Table Of ContentContents
I. Remarkable Facts of Complex Analysis
A. G. Vitushkin
1
, I
II. The Method of Integral Representations in Complex Analysis 9
G. M. Khenkin
li
19
III. Complex Analytic Sets
E. M. Chirka
117’ . .
IV. Holomorpbic Mappings and the Geometry of Hypersurfaces
A. G. Vitushkin
159
I
V. General Tbqory of Multidimensional I&&dues
P. Dolbeault ”
215
c Author’kdex -
243 (
Subjedt Index .
246
t
I. Facts jof Complex Analysis 1 ,
1’1 ,
A.G. Vitushkin
. :
( ,
Translated from. the Russian
by-PM. Gauthier
‘
/’
Contents
I
Introduction . . . . . : . .‘. . . ; . . . . . . . . . . . ..a.**. . . . . . . . ..i.*.... ‘1.
The Continuation Phenomenon . . . . . .. . . . i , . . . . . . . . . . . . . ‘. . . . 1,
Domains of Holomorphy . . . ‘. . . . ,‘. . . . , . . . . . . .t..;......*.. 3
Hoiomorphic Mappings., Classification Problems. . . , . . . . . . . . , . . 5
Integral Representations of Fun&ions . . . . -.’ . . . ...*...\ . . . . . . L. ‘6
Approximation of Functions. . :. . . . . . . . . , . . ; . . . . . . . . . . . . . .I 8
Isolating the Non-Holomoiphic Part of a Fynction. . , , , . . . . . . . . 10 ’
Construc‘tion of Functions with Given Zeros :, ‘. . . . . . . . . . . . . . . . 1’2
Stein ‘Manifolds. . . . . . . . . . . . . . . . . . . . , . . . . . . .* . . . .,:... :.. . * 14
Deformations of Complex Structure.. , . . . . . . . . . . . , . . . , . . . . . . . 16
?
/
b-
lrrirodubtion . ‘*
The present article gives a short survey of results in contemporary complex
analysis and its applications. The material presented is conccnt’rateda round
several pivotal facts whoseu nderstanding enables one to have a general view of
this area of analysis.
.*.
,.
§ 1. The Continyatik Phenqm@xa . , : ,
“.
The most impressive f&t from cmiplex analysis is the phenomenon of the ’
continuation of. functions (Hat-togs; E&6; Poinkr& 1907) We elucidate its
,2 A.G. Vitushkin I. Remarkable Facts of Compkx Adydd 3
I -.
an example. If a function f is defined and holomorp iI ic on the An uqohed pml~+m. Is a set in C” consisting of a finite number of pairwe,
l3 in n-dimensional complex spaceC ”(n 1 2), then it turns out disjoint b;tlls polynomially convex? If the number of balls is at, most 3, then the!
continued to a function holomorphic on the yhole ball B.: answer is boqitive; their union is polynomially convex (FaRin, /964).
Analogousliifor an arbitrary bounded dofniin whose complement is connected, .Another ’ varlint of the continuatiqn phenomenon 4 the theorem d
any functiob olomorphic on the boundaqy of such a domain]dmits a holo-! Bogolyubo;;,lnicknamedlnicknatmheed e dge-of-thi-wedge theorem (S,Ni Berqstein, 14%
morphic cantf puation to the domain-itself. Let us emphasizet hatthis holds only N.N, Bogoly+ov, I956; . . . , , V.V. Zharinov, 170). Let Cf be ti acute convy
for n z 2. Inkhe one $me,nsional case, this phenomenon clearly does not cone in Wt cogsisting of rays e,tianating fropl the origin. Let ,C- be the
,occur. Ind&$ for each set’ E c @’ atid each pohii z,i@\E,~.the function symmetric io C,’ with respect to the origin. Let Q be a dosp R”, and D)
l/(z -i$) is%opmorphic on E but cannot be holomotphically continued to the D- two wedge& i.e. domains in 4=” of the type’ i /
point zo. ‘i, . . - D+ = {~+tzC”: Rezen, Im zEC+} / ’
This$sco\;e 1 y TIlrl&d the beginning o/f-the systematics tudy of functions of
and
-
s&era1 complF$ v&rlables. Two fundamental notions,‘orig+ting in connection b- ‘= {zEC”:Rez~R, ImzcC-).
with this pro&y of holomorljhic fun&ions,, are “envelope bf holomorphy” and
‘Suppose f is a function ho!omorphic on D+ u,D- and suppose the functions
“domairi :of .h?$omorphy”. Let D be k domain or a comp& set in C”. The
envelo& I$ hokk~~7,+~d of the set D is the laqgest set to which all function& flo+ arid &- have boundary values which agree in the sense of distributions
holomorphic OIJ D extend’ holomorphically. The envelope of holomorphy of a along the edge of these cones, i.e. on the set Do = {z E C”: Re z E Q, Im z = 0).
