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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 to the subset Reg(X) of manifold points of X is holomorphic [15].

2 Proof of Proposition 1

The singularities that we implant at the points 2, 3,..., of C are generalized cusp singularities. Let p and q be integers ≥ 2 that are coprime. Consider the cusp like irreducible and locally irreducible complex curve C2 p q C2 Γ= {(z1, z2) ∈ : z1 = z2} ⊂ . Its normalization is C and π : C −→ Γ, t 7→ (tq, tp), is the normalization map. 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 (z1, z2) ∈ Γ,

m n z1 /z2 if z2 =06 , h(z1, z2) :=  0 if z2 =0. It is easily seen that h is continuous (as π is a homeomorphism, the continuity of h follows from that of h ◦ π, that equals id of C), h is holomorphic off its zero set (incidentally, here, this is the set of smooth points of Γ), and h is not holomorphic about (0, 0) (use a Taylor series expansion about (0, 0) ∈ C2 of a presumably holomorphic extension). Furthermore, hk is globally holomorphic provided that k ≥ (p − 1)(q − 1). (Because every integer ≥ (p − 1)(q − 1) can be written in the form αp + βq with α, β ∈ {0, 1, 2,...}, and since hp and hq are holomorphic being the restrictions of z2 and z1 to Γ respectively.) a b Also, z1 z2h 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 O0 at 0 is generated as an O0-module by the germs at 0 of 1,h,...,hr, where r = −1 + min{p, q}.. e Rad´otheorem for complex spaces 3

C2 k k+1 Now, for each integer k ≥ 2, let Γk := {(z1, z2) ∈ ; z1 = z2 }. As previously noted, Γk is an irreducible curve whose normalization map is k+1 k πk : C −→ Γk, t 7→ (t , t ), and the function hk : Γk −→ C defined for (z1, z2) ∈ Γk by z1/z2 if z2 =06 , h(z1, z2) :=  0 if z2 =0. has the following properties:

ak) The function hk is weakly holomorphic. k−1 bk) The power hk is not holomorphic.

k−1 k ck) The function z1 hk is holomorphic because it is the restriction of z2 to Γk. Here, with these examples of singularities at hand, we change the standard complex structure of C at the discrete analytic set {2, 3,...} by complex surgery in order to obtain an irreducible Stein complex curve X and a discrete subset Λ = {xk : k = 2, 3,...} such that, at the level of germs (X, xk) is biholomorphic to (Γk, 0). The surgery, that we recall for the commodity of the reader because in some monographs like [8] the subsequent condition (⋆) is missing, goes as follows, 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 that the 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−1 : U ′ \ A′ −→ U \ A extends to a continuous function ψ : U ′ −→ U such that ψ(A′)= A. In this case, if D 4 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−1(V ) and (⋆) follows immediately. (This process is employed, for instance, in the construction of the blow-up of a point in a complex manifold!) Coming back to our setting, consider Y = C, A = {2, 3,...} and for each k = 2, 3,..., let ∆(k, 1/3) be the disk in C centered at k of radius 1/3 that is mapped holomorphically onto an open neighborhood Uk of (0, 0) ∈ Γk through the holomorphic map t 7→ πk(z − k). Applying surgery, we get an irreducible Stein curve X and the discrete subset Λ with the aforementioned properties. It remains to produce the function f as stated. For this we let I ⊂ OX mk−1 be the coherent ideal sheaf with support Λ and such that Ixk = xk for m k = 2, 3 ..., where xk is the maximal ideal of the analytic algebra of the stalk of OX at xk. From the exact sequence 0 −→ I −→ O −→ O/I −→ 0, where O stands for the sheaf of germse ofe weakly holomorphic functions in X, we obtain a weakly holomorphic function f on X such that, for each e k =2, 3,..., at germs level f equals fk (mod Ixk ). By properties ak), bk) and ak) from above, it follows that no n ∈ N exists n k−1 for which f becomes holomorphic on X. (For instance, if f = fk + gk , for m k−1 certain gk ∈ xk , then f is not holomorphic about xk.)

3 Proof of Theorem 1

Before starting the proof, we collect a few definitions and facts. Let (A, a) be a germ of a local analytic subset in Cn. The third Whitney n cone Ca(A) consists of all vectors v ∈ C such that there is a sequence (aj)j of points in A converging to a and a sequence (nj)j of N such that

nj(aj − a) 7→ v as j 7→ ∞.

