Table Of ContentEncyclopaedia of
Mathematical Sciences
Volume 8
Editor-in-Chief: R. V. Gamkrelidze
G.M. Khenkin A.G. Vitushkin (Eds.)
Several
Complex Variables II
Function Theory in Classical Domains
Complex Potential Theory
With 19 Figures
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
A.A. Agrachev, AA Gonchar, E.F. Mishchenko,
N. M. Ostianu, V. P. Sakharova, A B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 8, Kompleksnyj analiz - mnogie peremennye 2
Publisher VINITI, Moscow 1985
Mathematics Subject Classification (1991):
32-02, 32A07, 32A27, 32A35, 32A40, 32F05
ISBN 978-3-642-63391-1
Library of Congress Cataloging-in-Publication Data
Kompleksnyi analiz-mnogie peremennye 2. English
Several complex variables II: function theory in c1assieal domains: complex potentialtheory /
G. M. Khenkin, A. G. Vitushkin (eds.)
p. cm. - (Encyclopaedia of mathematical sciences; v. 8)
Includes bibliographical references and indexes.
ISBN 978-3-642-63391-1 ISBN 978-3-642-57882-3 (eBook)
DOI 10.1007/978-3-642-57882-3
1. Functions of several complex variables. 1. Khenkin, G. M. II. Vitushkin, A. G. (Anatolii Georgievich)
III. Title. IV. Title: Several complex variables 2. V. Series.
QA33I.K7382513 1994 515'.94-dc20 92-45735
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Originally published by Springer-Verlag Berlin Heidelberg New York in 1994
Softcover reprint of the hardcover 1s t edition 1994
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List of Editors, Authors and Translators