Then, f has a holomorphic extension to some neighbourhood of the set Do. The
domain in c is a domain which in general “cannot fit” into C”, but rather is
theorem on C&onvex hull (V.S. Vladimirov, 1961) ‘ves an estimate on the size
a multi-sheqed domain over C” (Thullen, 1932). A domain D c C” iS called a
of this’ neighbourhopd. For example, if Q = U?“,r’ h en (D+ u Do u D-)-= Q=*
domain ofholomorphy if d =‘D, i.e. if there exists a holomorihic function on D /”
(Bochner, 1937). ’
which cannot be continued to any larger domain. Domains of holomorphy are
The theorem df Bogolyubov has been used to establish several relations in
also sometimes called holoyorphically conuex domains.
axiomatic quantum field theory. This theorem’ also laid the foundations of the
The theorem on discs( Hartogs, 1909)g ives an idea helpful idcqnstructing the
theory of hyperftmctions (Sato, .1959; Martineau, 1964; . . . , V.V. Napalkov,
envelope of holomorphy of a domain: if,a sequenceo f aqalptic discs,l ying in the
1974). For-&ore details, see artiicl~ II, III and volume 8,.article IV.
domain D, converges towards a disc whose boundary lies in D, then this entire
limit disc lies in the envelope of hdlomorphy of 4. An analytic disc is the /
/”
biholomorphic image of a closed disc. The technique of constructioc pf the
envelope of holomorphjl of a dompact set and in particular of a surfac&rePeso n
92. Demains of Holoeg@Y
. ” conglomeration of “attached” discs whose boundaries lie bn,the given surface
(Bishop, 1965).
Closely related tgthe &ion of envelope of holomorphy is the notion of hull
Domains of holomorphy are of interest because in suqh domains oqe can
with respect to son@ cl&s or other of functions, for example, the polynomial
solve traditional problems of a&y&. In certain of these domains holomorphic
hull, the rational,hull, etc. The polynomial hull of a set D c C” is the set of all
functions have integral representations ‘and admit approximation by poly-
ZEC” for which the following condition holds: for each polynomial P(c),
“. nomials. In domains of holomorphy the Cauchy-Riemann equations are solv-
able; it turns out to be possible to interpolate functions; the problem of division
P@N s ;s IJWI. ’ .
is solvable; etc.
: . Every, smooth curve is ‘holomorphica~ly coma, i.e. its envelope of ho]& Two of the simplest types of domains of holomorphy are polynomial poly-
morphy coincides with the curve itself. The polynomial hull of a curve is in hedra and strictly pseudoconvex domains. A polynomicrl polytion is a domain
gei~ral non-trivial. For example, if a smooth curve is closed and without self- givenbyasystemofthetypeIp’(z)l<l, jT1,2,...,,whereeachP’(z)isa .
/’
intemcetions, then its polynomial hull is either trivial or it is a one-dimensional pdynomiid in z. Polynomial polyhedra were introduced by Weil(l932) and are
complex analytic sebwhqst boundary coincides with the given curve (Wermer, also calkd Weil polyhedra. A domain is &led strictly pseuducorwex if in the
195s; Bishop, 1962). We recall that a set in C” is called analytic provided thit id neighbourhood of each of its boundary points the domain is strictly coayex for a
vicinity of each O[ its p&s it is defined by a finite bystem of equations suitable choice of cooidinates. Suppose the hypersurface bounding a domain is
l tk
* {/r(c) - O}, ~hwe Ifif. are holomorphiti functions.. given by an equation p(z, 5) = 0. If in each point of the h-u&ice the Levi
‘\
4 A.G. Vitushkin 1 , I. Remarkable Facts of Complex Analysis 5
I
form of the hypersurface is positive definite, then the domain in q$stion is. that on accobnt of the Weierstrass preparation Theorem (188S), the local
problem of d&&ion is always solvable. .
strictly pseudoconvex( E. Levi, 1910).T he Levi form is the for& & & dzi. & One can define the notion of holomorphic convexity in terms of plurisub-
restricted to the compIex tangent space to the hypersurface at ‘thepomt zo. harmonic functions. A function is called ‘plurisubharmonic if its restriction to
The solution of various forms of the ‘problem of Levi concern g the holo- each complex lind; is a subharmonic function. A domain D is a domain of
morphic convexity of strictly pseudocqnvex domains remain the central holomorphy if an only if. the function -In&) is plurisubharmonic on D,
problem of complex analysisf or several decades.O ka (1942) shoi wed that each where p(z) is the d&i rice from the point z to.the boundary of D (Lelong, 1945).
strictly pseudoconvex domain is holomorphically convex and conversely ‘each For further details see article II and Volume 8, article II.
domain of holomorphy can be exhausted from the interior by domains of this
type. Polynomial polyhedra are easily seen to be polynomially convex and
consequently holomorphically convex. $3. Hdomorphk Mappings. Classification Problems
Boundary points of a domain of holomorphy are not equivalent. A par-
ticularly important role is played by that part of the boundary which is called
the distinguished boundary or the Shilov boundary. The Shiloo bound&y of a By the Riemann Mapping Theorem, in C’ a-nyt wo proper simply-connected
bounded domam is the smallestc losed subset S(D) of the boundary of D ,such
domains are holomorphi&ly equivalent. In the multidimensional case, the
that, for each function f continuous on the closure of D and holomorphic in D situation is substantially different. For example, a ball and a polydisc are not
and for each point z E D the inequality 1. &)I s max { f(Q[ holds. For a ball the equivalent (Reinhardt, 1921). Moreover, almost any two randomly chosen
domains turn out to be non-equivalent (Burns, Shnider, Wells, 1978).