If A is pure k-dimensional, then Ca(A) is also a pure k-dimensional analytic set. This cone Ca(A) lies in the tangent space Ta(A) to A at a (Ta(A) is the sixth Withney cone; see [4] for more details). Ck Cm−k Assume that the analytic germ (A, 0) ⊂ z × w , with x = (z,w), is pure k-dimensional such that the projection π(z,w)= z is proper on it, and Rad´otheorem for complex spaces 5

−1 π (0) ∩ C0(A) = {0}. Then π is a branched covering on A with covering number d := deg0A and critical set Σ. Now, whenever x 6∈ Σ, we may define, for a non-constant c-holomorphic germ f :(A, 0) −→ (C, 0), the polynomial

ω(x, t)= (t − f(x′)). π(x′Y)=π(x)

Since f is holomorphic on the regular part Reg(A) of A and f is continuous on A, we obtain a distinguished Weierstrass polynomial of degree d with d d−1 holomorphic coefficients, W (z, t) = t + a1(z)t + ··· + ad(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(kzk1/d) as (z, t) → 0 meaning that there are positive constants M and ǫ such that, if W (z, t)=0 and max{|t|, kzk} < ǫ, then |t| ≤ Mkzk1/d. Therefore, if f : (A, 0) −→ (C, 0) is a non-constant c-holomorphic germ on a pure k-dimensional analytic germ at 0 ∈ Cm, then

|f(ζ)| = O(kζk1/d) as ζ → 0, where d equals the degree of the covering of the projection π satisfying −1 π (0) ∩ C0(A)= {0}. In other words, if f : (A, a) −→ (C, 0) is a non-constant c-holomorphic germ on a pure dimensional analytic germ A at a, we may define the order of flatness of f at a to be

α ordaf = max{α> 0 : |f(x)| = O(kx − ak ) as x → a}.

It follows from the above discussion that ordaf ≥ 1/d. Following Spallek [13], a germ function f :(A, a) −→ (C, 0) is said to be ON -approximable at a if there exists a polynomial P (z, z¯) of degree at most N − 1 in the variables zi − ai, zi − ai, i =1,...,n, such that

|f(z) − P (z, z¯)| = O(kz − akN ) as z → a.

Here we quote from Siu [12] the following result that improves onto Spallek’s similar one (loc. cit.). 6 Vˆıjˆıitu

Proposition 2 For every compact set K of a complex space X there exists a positive integer N (depending on K) such that, if f is a c-holomorphic function germ at x ∈ K and the real part ℜf of f is ON -approximable in some neighborhood of x, then f is a holomorphic germ at x. The above discussion concludes readily the proof of Theorem 1.

4 A final remark

Below we answer a question raised by Th. Peternell at the XXIV Conference on 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 for non reduced complex spaces. More specifically, let (X, OX ) be a not necessarily reduced complex space and 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, OX ) whose reduction Red(σ) equals f|X\A. Is it true that, for every relatively compact open subset D of X, there is n a positive integer n such that σ extends to a section in Γ(D, OX ) ? 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 obvious addition, and multiplication defined by

(r, m) · (r′, m′)=(rr′, rm′ + r′m).

This is the Nagata ring structure from algebra [9]. Now, if (X, OX ) is a complex space, and F a coherent OX -module, then H := OX ⊕F becomes a coherent sheaf of analytic algebras and (X, H) a complex space ([5], Satz 2.3). n The example is as follows. Let nO denotes the structural sheaf of C . The above discussion produces a complex space (C, H) such that H = 1O ⊕ 1O, that can be written in a suggestive way H = 1O + ǫ · 1O, where ǫ is a symbol with ǫ2 = 0. As a matter of fact, if we consider C2 with complex coordinates (z,w) and the coherent ideal I generated by w2, then H is the analytic restriction of the quotient 2O/I to C. The reduction of (C, H) is (C, 1O). A holomorphic section of H over an open set U ⊂ C consists of couple of ordinary holomorphic functions on U. Rad´otheorem for complex spaces 7

Now take 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 a singularity at 0, for instance g(z)=1/z. Obviously, the reduction of σ is the restriction of id on C⋆, and no power σk of σ extends across 0 to a section in Γ(C, H), since σk = id + ǫkg and g does not extend holomorphically across 0 ∈ C.

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Universit´ede Lille, Lab. Paul Painlev´e, Bˆat. M2 F-59655 Villeneuve d’Ascq Cedex, France E-mail: [email protected]