aa r X i v : . [ m a t h . C V ] J a n A Rad´o theorem for complex spaces
Viorel Vˆıjˆıitu
Abstract . We generalize Rad´o’s extension theorem from the complex planeto reduced complex spaces. : 32D15, 32D20, 32C15
Key words : Rad´o’s theorem, complex space, c-holomorphic function
A theorem due to Rad´o asserts that a continuous complex valued functionon an open subset of the complex plane is holomorphic provided that it isholomorphic off its zero set.Essentially this theorem was proved in [10]. Since then many other proofshave been proposed, e.g. [2], [3], [6], and [7]. The articles [1], [11] and [14]give some generalizations.Rad´o’s statement remains true for complex manifolds (or, more generally,for normal complex spaces) as well as in the complex plane.In this short note we investigate a natural extension of Rad´o’s theoremwhen the ambient space has (non normal) singularities.Complex spaces, unless explicitely stated, are assumed to be reduced andcountable at infinity. Let N = { , , . . . } .Here we state our main results. Proposition 1
There is an irreducible Stein curve X and a continuous func-tion f : X −→ C that is holomorphic off its zero set but no power f n , n ∈ N ,is globally holomorphic. Theorem 1
Let X be a complex space and Ω ⊂ X be a relatively compactopen set. Then, there is n Ω ∈ N such that, for every continuous function f : X −→ C that is holomorphic off its zero set, for every integer n ≥ n Ω ,the power f n is holomorphic on Ω . Vˆıjˆıitu
Recall the following definition. Let X be a complex space. A function f : X −→ C is c - holomorphic if it is continuous and the restriction of f tothe subset Reg( X ) of manifold points of X is holomorphic [15]. The singularities that we implant at the points 2 , , . . . , of C are generalizedcusp singularities.Let p and q be integers ≥ { ( z , z ) ∈ C : z p = z q } ⊂ C . Its normalization is C and π : C −→ Γ, t ( t q , t p ), is the normalizationmap. Note that π is a homeomorphism.A continuous function h : Γ −→ C that is holomorphic off its zero set,but fails to be globally holomorphic is produced as follows.Select natural numbers m and n with mq − np = 1, and define h : Γ −→ C by setting for ( z , z ) ∈ Γ, h ( z , z ) := (cid:26) z m /z n if z = 0 , z = 0 . It is easily seen that h is continuous (as π is a homeomorphism, the continuityof h follows from that of h ◦ π , that equals id of C ), h is holomorphic off itszero set (incidentally, here, this is the set of smooth points of Γ), and h is notholomorphic about (0 ,
0) (use a Taylor series expansion about (0 , ∈ C ofa presumably holomorphic extension).Furthermore, h k is globally holomorphic provided that k ≥ ( p − q − ≥ ( p − q −
1) can be written in the form αp + βq with α, β ∈ { , , , . . . } , and since h p and h q are holomorphic being therestrictions of z and z to Γ respectively.)Also, z a z b h is holomorphic on Γ provided that q ⌊ ( m + a ) /p ⌋ + b ≥ n, where ⌊·⌋ is the floor function.It is perhaps interesting to note that the stalk of germs of weakly holo-morphic functions e O at 0 is generated as an O -module by the germs at 0of 1 , h, . . . , h r , where r = − { p, q } .. ad´o theorem for complex spaces k ≥
2, let Γ k := { ( z , z ) ∈ C ; z k = z k +12 } .As previously noted, Γ k is an irreducible curve whose normalization map is π k : C −→ Γ k , t ( t k +1 , t k ), and the function h k : Γ k −→ C defined for( z , z ) ∈ Γ k by h ( z , z ) := (cid:26) z /z if z = 0 , z = 0 . has the following properties: a k ) The function h k is weakly holomorphic. b k ) The power h k − k is not holomorphic. c k ) The function z k − h k is holomorphic because it is the restriction of z k to Γ k .Here, with these examples of singularities at hand, we change the standardcomplex structure of C at the discrete analytic set { , , . . . } by complexsurgery in order to obtain an irreducible Stein complex curve X and a discretesubset Λ = { x k : k = 2 , , . . . } such that, at the level of germs ( X, x k ) isbiholomorphic to (Γ k , ⋆ ) is missing, goes asfollows,Let Y and U ′ be complex spaces together with analytic subsets A and A ′ of Y and U ′ respectively, such that there is an open neighborhood U of A in Y and ϕ : U \ A −→ U ′ \ A ′ that is biholomophic.Then define X := ( Y \ A ) ⊔ ϕ U ′ := ( Y \ A ) ⊔ U ′ / ∼ by means of the equivalence relation U \ A ∋ y ∼ ϕ ( y ) ∈ U ′ \ A ′ .Then there exists exactly one complex structure on X such that U ′ and Y \ A can be viewed as open subsets of X in a canonical way provided