Introduction to the theory of currents

Informations

• Publié par: Shied
• Nombre de lectures: 58
• Langue: English

Extrait

Introduction to the theory of currents
Tien-Cuong Dinh and Nessim Sibony
September 21, 200523
This course is an introduction to the theory of currents. We give here the main
notions with examples, exercises and we state the basic results. The proofs of
theorems are not written. We hope that this will be done in the next version.
All observations and remarks are welcome.
[e-mail] and nessim.sibony@math.u psud.fr4Contents
Notations 7
1 Measures 9
1.1 Borel -algebra and measurable maps . . . . . . . . . . . . . . . . 9
1.2 Positive measures and integrals . . . . . . . . . . . . . . . . . . . 12
1.3 Locally nite measures . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Outer measures and Hausdor measures . . . . . . . . . . . . . . 18
2 Distributions 21
2.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Operations on distributions . . . . . . . . . . . . . . . . . . . . . 23
2.3 Convolution and regularization . . . . . . . . . . . . . . . . . . . 25
2.4 Laplacian and subharmonic functions . . . . . . . . . . . . . . . . 27
3 Currents 31
3.1 Di erential forms and currents . . . . . . . . . . . . . . . . . . . . 31
3.2 Operations on currents and Poincare’s lemma . . . . . . . . . . . 34
3.3 Convolution and regularization . . . . . . . . . . . . . . . . . . . 37
4 Currents on manifolds 39
4.1 Di erentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Tangent and di erential forms . . . . . . . . . . . . . . . . 42
4.4 Currents and de Rham theorems. . . . . . . . . . . . . . . . . . . 44
5 Slicing theory 47
5.1 Normal and at currents . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Federer’s support theorems . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Slicing of at currents . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Currents in complex analysis 51
6.1 Positive forms and positive currents . . . . . . . . . . . . . . . . . 51
6.2 Plurisubharmonic functions . . . . . . . . . . . . . . . . . . . . . 54
6.3 Intersection of currents and slicing . . . . . . . . . . . . . . . . . 56
56 CONTENTS
6.4 Skoda’s extension theorem . . . . . . . . . . . . . . . . . . . . . . 59
6.5 Lelong number and Siu’s theorem . . . . . . . . . . . . . . . . . . 60
Bibliography 61Notation
00 N NX, X open sets in euclidian spacesR ,R or in other manifolds
NB (a,r):={x∈R : kx ak<r} open ball of radius r centered at aN
B (r) open ball B (0,r)N N
B unit open ball B (0,1)N N
B(X) Borel -algebra of X
Nx =(x ,...,x ) and y =(y ,...,y ) coordinates ofR1 N 1 N
00 0 0 Nx =(x ,...,x ) coordinates ofR01 N
R :=R∪{ ∞}
R :={x∈R, x0} andR :={x∈R, x0}+
R :=R ∪{+∞} andR :=R ∪{ ∞}+ +
Dirac mass at aa
N N
L Lebesgue measure onR

