Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

2.4 MB·English

by  Joël Merker

HOLOMORPHIC EXTENSION OF CR FUNCTIONS,
7
0 ENVELOPES OF HOLOMORPHY,
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2
n AND REMOVABLE SINGULARITIES
a
J
9 JOE¨LMERKERANDEGMONTPORTEN
1

]
V De´partement de Mathe´matiques et Applications, UMR 8553 du CNRS, E´cole Normale Supe´rieure, 45 rue
C d’Ulm,F-75230ParisCedex05,France.
[email protected] http://www.cmi.univ-mrs.fr/ merker/index.html
. ∼
h Department ofEngineering, PhysicsandMathematics, MidSwedenUniversity, CampusSundsvall, S-85170
t Sundsvall,Sweden [email protected]
a
m
[ Tableofcontents(7mainparts)

1 I.Introduction ....................................................................................I.
v II.Analyticvectorfieldsystems,formalCRmappingsandlocalCRautomorphismgroups ..........II.
1 III.Sussman’sorbittheorem,locallyintegrablesystemsofvectorfieldsandCRfunctions ...........III.
3 IV.HilberttransformandBishop’sequationinHo¨lderspaces ......................................IV.
5 V.HolomorphicextensionofCRfunctions .........................................................V.
1 VI.Removablesingularities ......................................................................VI.
0 [0+3+7+1+19+7 diagrams]
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IMRS International Mathematics Research Surveys
i
X
r Volume2006,ArticleID 28295,287pages
a
www.hindawi.com/journals/imrs
Date:2008-2-2.
2000MathematicsSubjectClassification. Primary:32-01,32-02.Secondary:32Bxx,
32Dxx, 32D10, 32D15, 32D20, 32D26, 32H12, 32F40, 32T15, 32T25, 32T27, 32Vxx,
32V05,32V10,32V15,32V25,32V35,32V40.
1
2 JOE¨LMERKERANDEGMONTPORTEN
I: Introduction
1.1. CR extension theory. In the past decades, remarkable progress has
beenaccomplishedtowardstheunderstandingofcompulsoryextendability
ofholomorphicfunctions,ofCRfunctionsandofdifferentialforms. These
phenomena, whose exploration is still active in current research, originate
from theseminalHartogs-Bochnerextensiontheorem.
InlocalCRextensiontheory,themostsatisfactoryachievementwas the
discoverythat,onasmoothembeddedgenericsubmanifoldM Cn,there
⊂
isaprecisecorrespondencebetweenCRorbits ofM andfamiliesofsmall
BishopdiscsattachedtoM. Suchdiscscoverasubstantialpartofthepoly-
nomial hull of M, and in most cases, this part may be shown to constitute
a globalone-sidedneighborhood (M) ofM, ifM isahypersurface, or
±
V
else a wedgelikedomain attached to M, if M has codimension > 2.
W
A local polynomialapproximation theorem, or a CR version of theKonti-
nuita¨tssatz (continuity principle) assures that CR functions automatically
extend holomorphically to such domains , which are in addition con-
W
tained in the envelope of holomorphy of arbitrarily thin neighborhoods of
M in Cn.
Tre´preau in the hypersurface (1986) case and slightly after Tumanov in
arbitrarycodimension(1988)establishedanowadayscelebratedextension
theorem: if M Cn is a sufficiently smooth ( 2 or 2,α suffices) generic
⊂ C C
submanifold,then at every point p M whoselocal CR orbit loc(M,p)
∈ OCR
has maximal dimension equal to dimM, there exists a local wedge of
p
W
edge M at p to which continuous CR functions extend holomorphically.
Several reconstructions and applications of this groundbreaking result, to-
getherwithsurveysaboutthelocalBishopequationhavealreadyappeared
in theliterature.
Propagational aspects of CR extension theory are less known by con-
temporary experts of several complex variables, but they lie deeper in the
theory. UsingFBI transformandconceptsofmicrolocalanalysis,Tre´preau
showed in 1990 that holomorphic extension to a wedge propagates along
curves whose velocity vector is complex-tangential to M. His conjecture
that extension to a wedge should hold at every point of a generic subman-
ifold M Cn consisting of a single global CR orbit has been answered
⊂
independentlybyJo¨rickeandbythefirstauthorin1994,usingtoolsintro-
duced previouslybyTumanov. Totheknowledgeofthetwoauthors,there
is nosurveyoftheseglobalaspects intheliterature.