thatthe following condition is satisfied:( ⋆ ) For every y ∈ ∂U and a ′ ∈ A ′ there are open neighborhoods D of y in Y , D ∩ A = ∅ , and B of a ′ in U ′ such that ϕ ( D ∩ U ) ∩ B ⊆ A ′ . Thus X is formed from Y by ”replacing“ A with A ′ .In practice, the condition ( ⋆ ) is fulfilled if ϕ − : U ′ \ A ′ −→ U \ A extendsto a continuous function ψ : U ′ −→ U such that ψ ( A ′ ) = A . In this case, if D Vˆıjˆıitu and V are disjoint open neighborhoods of ∂U and A in Y respectively, then B = A ′ ∪ ϕ ( V \ A ) is open in U ′ because it equals ψ − ( V ) and ( ⋆ ) followsimmediately. (This process is employed, for instance, in the construction ofthe blow-up of a point in a complex manifold!)Coming back to our setting, consider Y = C , A = { , , . . . } and foreach k = 2 , , . . . , let ∆( k, /
3) be the disk in C centered at k of radius 1 / U k of (0 , ∈ Γ k through the holomorphic map t π k ( z − k ). Applying surgery, we get anirreducible Stein curve X and the discrete subset Λ with the aforementionedproperties.It remains to produce the function f as stated. For this we let I ⊂ O X be the coherent ideal sheaf with support Λ and such that I x k = m k − x k for k = 2 , . . . , where m x k is the maximal ideal of the analytic algebra of thestalk of O X at x k .From the exact sequence0 −→ I −→ e O −→ e O / I −→ , where e O stands for the sheaf of germs of weakly holomorphic functions in X , we obtain a weakly holomorphic function f on X such that, for each k = 2 , , . . . , at germs level f equals f k (mod I x k ).By properties a k ), b k ) and a k ) from above, it follows that no n ∈ N existsfor which f n becomes holomorphic on X . (For instance, if f = f k + g k − k , forcertain g k ∈ m x k , then f k − is not holomorphic about x k .) Before starting the proof, we collect a few definitions and facts.Let (
A, a ) be a germ of a local analytic subset in C n . The third Whitneycone C a ( A ) consists of all vectors v ∈ C n such that there is a sequence ( a j ) j of points in A converging to a and a sequence ( n j ) j of N such that n j ( a j − a ) v as j
7→ ∞ . If A is pure k -dimensional, then C a ( A ) is also a pure k -dimensional analyticset. This cone C a ( A ) lies in the tangent space T a ( A ) to A at a ( T a ( A ) is thesixth Withney cone; see [4] for more details).Assume that the analytic germ ( A, ⊂ C kz × C m − kw , with x = ( z, w ), ispure k -dimensional such that the projection π ( z, w ) = z is proper on it, and ad´o theorem for complex spaces π − (0) ∩ C ( A ) = { } . Then π is a branched covering on A with coveringnumber d := deg A and critical set Σ. Now, whenever x Σ, we may define,for a non-constant c-holomorphic germ f : ( A, −→ ( C , ω ( x, t ) = Y π ( x ′ )= π ( x ) ( t − f ( x ′ )) . Since f is holomorphic on the regular part Reg( A ) of A and f is continuouson A , we obtain a distinguished Weierstrass polynomial of degree d withholomorphic coefficients, W ( z, t ) = t d + a ( z ) t d − + · · · + a d ( z ), such that W ( x, f ( x )) = 0 for x ∈ A .Note that W does not depend on w . Besides, if W ( t, z ) = 0, then | t | = O( k z k /d ) as ( z, t ) → M and ǫ such that, if W ( z, t ) = 0and max {| t | , k z k} < ǫ , then | t | ≤ M k z k /d .Therefore, if f : ( A, −→ ( C ,
0) is a non-constant c-holomorphic germon a pure k -dimensional analytic germ at 0 ∈ C m , then | f ( ζ ) | = O ( k ζ k /d ) as ζ → , where d equals the degree of the covering of the projection π satisfying π − (0) ∩ C ( A ) = { } .In other words, if f : ( A, a ) −→ ( C ,
0) is a non-constant c-holomorphicgerm on a pure dimensional analytic germ A at a , we may define the orderof flatness of f at a to beord a f = max { α > | f ( x ) | = O( k x − a k α ) as x → a } . It follows from the above discussion that ord a f ≥ /d .Following Spallek [13], a germ function f : ( A, a ) −→ ( C ,
0) is said to beO N - approximable at a if there exists a polynomial P ( z, ¯ z ) of degree at most N − z i − a i , z i − a i , i = 1 , . . . , n , such that | f ( z ) − P ( z, ¯ z ) | = O( k z − a k N ) as z → a. Here we quote from Siu [12] the following result that improves ontoSpallek’s similar one ( loc. cit. ). Vˆıjˆıitu
Proposition 2
For every compact set K of a complex space X there existsa positive integer N ( depending on K ) such that, if f is a c -holomorphicfunction germ at x ∈ K and the real part ℜ f of f is O N -approximable insome neighborhood of x , then f is a holomorphic germ at x . The above discussion concludes readily the proof of Theorem 1.