Editor-in-Chief
RV. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITI),
ul. Usievicha 20a, 125219 Moscow, Russia
Consulting Editors
G. M. Khenkin, Central Economic and Mathematical Institute of the Russian
Academy of Sciences, ul. Krasikova 32,117418 Moscow, Russia
A. G. Vitushkin, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
Authors
L. A. Aizenberg, Akademgorodok, Institute of Physics, 660036 Krasnoyarsk 36,
Russia
A. B. Aleksandrov, Petrodvorets, S1. Petersburg State University,
198904 S1. Petersburg, Russia
A. Sadullaev, Vuzgorodok, Tashkent State University, 700095 Tashkent,
Usbekistan
A. G. Sergeev, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
A. K. Tsikh, Akademgorodok, Institute of Physics, 660036 Krasnoyarsk 36, Russia
V. S. Vladimirov, Steklov Mathematical Institute, ul. Vavilova 42,
117966 Moscow, Russia
A. P. Yuzhakov, Akademgorodok, Institute of Physics, 660036 Krasnoyarsk 36,
Russia
Translators
P. M. Gauthier, Departement de Mathematiques et de Statistique, Universite de
Montreal, CP 6128-A, Montreal QC H3C 3J7, Canada
J. R. King, Department of Mathematics, GN-50, Seattle, WA 98195, USA
Contents
I. Multidimensional Residues and Applications
L. A. Aizenberg, A. K. Tsikh, A. P. Yuzhakov
1
ll. Plurisubharmonic Functions
A. Sadullaev
59
Ill. Function Theory in the Ball
A. B. Aleksandrov
107
IV. Complex Analysis in the Future Tube
A. G. Sergeev, V. S. Vladimirov
179
Author Index
255
Subject Index
258
I. Multidimensional Residues and Applications
L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov
Translated from the Russian
by 1.R. King
Contents
Chapter 1. Methods for Computing Multidimensional Residues
(A.P. Yuzhakov) ............................................ 3
Introduction .................................................. 3
§ 1. Leray Theory. Froissart Decomposition Theorem ............... 4
1.1. Leray Coboundary ..................................... 4
1.2. Form-Residue, Class-Residue, Leray Residue Formula ....... 5
1.3. Tests for Leray Coboundaries. Froissart Decomposition
Theorem .............................................. 6
1.4. Cohomological Lowering of Pole Order .................... 7
1.5. Generalization of the Leray Theory to the Case of Submanifolds
of Codimension q > 1 ................................... 9
§ 2. Application of Alexander-Pontryagin Duality and De Rham
Duality ................................................... 10
2.1. Application of Alexander-Pontryagin Duality ............... 10
2.2. Residues of Rational Functions of Two Variables ............ 11
2.3. Application of De Rham Duality .......................... 13
§ 3. Homological Methods for Studying Integrals that Depend upon
Parameters. Application to Combinatorial Analysis .............. 15
3.1. Analytic Continuation of Integrals Depending on Parameters.
Isotopy Theorem ....................................... 16
3.2. Foliation near a Landau Singularity. Picard-Lefschetz Formula 18
3.3. Some Examples of Integrals Depending on Parameters ....... 20
3.4. Application of Residues to Combinatorial Analysis .......... 22
Chapter 2. Multidimensional Logarithmic Residues and Their
Applications (L.A. Aizenberg) ................................ 24
§ 1. Multidimensional Logarithmic Residues ....................... 24
§ 2. Series Expansion ofImplicit Functions ......................... 31
2 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov
§ 3. Application of the Multidimensional Logarithmic Residue to Systems
of Nonlinear Equations ..................................... 33
§ 4. Computation of the Zero-Multiplicity of a Holomorphic Mapping. 37
§ 5. Application of the Multidimensional Logarithmic Residue to the
Theory of Numbers ......................................... 38
Chapter 3. The Grothendieck Residue and its Applications to Algebraic
Geometry (A.K. Tsikh) ...................................... 39
Introduction .................................................. 39
§ 1. Integral Definition and Fundamental Properties of the Local
Residue ................................................... 40
1.1. Definitions ............................................ 40
1.2. Representation of the Local Residue by an Integral over the
Boundary of a Domain .................................. 41
1.3. Transformation Formula for the Local Residue ............. 41
1.4. Local Duality Theorem .................................. 42
§ 2. Using the Trace to Express the Local Residue ................... 43
2.1. Definition of the Trace and its Fundamental Properties ....... 43
2.2. Algebraic Interpretation ................................. 44
§ 3. The Total Sum of Local Residues ............................. 45
3.1. The Total Sum of Residues on a Compact Manifold. The Euler-
Jacobi Formula ........................................ 45
3.2. Applications to Plane Projective Geometry ................. 47
3.3. The Converse of the Theorem on Total Sum of Residues ...... 47
3.4. Abel's Theorem and its Converse .......................... 48
3.5. Residue Theorem for Vector Bundles ...................... 50
3.6. The Total Sum of Residues Relative to a Polynomial Mapping
in cn .............•••.•.•..••••••••••...••••.•......•. 51
§ 4. Application of the Grothendieck Residue to the Algebra of
Polynomials and to the Local Ring (!)a ....•.....•.•..•••.••.••• 52
4.1. Macauley's Theorem .................................... 52
4.2. Noether-Lasker Theorem in ClPn .......................... 52
4.3. Verification of the Local Noether Condition ................ 53
4.4. A Consequence of Global Duality ......................... 54
Bibliography .................................................. 55
I. Multidimensional Residues and Applications 3
Chapter 1
Methods for Computing Multidimensional Residues
A.P. Yuzhakov
Introduction
One of the problems in the theory of multidimensional residues is the prob-
lem of studying and computing integrals of the form
(1)
where w is a closed differential form of degree p on a complex analytic manifold
X with a singularity on an analytic set SeX, and where Y is a compact p-
dimensional cycle in X\S. A special case of this problem is computing the
integral (1) when w is a holomorphic (meromorphic) form of degree p = n =
dime X; in local coordinates the form can be written as w = I(z) dz =
I(z l' ... , zn) dz 1 /\ ... /\ dzn, where 1 is a holomorphic (meromorphic) function.
According to the Stokes formula, the integral (1) depends only on the homology
class l [y] E Hp(X\S) and the De Rham cohomology class [w] E HP(X\S). Thus
in integral (1) the cycle Y can be replaced by a cycle Yl homologous to it (Yl '" y)
in X\S and the form w can be replaced by a cohomologous form W 1(W 1 '" w)
which may perhaps be simpler; for example, it could have poles of first order on
S (see § 1, Subsection 4). If {Yj} is a basis for the p-dimensional homology of the
manifold X\S, then by Stokes formula for any compact cycle Y E Zp(X\S) the
integral (1) is equal to
(2)
where the kj are the coefficients of the cycle Y as a combination of the basis
elements {yJ, Y '" Lj kjYj. Formula (2) shows that the problem of computing
integral (1) can be reduced to
1) studying the homology group Hp(X\S) (finding its dimension and a basis);
2) determining the coefficients of the cycle Y with respect to a basis;
3) computing the integrals over the cycles in the basis.
Solving problems 1) and 2) is a difficult topological problem in the multi-
dimensional case and requires the machinery of algebraic topology. In some
1 In this chapter we will denote by Hp the group of compact singular homology; this group was
denoted by H~ in the contribution of Dolbeault (Dolbeault, 1985) in Volume 7 of the Encyclopaedia
of Mathematical Sciences.
4 L.A. Aizenberg, A.K. Tsikh, A.P. Yuzhakov
cases, to solve this it helps to apply the dualities of Alexander-Pontryagin
and De Rham (§ 2). Simple and multiple Leray coboundaries (Subsection 1.1)
give a construction of standard cycles in X\S. The general structure of the
homology group Hp(X\S) is described in "good cases" by the decomposition
theorem of Froissard (Subsection 1.3). Integrals on coboundary cycles can be
reduced to integrals of lower degree by the simple and multiple Leray residue
formulas (Subsection 1.2). The computation of an important class of residues,
the Grothendieck residues, and a special case of them, the logarithmic residue,
is considered in Section 2 and Section 3 of this article; § 3 is devoted to the
application of residues to the study of integrals depending on parameters and to
combinatorial analysis.