Shilov boundary coincides with its topological c:idary. The Shilov boundary
Let us consider the class of strictly pseudoconvex domains having analytic
of the polydisc lzjl < 1, j = 1,2, . . . , n, is the n-dimensional torus lz,l = 1, boundary. In this situation any biholomorphic mapping from one domain onto
j-1,2,..., n. For domains whose boundary is C*, the Shilov boundary is the another ,extends to a biholomorphic correspondence between the boundaries
closure of the set of strictly pseudoconvex points (Basener, 1973). (Fefferman, 1974; S.I. Pinchuk, 1975),a nd by the same token, the classification
For domains of holomorphy, a strong maximum principle holds. If D is a problem for such domains reducest o that of classifyingl iypersurfaces.T here are
domairrof holomorphy and’f is non-constant, continuous on the closure of D, two approaches to this problem. The first is geometric; the hypersurface is’
holomorphic in D, and attains a local maximum at some point, then that point characterized by a systemo f differential-geometric invariants (E. Cartan, 1934;
lies in S(D) (Rossi, 196 simple casest he non-Shilov part of the boundary Tanaka, 1967;C hern, 1974).I n the second approach, the characterization is by a
has analytic structurti folliates into analytic sets. This was shown, for special equation, the sa-called normal form (Moser, 1974). Both of these
example, for domains in @* having C’. boundary (N.V. Shcherbina, 1982). constructions enable one to distinguish the infinite-dimensional-space of pair-
Concerning the topology of domains of holomorphy, it is known that the
wise nonequivalent analytic hypersurfaces.
homology groups Hk of order k are trivial for all k > rr. For polynomially In connection with the classification problem, .a description of mappings-
convex domains, the n-th homology group is also trivial (Serre, 1953;
realizing the equivalence between two surfacesh as been obtained The results
Andreotti and Narasimhan, 1962).
for mappings are described as for the’ case of functions by properties of
Several classical problems of analysis are solvable only ‘for domains of
continuation. In the case of mappings a new variant of this phenomenon
holomorphy. For example a domain is a domain of holomorphy if and only if
appears. For example, it turns out that a holomorphic mapping of a sphere to
each function holomorphic on a complex submanifold of the domain is the
itself given in a small neighbourhood of some point of the sphere can be
restriction of somef unction holomorphic on the whole domain (Oka, H. Cartan,
holomorphically extended to the entire sphere and moreover, is in fact a
1950). Analogously, a dqmain is a domain of holomorphy if and only if the
fractional linear transformation (Poincare, 1907;A lexander, 1974).I f the surface
problem of division is @vable’(Oka, H.,Cartan, 1950).T he problem ofdivision is
is not spherical, Le. cannot, by a local change of coordinates, be transformed into
said to be solvable in the domain D if for any functionsf,? . . . ,& holomorphic
the equation of a sphere, then the germ of such a mapping of the surface into
in D, and any holomorph{c function f in D whose zero set contains (taking into
itself can be continued, not only along the surface,b ut also,i n a direction normal
account m’ult@icities) the set of common zeros of the,functions.Ji, . . . ,A, there to the surface. Namely, if a strictly pseudoconvex analytic hypersurface is not
exist ~fnnctionsa i, . :, , gk; holomorphic in D, such that Chgi =$ We recall spherical, then the germ of any holomorphic mapping of this surface into itself
, . . _..‘( ; j + has a holomorphic continuation (with. an estimate on the norm) to a “large”
6 A.G. Vitushkin
I. Remarkable Facts of Complex Analysis 1
neighbourhood of the center of the germ. Moreover, a
function in a domain in terms of its boundary values. The second term isolates
the neighbourhood as well as for the constant esti
af
mined by the two characteristics of the surface, the non:hqlomorphic part of f and yields a solution to the &equation z = g.
analyticity of the .surface and its constant of non-spherici
For functions of. several variables, there does not exist such a simple and
1985). In particular, a surface of the indicated type has
universal formula, and hence it is suitable to consider the problem of integral
which all automorphisms of the surface extend. It is wort
formulas for : holomorphic functions and the solvability of the &equations
both examples we have presented, the mappings, in contr _’
not only to the envelope ,of hofomorphy of the domai separately.
For some .cl+sseso f domains in C”, there are explicit formulas which re-
defined,,but also to some domain lying outside the domain of holomorphy. The
produce a holomorphic function in terms of its boundary values. For poly-
theorem on germs of mappings concludes a lengthy chain of works on holo-
morphic mappings of surfaces (Alexander, 1974; Bums and Shnider, f976; S.I. nomial .polyh&a such a formula w& obtained by A. Weil (1932); for strictly
Pinchuk, 1978; V.K. Beloshapka and A.V. .tOboda, 1980; V.V. Ezhov and N.G. pseudoconvex domains, by GM. Khenkin (1968).S uch a formula was given for
the polydisc’by Cauchy (1841) and for the ball, by Bochner (1943). There is a
Kruzhilin, 1982).
formula of Bochner-Martinelli (1943) for smooth functions on arbitrary
From the Theorem on Germs, it follows that a stability group of a surface
domains having smooth boundary. In this formula, in contrast to the previous
(group of its automorphisms which leave a certain point fixed) is compact. _
ones; the kernel is not holomorphic, and this often makes it difficult to apply,
Hence, by Bochner’s theorem on the .linearization of a compact group of
For polynomial polyhedra there is still another formula which distinguishes
automorphisms (1945), one obtains that a stability group of a non-spherical
itself from the Weil formula and other formulas in. that its kernel is not only
surface can bc linearized, i.e. by choosing appropriate coordinates, every auto-
holomorphic but also integrable (A.G. Vitushkin’, 1968).
morphism can be written as a’ linear transformation (N.G. Kruzhilin and A.V.
I,et us introduce the formulas for the polydisc and the ball. If f is holo-
Loboda, 1983):Tdgether with the theorem of Poincare, this means that for each
morphic on the closure of the polydisc D”, then
pair of locally given strictly pseudoconvex analytic hypersurf&s, every map -,
ping sending one hypersurface into the other can be written as a fractionai-fine&
transformation by an appropriate choice of coordinates in the image and
preimage. The-problem on the linearization of mappings of surfaces having a
I If f,is holomorphic on the closed ball B: lil 5.1, then inside the ball,
non-positive”Levi form remains open. For further details see art@e IV and
Volume. 9, articles V and VI.
We have considered here only one aspect of the problem of classification.