H Hausdor measure of dimension
C(X) space of continuous functions on X
C (X) space of continuous functions with compact support in Xc
1
L () space of functions integrable with respect to
1
L () space of locally integrable with respect to loc
1
L (X) space of functions integrable with respect to the Lebesgue measure on X
1
L (X) space of functions locally integrable with respect to the Lebesgue mea-loc
sure on X
k
E (X) space of functions of classesC on X[k]
∞
E(X) space of of classesC on X
p k
E (X) space of p-forms of classesC on X
[k]
p ∞
E (X) space of p-forms ofC on X
p,q k
E (X) space of (p,q)-forms of classesC on X
[k]
p,q ∞
E (X) space of (p,q of classesC on X
k
D (X) space of functions of classC with compact support in X[k]
∞
D(X) space of of classC with co support in X
p k
D (X) space of p-forms of classC with compact support in X
[k]
p ∞
D (X) space of p-forms of classC with compact support in X
p,q k
D (X) space of (p,q)-forms of classC with compact support in X[k]
p,q ∞
D (X) space of (p,q of classC with compact support in X
+
M (X) cone of positive locally nite measures on X
78 CONTENTS
0
D (X) space of distributions on X
0
D (X) space of currents of degree p on Xp
0
E (X) space ofts of p with compact support in Xp
0
D (X) space of currents of bidegree (p,q) on Xp,q
0
E (X) space ofts of (p,q) with compact support in Xp,q
+log ():=max log(),0
N () normal semi-norm on KK
p
N (X) space of normal p-currents on X with support in KK
p
N (X) space of normal p-currents on X
p
N (X) space of locally normal p-currents on Xloc
F () at semi-norm on KK
p
F (X) space of p-currents on X at on K
K
p
F (X) space of at p-currents on X
p
F (X) space of locally at p-currents on X
locChapter 1
Measures
NWe will recall the fundamental notions in measure theory onR . We will de ne
the Lebesgue and Hausdor measures and the operations on measures: multipli-
cationbyafunction,push-forwardbyacontinuousmapandproductofmeasures.
We will also give some basic properties of locally nite measures: compactness
and approximation by Dirac masses.
1.1 Borel -algebra and measurable maps
NRecall that a subset X of R is open if for every point a in X there is a ball
B (a,r) which is contained in X. In other words, open sets are unions of openN
balls. Complementsofopensetsaresaidtobeclosed. Roughlyspeaking,anopen
NsetinR doesnotcontainanypointofitsboundaryandinconstrastaclosedset
Ncontains all points of its boundary. Only∅ andR are open and closed. Unions
and nite intersections of open sets are open. Intersections and nite unions of
closed sets are closed.
NFrom now on, X is a non-empty open set inR . An open subset of X is the
Nintersection of X with an open set ofR , a closed subset of X is the intersection
N Nof X with a closed subset ofR . So, open subsets of X are open inR , but in
Ngeneral closed subsets of X, for example X itself, are not closed inR . Compact
Nsubsets of X are closed in X and in R . If E is a subset of X the closure E of
E in X is the intersection of the closed subsets containing E. We say that E is
relatively compact in X and we write Eb X if E is a compact subset of X, i.e.
for every sequence (a )E one can extract a subsequence (a ) converging to an ni
point of E.
De nition 1.1.1. A familyA of subsets of X is called an algebra if
1. ∅ is an element ofA.
2. If A is an element ofA then X\A∈A.
910 CHAPTER 1. MEASURES
3. A is stable under nite union: if A , n =1,...,m, are elements ofA thenn
Sm
A is inA.nn=1
An algebraA is said to be -algebra if
3’. A is stable under countable union: if A , n ∈N, are elements ofA thennS
A is inA.nn∈N
IfA is an algebra, then X is an element ofA andA is stable under nite
intersection. IfA is a -algebra, then it is stable under countable intersection.
The smallest -algebra is {∅,X} and the largest -algebra is the family of all
the subsets of X. The intersection of a family of algebras (resp. -algebras) is
an algebra (resp. -algebra).
De nition 1.1.2. The Borel -algebra of X is the smallest -algebraB(X) of
subsets of X which contains all the open subsets of X. An element ofB(X) is
called a Borel subset of X or simply a Borel set.
Closed subsets of X are Borel sets. To get the picture, we can construct
Borel sets as follows. We start with open and closed sets, then we take their
countable unions or intersections. We can produce new Borel sets by taking
countable unions or intersections of the sets from the srt step and iterate the
process. Borel sets could be very complicated and there exist sets which are not
Borel sets. In practice, except for some very exceptional cases, all sets that one
constructs are Borel sets.
NLet x = (x ,...,x ) denote the coordinates of R . Let I be an interval in1 N i
R not necessarily open, closed or bounded. The set
N
Y
NW := I = x∈R , x ∈I, i=1,...,Ni i i
i=1
is called N-cell.
Proposition 1.1.3. The familyA (X) of all nite unions of mutually disjoint0
N-cells in X is an algebra which generatesB(X). More precisely,B(X) is the
smallest -algebra containingA (X).0
0 1De nition 1.1.4. A map f : X → X is measurable if f (B) is a Borel set for
0every Borel set B in X .
1To verify that f is measurable, it is su cient to verify that f (B) is a Borel
0set for every open ball B in X . It is easy to show that continuous maps and
compositionsofmeasurablemapsaremeasurable. Inpractice, mostofmapsthat
one constructs are measurable.