§ 1. Leray Theory. Froissart Decomposition Theorem
Here we will pause to study in more detail the computational side of the
Leray theory of residues expounded in (Dolbeault (1985)). To start with, we
consider the case of co dimension 1.
1.1. Leray Coboundary. We give a constructive description of the
coboundary homomorphism b which was introduced in (Dolbeault (1985), Sect.
0.3). In the one-dimensional case the simplest cycle (contour) of integration is a
circle of sufficiently small radius around an isolated singular point. Leray (1959)
constructed the analog of this for complex analytic manifolds, the co boundary
homomorphism b. The construction of b-1 was first considered by Poincare (1887).
Let X be a complex analytic manifold of complex dimension n. Let S be a
complex-analytic submanifold of X of co dimension 1. We consider a tubular
neighborhood V of the submanifold S, which is a locally-trivial fiber bundle
with base S and fiber Ya, a E S, homeomorphic to the disk. In order to construct
such a fiber bundle we choose a Riemannian metric on X and take as Ya the
union of geodesic segments of length p(a), beginning at a and orthogonal to S,
where p(a) is sufficiently small. We assume that the function p(a) is smooth; this
implies the smoothness of OV To each (p - I)-dimensional element of a chain (a
simplex, a rectangle) <Tp- 1 in S we associate a p-dimensional chain in X\S. The
chain is b<TP _ 1 = UaE!up_tI ba where ba = oYa; it is homeomorphic to oYa x <Tp - 1
with the natural orientation. Thus a homomorphism of homology groups is
defined,
b : Hp- 1( S) ~ Hp(X\S),
since Ob = -bO. Then the Leray homology exact sequence is defined:
ro a i
... ~ Hp+1(X) ~ Hp_1(S) ~ Hp(X\S) ~ Hp(X) ~... (3)
where i is the homomorphism induced by the inclusion X\S c X and OJ
in induced by the intersection of chains in X, transversal to S, with the
submanifold S.
I. Multidimensional Residues and Applications 5
If a family of S l' ... , Sm of submanifolds of codimension 1 is in general
position, the multiple Leray co boundary is defined:
which is anticommutative with respect to the order of S l' ... , Sm (for the
cohomological multiple coboundary, see (Dolbeault (1985), Sect. 03).
1.2. Form-Residue, Class-Residue, Leray Residue Formula. As was pointed
out in (Dolbeault (1985), Sect. 03), if rfJ is a closed regular differential form in
X\S with a pole of first order on S, then in some neighborhood of any point
a E S the form rfJ can be represented as
ds
rfJ = - 1\ t/J + e, (4)
s
where s = sa(z) is the defining function of the manifold Sin Ua and t/J, ea re forms
which are regular on Ua. Here the form 1/1 Is is globally defined, is closed, and is
uniquely determined by the form rfJ. This restriction 1/1 Is is called the form-residue
of the form rfJ and is denoted by res[rfJJ. We remark that if rfJ is holomorphic on
X\S, then the form residue res[rfJJ is holomorphic on S.
Example 1. Let X = cn, with S = {z E Cn : s(z) = O} and rfJ = f(z) dZ 1 1\
... 1\ dzn/s(z). Since rfJ = ( _1)j-1 ds 1\ dZ!J1/s· S~j' where dZ!J1 = dz 1 1\ •.• 1\
dZj - 1 1\ dZj +1 1\ ... 1\ dzn, then res[rfJJ = (-1y-1f(z) dZ!J1/s~)s at the points
where s~. =1= O.
J
Remark. The map
f(z) dz/s(z) --+ (-1y-1f(z) dZ!J1/s;)s
is called the Poincare residue map and is denoted by P.R. If we denote by ,gx,
,gx(S), ,gs-I, the sheaves of germs, respectively, of holomorphic n-forms on X,
meromorphic n-forms having only simple poles on S, and holomorphic (n - 1)-
forms on S, then there is an exact sequence of sheaves
o --+ ,gx --+ ,gx (S) --+ ,gs -1 --+ 0
which defines an exact sequence on cohomology
HO(X, ,gx(S)) P~. HO(S, ,gS-1) ~ H1(X, ,gx).
Therefore, the Poincare residue map is surjective on global sections if
H1(X, ,gx) = O. In particular, this is true for projective space X = cpn, n> 1.
Thus for n > 1 every holomorphic form of degree n - 1 on the submanifold S is
the Poincare residue of a merom orphic n-form on cpn.
By the theorem of (Dolbeault (1985), Sect. 0.3), for every closed regular differ-
entiable form rfJ on X\S, there is a form ~ cohomologous to it which has a pole
of first order on S. In this case the cohomology class of the form res[~J depends
only on the cohomology class of the form rfJ.