Large sections of complex analysis are concerned with the study of invariant
metrics (Klhier, 1933; Caratheodory, 1927 Bergman, 1933; Kobayashi, 1967; where V is the (2n- l)-dimensional volume of the sphere aB and dV is its
element of volume.
Fefferman, 1974, . . . ); classification of manifolds (i-lodge, Kodaira, 1953; Yay
All of the formulas which we have mentioned above differ from one another in
Siu, 1980; . . . ); description of singularities of complex surfaces ‘(Milnor, 1968;
appearance, The appearance of the formula depends on the type of domain.
Brieskom, 1966, Malgrange, 1974; A.N. Varchenlco, 1981; . . . ).
There is a formula due to Fantappit-Leray (1956) which gives a general scheme
for writing such formulas. Let D be a domain in C:, where z = (zl, . .,. , z,) is a
, set of coordinate functions, and let f be holomorphic on the closure of D. Then
$4. Integral Representations of Functions
fiz) r (n- I)! j f(r)
WY 7C ttlKl -z1)+ * * * +tt&-ZJI”
A smooth function in a closed domain I!j c C can be expressed using the
Cauchy-Green formula
where y is a (2n”-- l)-dimensional cycle in the space @Fx CE lying over the
boundary of the domain D c @Fa nd covering it once. By choosing suitably the
The first term on the righr side is the formula which reproduces a holomorphic ,1f orm of the cycle y, having chosen q as a function of C, one can obtain any of
the preceding integral formulas.
8 A.6 Vitushkin I. Remarkable Facts of Complex Analysis 9 4
One of the applications of integral formulas is in solving the problem of any holomorphic function can be represented as the sum of a series of
interpolation with estimates.I f a complex submanifold M of the ball B crpsses polynomials. Runge’s Theorem reduces the question of the possibility of ap-
the boundary of the ball transversally, then every function holom&phic and proximating functions by polynomials to that of constructing holomorphic ap-
bounded on M can be continued to a function holomorphic and bounded in the proximations of functions. The criterion forthe possibility of approximation by
entire ball (G.M. Khenkin, 1971).T he extension is constructed ad follows. The holomorphic functions (A.G. Vitu’shkin, 1966) is formulated as follows, The
functionf(z) for z E M can be written as an integral I(z) off on the boundary of assertion that each -function in U-Z(E), where E’? C’, can be uniformly ap-
M. Moreover, it turns out that the function I(z) is defined for all z E B, and from proximated with ‘arbitrary accuracy by functions holomorphic on E is equiv-
the explicit formula for I(z), one obtains that the extended functionf(z) = I(z) is alent to the follo.wing condition on the compact set E: for each disc K, a(K\E) = .
holomorphic and bounded on B. * a(K\E), where E denotes the interior of E, and a(M) is the continuous analytic
The problem on the possibility of division with uniform estimates remains cbpdcity of a set M. By definition
open. Namely, it is not known whether for,each set of functions fr, , , . , &, *
f
a(M) = ,sup’ lim zf(z) .’
holomorphic and bounded in the ball B c C,” and such that :nfffr 1& (()I # 0, ML;/I z-00 I’.
e =
The supremum is taken over all compact setsI wc i M and all functions f which
there exist functions gl, . . . , & bounded and holomorphic on B such that vi
are everywhere continuous on C’, bounded in modulus by I and holomorphic
fig, = l.‘This is a modified formulation of the famous “corona” problem. In :ii outside of M*. In particular, approximation is possible if the inner boundary of
one dimensional case,t his problem was solved by CarIeson (1962).T he answer is” E is empty, i.e. each boundary point of E belongs to the boundary of some
positive: in the maximal ideal space for the algebra of bounded halomorphic complementary component of E. For example, all compact setsw ith connected
‘functions in the one-dimensional disc,t he set of ideals, corresponding to points complement belong to this class.T he above criterion emerged as a result of a
of the disc, is everywhere dense. long series of works on approximation (Walsh, 1926; Hartogs and Rosenthal,
The above enumerated formulas are forbounded domains. In the present time 1931; M.A. Lavrentiev, 1934; M.V. KeVysh, 1945; S.N. Mergelyan, 1951 and
analysis on unbounded domains is also flourishing. In part&rIar, integral others). l
formulas have been constructed for such domains: There are explicit formulas The notion of analytic capacity is useful not only in’approximation. It appears
for tubular domains over a cone (Bochner, 1944), on Dyson domains (Jost, along with its analogues in integral estimates (M.S. Mel’nikov, 1967). Such
Lehmann, Dyson, 1958; V.S. Vladimirov) and Siegel domains (S.G. Gindikin, capacitiesa re used for depcribing the set of removeable singularities of a function
1964). Weighted integral representations for entire+f unctions have also been (Ahlfors, 1947; . . . E.P. Dolzhenko, 1962; . . . Mattila, 198s). Among the un-
constructed (Berndtsson, 1983).F or further results see’articleI I and Volume 8, solved problems, we draw attention to the problem of the subadditivity of
articles I, II and, IV. analytic capacity: is it true that for any two compact sets,t he capacity of their
union is no greater than the.s um of their capacities?
The integral formula of Weil is a generalization of Hilbert’s construction..