Below we answer a question raised by Th. Peternell at the XXIV Conferenceon Complex Analysis and Geometry, held in Levico-Terme, June 10–14, 2019.He asked whether or not a similar statement like Theorem 1 does hold fornon reduced complex spaces.More specifically, let ( X, O X ) be a not necessarily reduced complex spaceand f : X −→ C be continuous such that, if A denotes the zero set of f ,then X \ A is dense in X and there is a section σ ∈ Γ( X \ A, O X ) whosereduction Red( σ ) equals f | X \ A .Is it true that, for every relatively compact open subset D of X , there isa positive integer n such that σ n extends to a section in Γ( D, O X ) ?We show that the answer is ”No“.Recall that, if R is a commutative ring with unit and M is an R -module,we can endow the direct sum R ⊕ M with a ring structure with the obviousaddition, and multiplication defined by( r, m ) · ( r ′ , m ′ ) = ( rr ′ , rm ′ + r ′ m ) . This is the Nagata ring structure from algebra [9].Now, if ( X, O X ) is a complex space, and F a coherent O X -module, then H := O X ⊕ F becomes a coherent sheaf of analytic algebras and ( X, H ) acomplex space ([5], Satz 2.3).The example is as follows. Let n O denotes the structural sheaf of C n . Theabove discussion produces a complex space ( C , H ) such that H = O ⊕ O ,that can be written in a suggestive way H = O + ǫ · O , where ǫ is a symbolwith ǫ = 0. As a matter of fact, if we consider C with complex coordinates( z, w ) and the coherent ideal I generated by w , then H is the analyticrestriction of the quotient O / I to C .The reduction of ( C , H ) is ( C , O ). A holomorphic section of H over anopen set U ⊂ C consists of couple of ordinary holomorphic functions on U . ad´o theorem for complex spaces f the identity function id on C , and the holomorphic section σ ∈ Γ( C ⋆ , H ) given by σ = id + ǫg , where g is holomorphic on C ⋆ having asingularity at 0, for instance g ( z ) = 1 /z .Obviously, the reduction of σ is the restriction of id on C ⋆ , and no power σ k of σ extends across 0 to a section in Γ( C , H ), since σ k = id + ǫkg and g does not extend holomorphically across 0 ∈ C . References [1] B. Aupetit: Une g´en´eralisation du th´eor`eme d’extension de Rad´o,
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Math. Ann. (1967), 282.[8] L. Kaup and B. Kaup: Holomorphic functions of several variables, deGruyter Studies in Mathematics , Berlin, 1983.[9] M. Nagata: Local rings. New York: Interscience 1962.[10] T. Rad´o: ¨Uber eine nicht fortsetzbare Riemannsche Mannigfaltigkeit, Math. Z. (1924), 1–6.[11] J. Riihentaus: A note concerning Rad´o’s theorem, Math. Z. (1983),159–165.
Vˆıjˆıitu [12] Y.T. Siu: O N -approximable and holomorphic functions on complexspaces, Duke Math. J. (1969), 451–454.[13] K. Spallek: Differenzierbare und holomorphe Funktionen auf analytis-chen Mengen, Math Ann. (1965), 143–162.[14] E. L. Stout: A generalization of a theorem of Rad´o,
Math. Ann. (1968), 339–340.[15] H. Whitney: Complex analytic varieties, Addison–Wesley, 1972.Universit´e de Lille, Lab. Paul Painlev´e, Bˆat. M2F-59655 Villeneuve d’Ascq Cedex, FranceE-mail:(1968), 339–340.[15] H. Whitney: Complex analytic varieties, Addison–Wesley, 1972.Universit´e de Lille, Lab. Paul Painlev´e, Bˆat. M2F-59655 Villeneuve d’Ascq Cedex, FranceE-mail: