)2 i,j=1 ∂zi ∂ z

(2.2.5)

On the right-hand side of (2.2.5) we have the sum of the second order terms in the Taylor expansion of f (P + w) and f (P + iw) respectively. When we apply (2.2.4) and sum the corresponding terms we get cancellation of terms where |α| = 2 or |β| = 2 and only the terms with |α| = |β| = 1 remain producing the left-hand side of (2.2.5).

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3 3.1

3 DIFFERENTIAL FORMS

Differential forms Real forms

Let Ω ⊂ Rm be open. We say that ω is a p-form in Ω if X

ω(x) =

aj1 ...jp (x) dxj1 ∧ . . . ∧ dxjp .

1≤j1