Using this formula, A. Weil (1932) showed that on any polynomially convex
$5. Approximation of Functions compact set in~Q=“e, ach,h olomorphic function can be approximated by poly-
*
‘_ nomial.+?T hus in C” as in C’, polynomial approximation reduces to holo-
morphrc approximation. The integral formula of G.M. Khenkin emerged as a
Let us denote by CH(E) the set of all continuous funchons on the comnact set result of attempting to construct holomorphic approximations on arcs.,W hile
E c C” which are holomorphic at interior points of E. It is clear that f&&c& developing such approximations, the technique of integral formulas: ,found
which can be uniformly approximated on E with arbitrary accuracyb y complex various applications. Nevertheless, the initial question on the possibility of
polynomials or by functions holomorphic on E belong to the classC HQ. When approximating continuous functions on polynomially convex arcs by poly-
we speak of the possibility of approximating functions on the compact set E, we nomials remains open.
shall mean the following each function in W(E) can be approximated uni- The possibility of holomorphic approximation has been established for the
formly with arbitrary precision by functions holomor’phic~oir~E. following casesa: rcs having nowhere dense projection on the coordinate planes
If a compact set E in C’ has a connected complement, “then each function (E.M. Chirka, 1965);s trictly pseudoconvex domains (G.M. Khenkin, 1968);n on’
holomorphic on E can be approximated by polynomials(Rurigc, ,J885).T his is , degenerate Weil polyhedra (A.I. Petrosyan, 1970); and C.R.-manifolds
equivalent to a theorem of Hilbert (1897):o n each polyno& polyhedron in @*, (Baouendi and T&es, 1981). There are several examples of compact sets on
0
10 A.G. Vitushkin I. Remarka& Facts of Complex Analysis 11
which approximation is not possible. Diederich and Fornaess (19%)* con- Dolbeault, 1953). If the domain is bounded and gE L2, then there exists a ’
structed a domain of holomorphy in C2, with C”S-boundary, whose closure 6 solution ‘to- the C.-R. equations which ties in L, and is orthogonal to the
not a compact set of holomorphy, i.e. it cannot be represented as the intersection subspace of &closed (p,q- I)-forms (Morrey, Kohn, Htirmander, 1965). For
of a decreasing,sequence of domains of hoiomorphy. Moreover, on this domain, sirictly pseudoconvex domains thereare explicit formulas for the solution of
one can define a holomorphic function, infinitely differentiable ‘up to the I? these equations and estimates on the solution in the uniform norm and in
boundary of the domain, which cannot be approximated by functions holo- several other metrics (G.M. Khenkin, Grauert, Lieb, 1969). ’
morphic on the cIosure of the domain. For some simple domains, the question of the possibility of solving the
:. .I
For related results, see papers II and III. . &equations with uniform estimates remains open. For example there are no
Above we discussed only the possibility of approximation. There is a lengthy such estimates on a Siegel domain, also called a generalized unit disc. This is the
series of works devoted to the explicit construction of approximating functions. . domain,‘in the n2dimensional space, of square matrices 2 determined by the
In recent years in connection with applications, there has been a renewed condition E-Z-Z* 3~ 0, i.e., consisting of matrices Z, for which the indicated
interest in classical rational approximation (continuous fractions, Pad& ap- expression is a positive d&&e matrix.
proximation, etc.). We mention one example concerning rational approximation To every complex manifold is associated a system of cohomology groups
in connection with the holomorphic continuation of functions. Letf be holo- called the Dolbeuult cohomology (1953). The Dolbeault group of type (p, q) is the
quotient of the group of &closed (p, q)-forms by the group of &exact
morphic on the ball B c C”, and set rk(f) i inf sup 1f (z)- cp(z)l, where the
(p, q)-forms. In many cases (for example, for compact Kihler manifolds), these
infimum is taken over all rational functions cp of dag;i:k. Then; if for each q > 0, groups can be calculated using de Rham cohomology. However, on domains of
holomorphy, the Dolbeault cohomology is trivial while the de Rham cohom-
lim rJf)q-t = 0, then the global analytic function, generated by the elementf,
ology may be non-trivial.
k-m
turns out to be single-valued, i.e., its domain of existence is single-sheeted over Interest in the &equations is also connected to the phenomenon that there is a
C” (A.A. Gonchar, 1974). See. Vol. 8, paper II. wide class of differential equations which by a change of variables are trans-
formed to the &equations, and in many cases this yields the possibility of
characterizing the sol‘utions of the initial equations in one form or the other. In
the general situation, this change of variables leads to the &equations on a
surface (the tangential Cauchy-Riemann equations). In these situations the
96: Isolating the Non-Holomorphic Part of a Function &quations are to be understood as follows: f is called a solution to the
equation @= g on the surface M if this equation is fulfilled for all vectors lying
in the complex tangent space to M. Each system of linear differential equations
Sometimes in order to construct a holomorphic function with given in general position, with analytic coefficients, and one unknown function, can be
properties,’ one proceeds as follows. One constructs some smooth function cp transformed by an analytic change of coordinates to the z-equations (of type
roperties and then one breaks up cpa s the sum of two functions (0,l)) on an analytic surface (Rossi, Andreotti, Hill, 1970). Such equations
holomorphic while the second is insome sense small. In this satisfying the natural compatibility conditions, are locally solvable-@pencer,
situation, the first function may turn out to be the function we require. The V.P. Palamodov, 1968). If the right-hand side is not analytic, then, such
second term is sought in the form of a solution to the equation af= g, where equations are, generally speaking, not solvable. For example, on the sphere in
II
C2, one can give an infinitely differentiable (0, l)-form such that the equation
and g = 53. This scheme is used for constructing
@ = g turns out to be not lo&lly solvable (H. Lewy, 1957). Explicit integral
in approximation, etc. Equations of the type formulae for solutions to the &quations yield criteria for solvability (G.M.
af= g are called the Cauchy-Riemann equations or &equations. Khenkin 1980). Systems of equations with smooth coefficients are, generally
Let us consider a more general case of the equation aj= g, namely, we shall speaking, not reduceable to &equations (Nirenberg, 1971). If we extend the class’
fake for Q’ a differential (p, q)-form, i.e., a form, having degree p 2 0 in & and of transformations acting on these equations, namely, by adding homogeneous
degree 4 2 1 in d2. A necessary condition for the solvability of this equation is simpletic transformations in the cotangent bundle, hen almostall linear systems
that the form g be &losed, i.e. & = 0. This is a necessary compatibility of equations with analytic coefficients can be locally to &equations on
condition and SO it is always assumed to be satisfied. The Cauchy-Riemann standard surfaces (!&to, Kawai, Kashiwara, For further results! see
equations ‘are solvable on each domain of holomorphy (Grothendieck, paper II.