Thefirstmainobjectiveofthepresentsurveyistoexposethetechniques
underlyingtheseresults in acomprehensiveand unified way, emphasizing
HOLOMORPHICEXTENSIONSANDREMOVABLESINGULARITIES 3
propagational aspects of embedded CR geometry and discussing optimal
smoothness assumptions. Thus, topics that are necessary to build the the-
ory from scratch will be selected and accompanied with thorough proofs,
whereas other results that are nevertheless central in CR geometry will be
presented in concisesurveystyle,withoutanyproof.
The theory of CR extension by means of analytic discs combines var-
ious concepts emanating mainly from three (wide) mathematical areas:
Harmonicanalysis,PartialdifferentialequationsandComplexanalysisin
severalvariables. As the project evolved, we felt the necessity of being
conceptional, extensive and systematic in the restitution of (semi)known
results, so that various contributions to the subject would recover a cer-
tain coherence and a certain unity. With the objective of adressing to a
younger audience, we decided to adopt a style accessible to doctoral can-
didates workingon adissertation. Parts III, IV and V present elementarily
general CR extension theory. Also, most sections of the text may be read
independentlybyexperts,as quantaofmathematicalinformation.
1.2.Concisepresentationofthecontents. Thesurveytextisorganizedin
six main parts. Actually, the present brief introductionconstitutesthe first
and shortest one. Althoughthe reader will find a “conceptional summary-
introduction” at the beginning of each part, a few descriptive words ex-
plainingsomeofouroptionsgoverningthereconstructionofCRextension
theory (Parts III, IV and V)arewelcome.
The next Part II is independent of the others and can be skipped in a
first reading. It opens the text, because it is concerned with propagational
aspects ofanalyticCR structures,betterunderstoodthanthesmoothones.
In Part III, exclusivelyconcerned with thesmoothcategory, Sussmann’s
•
orbit theorem and its consequences are first explained in length. Involu-
tivestructuresandembeddedCRmanifolds,togetherwiththeirelementary
properties,areintroduced. Structuralpropertiesoffinitetypestructures,of
CR orbits and of CR functions are presented without proofs. As a collec-
tionofbackground material,thispart shouldbeconsultedfirst.
In Part IV, fundamental results about singular integral operators in the
•
complex plane are first surveyed. Explicit estimates of the norms of
the Cauchy, of the Schwarz and of the Hilbert transforms in the Ho¨lder
spaces κ,α are provided. They are useful to reconstruct the main Theo-
C
rem 3.7(IV), due to Tumanov, which asserts the existence of unique so-
lutions to a parametrized Bishop-type equation with an optimal loss of
smoothness with respect to parameters. Following Bishop’s constructive
philosophy, the smallness of the constants insuring existence is precised
4 JOE¨LMERKERANDEGMONTPORTEN
explicitly, thanks to sharp norm inequalities in Ho¨lder spaces. This part
is meant to introduce interested readers to further reading of Tumanov’s
recent works about extremal (pseudoholomorphic) discs in higher codi-
mension.
InPartV,CRextensiontheoryisfirstdiscussedinthehypersurfacecase.
•
A simplified proof ofwedge extendabilitythat treats both locally minimal
andgloballyminimalgenericsubmanifoldsonthesamefootingconstitutes
the main Theorem 4.12(V): If M is a globally minimal 2,α (0 < α <
C
1) generic submanifold of Cn of codimension > 1 and of CR dimension
> 1, there exists a wedgelike domain attached to M such that every
W
continuous CR function f 0 (M) possesses a holomorphic extension
∈ CCR
F ( ) 0(M )withF = f. Thefiguresareintendedtoshare
M
∈ O W ∩C ∪W |
thegeometricinsightofexpertsinhighercodimensionalgeometry.
In fact, throughout the text, diagrams (33 in sum) facilitating readabil-
ity (especially of Part V) are included. Selected open questions and open
problems (16 in sum) are formulated. They are systematically inserted in
the right place of the architecture. The sign “[ ]” added after one or sev-
∗
eral bibliographical references in a statement (Problem, Definition, Theo-
rem, Proposition, Lemma, Corollary, Example, Open question and Open
problem, e.g. Theorem 1.11(I)) indicates that, compared to the existing
literature, a slight modification or a slight improvement has been brought
bythetwoauthors. Statementscontainingnobibliographicalreferenceare
originaland appear hereforthefirst time.
WeapologizeforhavingnottreatedsomecentraltopicsofCRgeometry
that also involve propagation of holomorphicity, exempli gratia the geo-
metric reflection principle, in the sense of Pinchuk, Webster, Diederich,
Fornæss,ShafikovandVerma. Bylackofspace,embeddabilityofabstract
CR structures, polynomialhulls, Bishop discs growingat ellipticcomplex
tangencies, filling by Levi-flat surfaces, Riemann-Hilbert boundary value
problems, complex Plateau problem in Ka¨hler manifolds, partial indices
of analytic discs, pseudoholomorphic discs, etc. are not reviewed either.
Certainly, betterexperts willfill thisgapin thenear future.
Toconcludethisintroductorypresentation,webelievethat,althoughun-
easytobuild,surveysandsynthesesplayadecisiveroˆleintheevolutionof
mathematicalsubjects. Forinstance,inthelastdecades,theremarkablede-
velopment of ∂ techniques and of L2 estimates has been regularly accom-
panied by monographs and panoramas, some of which became landmarks
in the field. Certainly, the (local) method of analytic discs deserves to be
HOLOMORPHICEXTENSIONSANDREMOVABLESINGULARITIES 5
known by a wider audience; in fact, its main contributors have brought it
to thedegreeofachievementthatopened theway to thepresent survey.
1.3. Further readings. Using the tools exposed and reconstructed in this
survey, the research article [MP2006a] studies removable singularities on
CR manifolds of CR dimension equal to 1 and solves a delicate remain-
ing open problem in the field (see the Introduction there for motivations).
Recently also, the authors builtin [MP2006c]a new, rigorous proofof the
classical Hartogs extension theorem which relies only on the basic local
Levi argument along analytic discs, hence avoids both multidimensional
integral representation formulas and the Serre-Ehrenpreis argument about
vanishingof∂ cohomologywithcompact support.
6 JOE¨LMERKERANDEGMONTPORTEN
II: Analytic vector field systems
and formal CR mappings
Tableofcontents
1.AnalyticvectorfieldsystemsandNagano’stheorem ........................7.
2.AnalyticCRmanifolds,Segrechainsandminimality ......................19.
3.FormalCRmappings,jetsofSegrevarietiesandCRreflectionmapping ...28.
[3 diagrams]
According to the theorem of Frobenius, a system L of local vector fields hav-
ing real or complex analytic coefficients enjoys the integral manifolds property,
provided itisclosed under Liebracket. IftheLiebrackets exceed L,considering
the smallest analytic system Llie containing Lwhich isclosed under Liebracket,
Nagano showed that through every point, there passes a submanifold whose tan-
gent space is spanned by Llie. Without considering Lie brackets, these submani-
foldsmayalsobeconstructedbymeansofcompositionsoflocalflowsofelements
of L. Such a construction has applications in real analytic Cauchy-Riemann ge-
ometry, inthereflection principle, informal CRmappings, inanalytic hypoellip-
ticitytheoremsandintheproblemoflocalsolvability andoflocaluniqueness for
systemsoffirstorderlinearpartialdifferential operators (PartIII).
For a generic set of r > 2 vector fields having analytic coefficients, Llie has
maximalrankequaltothedimensionoftheambientspace.
The extrinsic complexification of a real algebraic or analytic Cauchy-
M
Riemann submanifold M ofCn carries two pairs ofintrinsic foliations, obtained
bycomplexifying theclassicalSegrevarietiestogetherwiththeirconjugates. The
Nagano leaves of this pair of foliations coincide with the extrinsic complexifica-
tionsoflocalCRorbits. IfM is(Nash)algebraic, itsCRorbitsarealgebraic too,
because theyareprojections ofcomplexifiedalgebraic Naganoleaves.
A complexified formal CR mapping between two complexified generic sub-
manifolds must respect the two pairs of intrinsic foliations that lie in the source
and in the target. This constraint imposes strong rigidity properties, as for in-
stance: convergence, analyticity or algebraicity of the formal CR mapping, ac-
cording to the smoothness of the target and of the source. There is a combina-
torics of various nondegeneracy conditions that entail versions of the so-called
analyticreflectionprinciple. The concept of CRreflectionmappingprovides a
unifiedsynthesis ofrecentresultsoftheliterature.
HOLOMORPHICEXTENSIONSANDREMOVABLESINGULARITIES 7
1. ANALYTIC VECTOR FIELD SYSTEMS AND NAGANO’S THEOREM
§
1.1. Formal, analytic and (Nash) algebraic power series. Let n N
∈
with n > 1 and let x = (x ,...,x ) Kn, where K = R or C. Let K[[x]]
1 n
∈
betheringofformalpowerseriesin(x ,..., x ). Anelementϕ(x) K[[x]]
1 n
∈
writes ϕ(x) = ϕ xα, with xα := xα1 xαn and with ϕ K, for
α Nn α 1 ··· n α ∈
every multiindexα∈:= (α ,...,α ) Nn. Weput α := α + +α .
1 n 1 n
P ∈ | | ···
On the vector space Kn, we choose once for all the maximumnorm
x := max x and, for any “radius” ρ satisfying 0 < ρ 6 , we
16i6n i 1 1
| | | | ∞
define theopencube
(cid:3)n := x Kn : x < ρ
ρ1 { ∈ | | 1}
as a fundamental, concrete open set. For ρ = , we identify of course
1
∞
(cid:3)n withKn.
∞If the coefficients ϕ satisfy a Cauchy estimate of the form ϕ 6
α α
| |
Cρ−2|α|, C > 0, for every ρ2 satisfying 0 < ρ2 < ρ1, the formal
power series is K-analytic ( ω) in (cid:3)n . It then defines a true point map
C ρ1
ϕ : (cid:3)n K. Such a K-analytic function ϕ is called (Nash) K-algebraic
ρ1 →
ifthereexistsanonzeropolynomialP(X,Φ) K[X,Φ]in(n+1)variables
∈
such thattherelation P(x,ϕ(x)) 0holdsin K[[x]], henceforall xin (cid:3)n .
≡ ρ1
ThecategoryofK-algebraicfunctionsandmapsisstableunderelementary
algebraicoperations,underdifferentiationandundercomposition. Implicit
solutionsofK-algebraic equationsareK-algebraic ([BER1999]).
1.2. Analyticvector field systemsandtheir integral manifolds. Let
L0 := La 16a6r, r N, r > 1,
{ } ∈
be a finite set of vector fields L = n ϕ (x) ∂ , whose coefficients
a i=1 a,i ∂xi
ϕ are algebraic or analytic in (cid:3)n . Let A denote the ring of algebraic
a,i ρ1 P ρ1
oranalyticfunctionsin(cid:3)n . Thesetoflinearcombinationsofelementsof
ρ1
L0 withcoefficients inA willbedenoted byL(orL1)and willbecalled
ρ1
theA -linearhullofL0.
ρ1
If p is a point of (cid:3)n , denote by L (p) the vector n ϕ (p) ∂ . It
ρ1 a i=1 a,i ∂xi p
is an elementofT (cid:3)n Kn. Define thelinearsubspace
p ρ1 ≃ P (cid:12)
(cid:12)
L(p) := SpanK La(p) : 1 6 a 6 r = L(p) : L L .
{ } { ∈ }
Noconstancyofdimension,nolinearindependencyassumptionaremade.
Problem 1.3. FindlocalsubmanifoldsΛpassingthroughtheoriginsatis-
fying T Λ L(q) for everyq Λ.
q
⊃ ∈
8 JOE¨LMERKERANDEGMONTPORTEN
By thetheorem of Frobenius ([Stk2000]; original article: [Fr1877]), if the
L are linearly independent at every point of (cid:3)n and if the Lie brackets
a ρ1
[La,La′] belong to L, for all a,a′ = 1,...,r, then (cid:3)nρ1 is foliated by r-
dimensionalsubmanifoldsN satisfyingT N = L(q) foreveryq N.
q
∈
Lemma 1.4. IfthereexistsalocalsubmanifoldΛpassingthroughtheori-
gin and satisfying T Λ L(q) for every q Λ, then for every two vector
q
⊃ ∈
fields L,L′ L, the restrictionto Λ of theLie bracket [L,L′]is tangent to
∈
Λ.
Accordingly, set L1 := L and for k > 2, define Lk to be the A -
ρ1
linearhullofLk−1+ L1,Lk−1 . Concretely,Lk isgeneratedbyAρ1-linear
combinations of iterated Lie brackets [L ,[L ,...,[L ,L ]...]], where
1 2 k 1 k
L ,L ,...,L ,L (cid:2) L1. Th(cid:3)eJacobi identityinsures−(by induction)that
1 2 k 1 k
Lk1,Lk2 L−k1+k2.∈Define then Llie := k>1Lk. Clearly, [L,L′] Llie,
⊂ ∪ ∈
for everytwovectorfields L,L′ Llie.
(cid:2) (cid:3) ∈
Theorem 1.5. (NAGANO [Na1966, Trv1992, BER1999, BCH2005])
There exists a unique local K-analytic submanifold Λ of Kn passing
throughtheoriginwhichsatisfiesL(q) T Λ = Llie(q),foreveryq Λ.
q
⊂ ∈
Adiscussionaboutwhathappensinthealgebraiccategoryispostponed
to 1.12. In Frobenius’ theorem, Llie = L and the dimension of Llie(p)
§
is constant. In the above theorem, the dimension of Llie(q) is constant for
q belonging to Λ, but in general, not constant for p (cid:3)n , the function
∈ ρ1
p dimKL(p) beinglowersemi-continuous.
7→
Nagano’s theorem is stated at the origin; it also holds at every point
p (cid:3)n . The associated local submanifold Λ passing through p with the
∈ ρ1 p
property that T Λ = Llie(q) for every q Λ is called a (local) Nagano
q p
∈
leaf.
In the ∞ category, the consideration of Llie is insufficient. Part III
C
handles smooth vector field systems, providing a different answer to the
search ofsimilarsubmanifoldsΛ .
p
Example 1.6. In R2, take L0 = {L1,L2}, where L1 = ∂x1 and L2 =
e−1/x21 ∂x2. ThenLlie(0)isthelineR∂x1|0,whileLlie(p) = R∂x1|p+R∂x2|p
ateverypointp R 0 . Hence,therecannotexista ∞ curveΛpassing
6∈ ×{ } C
through0 withT0Λ = R∂x1|0 and TqΛ = Llie(q) foreveryq ∈ Λ.
Proof of Theorem 1.5. (May be skipped in a first reading.) If n = 1,
the statement is clear, depending on whether or not all vectorfields in Llie
vanish at the origin. Let n > 2. Since L(q) Llie(q), the condition
⊂
HOLOMORPHICEXTENSIONSANDREMOVABLESINGULARITIES 9
T Λ = Llie(q) implies the inclusion L(q) T Λ. Replacing L by Llie if
q q
⊂
necessary, we may therefore assume that Llie = L and we then have to
provetheexistenceofΛ withT Λ = Llie(q) = L(q), foreveryq Λ.
q
∈
Wereasonbyinduction,supposingthat,indimension(n 1),forevery
−
Aρ1-linearsystemL′ = (L′)lieofvectorfieldslocallydefinedinaneighbor-
hood of the origin in Kn−1, there exists a local K-analytic submanifoldΛ′
passingthroughtheoriginandsatisfyingTq′Λ′ = L′(q′),foreveryq′ Λ′.
∈
If all vector fields in L = Llie vanish at 0, we are done, trivially. Thus,
assume there exists L L with L (0) = 0. After local straightening,
1 1
∈ 6
L1 = ∂x1. Every L ∈ L writes uniquely L = a(x)∂x1 + L, for some
a(x) ∈ K{x}, with L = 26i6n ai(x)∂xi. Introduce the space L := {L :
L ∈ L} of such vector fiPelds. As ∂x1 belongs to L and as L iseAρ1-linear,
L = L − a(x)∂x1 beelongs to L. Since [L,L] ⊂ L, we have Le,L ⊂eL.
On the other hand, we observe that the Lie bracket between two elements
(cid:2) (cid:3)
oefL doesnotinvolve∂x1: e e
e L1,L2 = a1i2∂xi2, a2i1∂xi1
(1.7) (cid:2) (cid:3) h26Xi26n 26Xi16n i
e e ∂a2 ∂a1
= a1 i1 a2 i1 ∂ .
i2 ∂x − i2 ∂x xi1
26Xi16n (cid:16)26Xi26n h i2 i2i(cid:17)
We deducethat L,L L. In otherwords, Llie = L. Next, wedefine the
⊂
restriction
(cid:2) (cid:3)
e e e e e
L′ := L′ = L : L L ,
x1=0 ∈
{ }
and we claim that (L′)lie =(cid:8) L′ aleso(cid:12)(cid:12) holds treue. eIn(cid:9)deed, restricting (1.7)
aboveto x = 0 ,weobservethat
1
{ }
L ,L = L ,L ,
1 x1=0 2 x1=0 1 2 x1=0
{ } { } { }
(cid:2) (cid:12) (cid:12) (cid:3) (cid:2) (cid:3)(cid:12)
since neither L1 neor(cid:12)L2 invoelv(cid:12)es ∂x1. Thisesheows(cid:12) that L′,L′ ⊂ L′, as
claimed.
(cid:2) (cid:3)
Since(L′)liee= L′,teheinductionassumptionapplies: thereexistsalocal
K-analytic submanifold Λ′ of Kn−1 passing through the origin such that
Tq′Λ′ = L′(q′), for every point q′ Λ′. Let d denote its codimension.
∈
If d = 0, i.e. if Λ coincides with an open neighborhood of the origin in
′
Kn−1, it suffices to chosefor Λ an open neighborhoodoftheorigin in Kn.
Assumingd > 1,wesplitthecoordinatesx = (x1,x′) K Kn−1 andwe
∈ ×
letρj(x′) = 0,j = 1,...,d,denotelocalK-analyticdefiningequationsfor
10 JOE¨LMERKERANDEGMONTPORTEN
Λ′. We claim that it suffices to choose for Λ the local submanifold of Kn
with thesameequations,hence havingthesamecodimension.
Indeed,sincetheseequationsareindependentofx ,itisfirstofallclear
1
that the vector field ∂x1 ∈ L is tangent to Λ. To conclude that every L =
a∂x1 +L ∈ L is tangent to Λ, we thus have to prove that every L ∈ L is
tangent toΛ.
Let Le= 26i6n ai(x,x′)∂xi ∈ L. As apreliminaryobservatioen: e
P ∂a ∂
e (ad∂ )L := ∂ ,Le= i(x ,x) ,
x1 x1 ∂x 1 ′ ∂x
1 i
26i6n
(cid:2) (cid:3) X
e e
and moregenerally,forℓ N arbitrary:
∈
∂ℓa ∂
(ad∂ )ℓL = i(x ,x) .
x1 ∂xℓ 1 ′ ∂x
26i6n 1 i
X
e
Since L is a Lie algebra, we have (ad∂x1)ℓL ∈ L. Since (ad∂x1)ℓL does
not involve ∂x1, according to its expression above, it belongs in fact to L.
Also, after restriction (ad∂x1)ℓL x1=0 ∈ L′.eBy assumption, L′ is teangent
to Λ′. Wededucethat,foreveryℓ(cid:12)∈ N, thevectorfield e
e(cid:12)
∂ℓa ∂
L := (ad∂ )ℓL = i(0,x)
′ℓ x1 x1=0 26i6n ∂xℓ1 ′ ∂xi
(cid:12) X
e(cid:12)
is tangent to Λ. Equivalently, [L ρ ](x) = 0 for every x Λ. Letting
′ ′ℓ j ′ ′ ∈ ′
(x ,x) Λ,whencex Λ, wecompute:
1 ′ ′ ′
∈ ∈
∂ρ
Lρ (x ,x) = a (x ,x) j(x)
j 1 ′ i 1 ′ ∂x ′
i
26i6n
(cid:2) (cid:3) X
e ∞ xℓ ∂ℓa ∂ρ
= 1 i(0,x) j(x) [Taylordevelopment]
′ ′
ℓ! ∂xℓ ∂x
26i6n ℓ=0 1 i
X X
∞ xℓ
= 1 L ρ (x) = 0,
ℓ! ′ℓ j ′
ℓ=0
X (cid:2) (cid:3)
so L is tangent to Λ. Finally, the property Tx1,x′Λ = L(x1,x′) follows
immediately from Tx′Λ′ = L′(x′) and the proof is complete (the Taylor
deveelopment argument above was crucially used, and this enlightens why
thetheorem doesnotholdin the category).
∞
C