I
12 A.G. Vitushkin I. .Kcmarkab!e Facts of Compiex Asalyzis 13
inverse Fourier transform be defined, one must solve the division problem with
$7. Construc.tion of Functions with Given Zeros
estimates on- the growth of the solution at infinity.
Each meromorphic function on C” can be represented as the quotient of two
Let us consider several examples from which it will be’clear how the problems entire functions. This assertion was proved by Weierstrass (1874) for C:‘, by
on the zeros of functions arise. The first example is the use of the Weierstrass Poincare (1883) for @‘,.and by Cousin (1895) in general. This was essentially the
Theorem on the representation of an entire function of one variable in the form first series of works on several variables and it Iayed the foundations of several
of an infinite product. This theorem has applications in information theory. The directions in complex analysis. The modern theory of cohomology comes from
formula, regenerating an entire function from its zeros, is used in the problem of the work of Cousin, while potential theory and the theory of currents stems from
encoding signals having finite spectrum. A signal with finite spectrum is a the work of Poinsarb.
function of time, whose Fourier transform is a function of compact support, i.e. The statements of Cousin concerning the solvability of certain probicms have
an entire function of time of finite type. The most economic code for such signals come ‘to be called the first and second- Cousin problems. The first Cceosin
is constructed in the following form. It is necessary to compltxify .time and to problem is said to be solvable on some domain or other if it is possible on this
calculate the zeros of the function. As a code,, the function takes the coordinates domain to construct a meromorphic function with given poles. The second
of its zeros. Using the zeros we write the infinite product which gives the Cousin problem is said to be solvable if it is possible to construct, on this
function and this is the formula regenerating .the original function. It has been domain, a holomorphic function withgiven zeros. The first Cousin problem is
shown’that for such coding systems which don’t increase thedensity of the code, solvable on each domain of holomorphy (Oka, 1937), and’if the group of second
it is possible to broaden without limit the dynamic range of the connecting cohomology with integer coefficients for the domain is trivial, then the second
channel or of the reproducing system (V.I. Buslaev, A.G. Vitushkin, 1974). The problem is also solvable (Oka, 1939; Serre; 1953). In C” the second cohomology
dynamic range is the ratio between the maximal and minimal signals which are is trivial, and so, the second problem of Cousin is solvable. Consequently, any
reproduced with a specified accuracy. Codes which specify a wide dynamic meromorphic function in 6” can be represented as the quotient of entire
range for the amplitude of a signal are required for example in sound functions.
recordings. If M is the set of zeros of a function f; then ,f satisfies the equation of
.
It is not known whether there exists an analogous coding system for entire
PoincarCLelong: k #lnl f 1 = [M],.where [M] is a certain closed (1, 1)-current
functions .of two variables. In this vein we propose a problem. Let us denote by
K, the colle:tiion of all sets in C2 each of which is thkintersection of the zero set of integration on M, taking into account the multiplicity of the dkisor M. A
of some polynomial of degree at most n with the ball lzl I 1. As a metric on K, current is a generalized differential form, i.e. a linear functional on forms of
we take‘the HausdorfT distance between sets. The problem is to calculate the compact support of complementary degree. A current is called closed if it is zero
entropy H,(K,). By definition, H,(K,) = log2N,(K,), where N, is the number of on exact forms of compact support. The current [M] is integration on M of the
elements for a minimal s-net of the compact space K,. The conjecture is that for product of a test function and the multiplicity of the divisor M. The formula of
. Lelong, giving a solution of this equation in @“, yields a completely accurate
small E and large n, W,(K,) x 5 n2 log, i.
. ’ estimatebn’the speed of growth of an entire function depending on the density of
The next example is related to differential equations. Suppose we are given a its set of zeros (1953). For example, dn (n -. 1)-dimensional closed analytic subset
system,of linear diK&ential equations, with constant coefficients and smooth of @” 1sa lgebraic (i.e. is the zero set of a polynomial) if and only if the (2n -2)
right hand side, defined on a convex domain. Then,, provided certain necessary dimensional measure of the intersection of this set with an arbitrary ball of
conditions (of compatibility type) are satisfied by the right-hand side, the system radius r can be estimated from above by the quantity C*;2n-2, where C is
admits a solution on this domain (Ehrenpreis, V.P. Palamodov, Malgrange, independent of j (Rutishiiuser, Lelong, Stoll, 1953).
1963). If, for.example, the right-hand side is of compact support then the Fourier Currents were introduced by de Rham. The Lelong theory of closed currents
transform carries this system to a system of the type CX = F, where F is a vector was one of the fundamental tools in the research on analytic sets (Griffiths,
of entire functions. Here, C is a matrix of polynomials and X is an unknown 1973;..., E.M. Chirka, 1982). and plurisubharmonic functions (Josefson, 1978;
vector function, By solving this problem and using the inverse Fourier trans- Bedford, 1979; A. Sadullaev, 1981). Currents are practical in that they allow one
form, we obtain from X the solution to the oiiginal system. If the system has a to carry delicate problems on analytic sets over to standard estimates on
solution of compact support, then’F must be divisible by C and this gives the integrals. Using this technique, Marvey and Lawson (1974) showed that if a
form of the compatibility conditions for the right-hand side. In order that the (2k + l)-dimensional smooth submanifold M c C” has at each point a complex
14 A.G. Vitushkin
I. Remarkable Facts of Complex Analysis 15
tangent space of maximal possible dimension, ie. dimension UC, and if M is
pseudo-convex, then M is the boundary of a (2k + 2)dimensional analytic subset 6 theme for the case of solving the a-problem of type (0,l). First of all one solves
which (together with M) is the envelope of holomorphy of M. the problem locally, i.e. we fix a covering of the manifold such that the equations
For related results, see articles II, III; Vol. 8, article II; and Vol. 9, articles’& II! ‘- are solvable on each set of the covering. If an element of the covering is, for
I example, a ball or a polydisc, then one can give an explicit formula for the
solution. On the intersection of two elements of the cover, the local solutions
may not agree, i.e., their difference may not be zero. The next stage in con-
$8. Stein’Manifolds structing a solution consists in determining “correcting factors”, in this par-
ticular situation, holdmorphic functions, defined on elements of the cover and
who&difference on any intersection of elements of the cover is the same as for
A great deal of what we have discussed on domains of holomorphy carries the local solutions constructed above. If such “correcting factors” exist, then
over to manifolds which are called Stein manifolds (195 1). A complex manifold subtracting these correcting factors from the corresponding local solutions we
M is called a Stein manijbld ic first of all, it is holomorphicaliy convex, i.e. if the obtain new local solutions which agree on intersecting elements of the cover and
holomorphically convex hull of each compact set in M is compact in M, and hence yield a global solution to the equation. The difference of two local
secondly, if there exists on M a’finite set of holomorphic functions such that each solutions on the intersection of two elements of the cover is a one dimensional
point of M has a neighbourhood in which these functions separate points. Each closed cocycle of holomorphic functions. The existence of the desired correctind
domain of holomorphy is clearly a Stein manifold. Closed complex submani- factors amounts to the exactness of this cocycle. Thus, on a given manifold, the
folds of C” are also Stein manifolds. Conversely, each Stein manifold M can be &equation is solvable for any choice of the right hand side if and only if the one-
realized as such a submanifold, i.e., M can be imbeddcd into C” by a proper dimensional cohomology with holomorphic coefficients (or, as they say, with
holomorphic ‘mapping (Remmert, 1957). coefficients in the sheaf,of germs of holomorphic functions) is trivial.
It is not hard to see that a bounded Weil polyhedron can be imbedded in the The triviality of this group as well as that of other one-dimensional cohom-
polydisc in such a way that its boundary lies on the boundary of the polydisc. It ology groups, corresponding to the above enumerated problems, follows from a
turns out that the ball can also be realized as a closed complex submanifold of a theorem of H. Cartan (1953): a complex manifold is a Stein manifold if and only
polydisc. (A.B. Aleksandrov, 1984). However, there exists a bounded domain if its one dimensional cohomology group with coefficients in an arbitrary
with smooth boundary not admitting such a realization (&bony, 1985). , coherent analytic sheaf is always trivial. Locally a coherent analytic sheaf is a
A great achievement in complex analysis was the solution of Whitney’,s special type of subspace of the space of germs of holpmorphic vector-valued
problem. It has been shown that any real analytic manifold can be analytically functions on the given manifold or on some submanifold thereof. It can be, for
imbedded in a real Euclidean space of sufficiently high dimension (Morrey, example, the space of germs of vector-valued holomorphic functions itself, the
Grauert, 1958). On a given analytic manifold S we fix an atlas After complex-, subspace of germs having a given set of zeros, the quotient space of the first sheaf
ifying the charts of this atlas, that is, allowing the coordinates to take not only by the second, etc. The theorem of Cartan systematizes the material on domains
real but also complex values with small imaginary part, we may consider S as a Rf, holomorphy accumulated till the early 50’s. &tan’s theorem successfully
submanifold of some con$ex manifold M. For an appropriate choice of metric combines the results and techniques of Oka with Leray’s theory of analytic
on M, it turns out that Oneighbourhoods .M, of S are strictly pseudoconvex sheaves (1945). The next step in the development of cohomology theory was the .
domains for small E. The crucial moment in the construction is the general, theorem of Grauert (1958) which has come to be called the Oka-Grauert
ization of Oka’s theorem. Namely it is shown that on a complex @fold, each principle.
strictly pseudoconvex domain is holomorphically eonvex. From I: he hc$omor- Let M be a complex manifold. Let o be a collection of domains in M forming
phic convexity of M,, it follows easily that for small s, M, turns out to be a Stein a cover of M. For each two intersecting elements of this cover c(,p E o, let C,, be
manifold. By Remmert’s theorem, Me-can be imbe-dded in C”, and by the same a non-singular square matrix of degree n consisting of functions defined and
token, S turns out to be imbedded m R*“. holomorphic on the intersection a 6 /?. We suppose that the collection {Car)i
On Stein manifolds just as on domains of holomorphy the‘ problems 4 forms a cocycle; in other words, they are compatible on a triple intersection, i.e.’
interpolation and division are solvable, the &problem is solvable for arbitrae Caa c,, cw = I on an /? n y. Such a collection of matrices of functions is calle
type (p, $); the.first Cousin problem is solvable and the second Cousin problem initial data for the Cousin problem for.matrices. We say that the second Cousi i
is also solvable provided the second integer cohomology group is trivial. A.U of problem with initi’al data {Cma) is solvable if one cab. find a collection (C,} of
these problems are solved by one and the same scheme. Let us look, at this matrices of functions defined on o and such that on each non-empty intersection
an/3,wehaveC;Cg’ = Car. The Oka-Grauert printiple states that on a Stein
ic : /
I. Remarkable Facts of Complex Andysis 17
16 A.G. Vitushkin
tuples zo, . . . , z, on n-dimensional projective space CP” cai be considered as a
manifold, the second Cousin problem with given initial data has a solution {CA}, one-dimensional holomorphic bundle over CP”. In this case a fibre is the
where the C, are matrices of holomorphic functions, provided it has a solution collection of all tuples which can be obtained from each other by.multiplication
where the C, are matrices of continuous functions. This was proved by Oka by a complex number. Two bundles X and X*..on one and the same base M are
(1939) in the case where n = 1 and M is a,domain of holomarphy. He made use called equivalent if one can find a homeomorphic mapping of X into X*,
of this construction in order to find functions with given zeros. carrying fibres to fibres, acting linearly on each fibre, and fixing each point of M.
. Grauert’s theorem has various applications. For example, Griffiths (1975) If there exists a holomorphic mapping having the above properties, then the
while working on a problem of Hodge obtained from this theorem that on a bundles are called holomorphically equivalent. A deformation of structure of a
Stein’ manifold every class of even-dimensional cohomology with rational .bundle is a new bundle not holomorphically equivalent to the given one but ,
coefficients can be realized as a closed complex submanifold. For related results, obtained from it by varying the transition matrices. From the Oka-Grauert i
see Vol. 10, articles I and II. principle it follows that a bundle whose base is Stein has a rigid structure and
moreover from the topological%equivalence of such bundles follows the holo-
morphic equivalence (Grauert, 1959). On Stein manifolds there are, so to speak,
59. Deformations of Complek Structure as many holomorphic bundles as continuous ones. If the base is not Stein, then it i
is usually not so; and this is good..Sometimes a complicated manifold can be
interpreted as a space of deformations of a bundle thus yielding significant - ’
To specify a complex &&re on a manifold means to specify an atlas with information concerning the initial object. II
holomorphic transition functions. A deformation of complex structure is a new The twistor theory of Penrose(l967) is founded on such reductions. The ideal
complex structure obtained from the given one by modifying the trapsition of the genera1 plan of Penrose can be taken as follows. Several notions 04
functions. For example, the extended complex plane can be considered as the mathematical physics can be interpreted in terms of complex structure. For’
two-dimensional sphere with a complex structure. This complex structure does example, the metric on Minkowski space satisfying the Einstein equations, i.e.
not admit*deformation, i.e. on S, the complex structure is unique. On the. the gravitational field can be interpreted as a holomorphic structure on a
Zn-dimensional sphere S2,,, for n # 1,3, it is in general not possible to introduce domain in CP3. More precisely, there is a one-to-one correspondence between
a complex structure (Borel, Serre, 1951). It remains unknown, whether one can conformal classes of autodual solutions of the Einstein equations and ‘defor-
introduce a complex structure on the six-dimensional sphere. mations of structure of domains in @P” (Penrose, 1976). A structure of a domain
The various structures of a compact complex one-dimensional manifold’ of is the same as a choice of functions of three variables. The Penrose trans- 1
genus g(g > 1) form a manifold of real dimension 6g - 6 (Riemann, 1857). This formation, associating to each choice of functions a solution of the Einstein
manifold of structures has itself acomplex structure which can be obtained by equations, has a sufficiently simple form and hence allows one to write down
factoring by a discrete group on a bounded domain of holomorphy in CJBs3 many solutions to these equations.
(Ahlfors, 1953). There is an analogous correspondence between autodual solutions of the
Small deformations of the structure of a compact complex manifold of Yang-Mills equations and bundles of rank two over domains in CP’ (Ward, ___ ~_
arbitrary dimension can also be parametrized as the points of a complex space 1977). In this direction the class of so-called instanton and monopole solutions -
which can be realized as an analytic subset of C” (Kuranishi, 1964). Compact to the Yang-Mills’equations have been obtained which present interest for
manifolds, obtained by factoring some simple domain (for example, a ball) by a theoretical physics (Ward, Hitchin, Atiyah, . . . , Ju. I. Manin, 1978). Because of
discrete subgroup, have a rigid structure, i.e. do not admit small deformations of this series of works, this area of mathematics, which Penrose calls “the complex
structure. Moreover, if two such spaces are topologically equivalent, then they geometry of the real world”; has become very popular. For related results see
.are also holomorphically equivalent (Mostov, 1973). For non-compact mani- Vol. 9, article VII and Vol. 10, articles II and III. %
folds, the space of structures is as a ‘rule infinite dimensional. Within the limits of this article, we have restricted ourselves to discussing only
In order ‘to discuss deformations of bundles, we recall that an n-dimensional a few of the outstanding facts from complex analysis. Unfortunately, we have
vector bundle is a bundle for which th.e fibre is the space C”. The structure of not touched upon several major areas: the theory of residues (see article V and
such a bundle is ,g+en by the cocycle of transition matrices {C., f (of dimension Vol. 8, article l), the theory of singularities (see Vol. 10, article III), and value
n), defined on the intersections {a n R} of elements of a cover of the base. If the distribution (see Vol. 9, articles II-IV). A broad overview with the corresponding
base is a complex manifold and the.transition matrices { Caa} are holomorphtc, bibliographies for the various areas of multidimensional complex analysis is
then the bundle is called holomorphic. For example, the tangent bundle of a given in the series.of articles in volumes 7-10 of this series.
complex manifold M is a holomorphic vector bundle of rank n. The space of
