Singular points of weakly holomorphic functions

Maciej P. Denkowski^{1}

IMUJ PREPRINT 2006/10 December 13th 2006

Abstract. In this paper we are interested in two kinds of singular points of weakly holomorphic functions. Points where a weakly holomorphic function is not holomorphic and points at which it just is not continuous. The latter are closely connected to points of irreducibility of the given analytic set. We investigate the structure of such points proving they form analytically constructible sets. We prove also that non-holomorphicity points of a given weakly or c-holomorphic function form an analytic subset of the singularities.

1. Introduction

Throughout this paper A ⊂ ℂ^{m} is a locally analytic set. When trying to figure out which complex functions defined on A are the best

generalization of the notion of a holomorphic function one comes across two natural notions. The first one is due to R. Remmert.

Definition 1.1. (cf. [Wh]) A mapping f: A → ℂ^{n} is called c-holomorphic if it is continuous and the restriction of f to the subset RegA of regular points is holomorphic. We denote by O_{c}(A, ℂ^{n}) the ring of c-holomorphic mappings, and by O_{c}(A) the ring of c-holomorphic functions.

A well-known theorem states that a mapping defined in an open set is holomorphic if and only if it is continuous and its graph is an analytic set (it is then a submanifold). We have a similar result for c-holomorphic mappings (cf. [Wh] 4.5Q), which motivates this generalization:

1Kraków 1991 Mathematics subject classification. 32B30 Key words and phrases. Complex analytic sets, weakly holomorphic functions, cholomorphic functions, fibred products.

Date: September 23rd 2006

Theorem 1.2. A mapping f: A → ℂ^{n} is c-holomorphic iff it is continuous and its graph Γ_{f} := {(x, f(x)) | x ∈ A} is locally analytic in ℂ^{m} × ℂ^{n}.

On the other hand, there is the notion of weakly holomorphic functions which usually people are much more acquainted with. They were introduced by H. Cartan.

Definition 1.3. A holomorphic function f: RegA → ℂ is called weakly holomorphic if it is locally bounded on A (i.e. for any a ∈ A there exists a neighbourhood U of a such that f is bounded on U ∩ A). A mapping f: RegA → ℂ^{n} is called weakly holomorphic if all its components are weakly holomorphic. We denote by O_{w}(A, ℂ^{n}) the ring of weakly holomorphic mappings and put O_{w}(A) := O_{w}(A, ℂ).

One checks as in [Wh] that if A = ⋃ A_{ι} is the decomposition of A into irreducible components, then f has a unique extension onto RegA_{ι} for each ι (and that works in fact for germs). Then f is weakly holomorphic iff it is such on each irreducible component of A.

It is useful to observe that for locally irreducible sets weak holomorphic and c-holomorphic functions are just the same (see [Wh], c-holomorphic stands in fact for continuous weak holomorphic).

Recall that an analytically constructible subset of some open set Ω ⊂ ℂ^{m} is a set which can be written locally in Ω in the form ⋃_{ι=1}^{p} ⋂_{j=1}^{q_{j}}{F_{ιj} ∗_{ιj} 0}, where ∗_{ι,j} ∈ {=, ≠} and F_{ιj} are holomorphic (see [Ł]). In particular the difference of two analytic sets is analytically constructible and the closure of a constructible set is analytic.

In [D2] we proved a weakly holomorphic counterpart of theorem 1.2. It seems strange that it was not stated anywhere till now. It clearly implies theorem 1.2. For convenience sake we recall it with its proof:

Theorem 1.4. ([D2]) Let f: RegA → ℂ^{n} be a mapping locally bounded on an analytic subset A of an open set Ω ⊂ ℂ^{m}. The following three conditions are then equivalent:

(1) f ∈ O_{w}(A, ℂ^{n});

(2) \overline{Γ_{f}} is analytic in Ω × ℂ^{n};

(3) Γ_{f} is analytically constructible in Ω × ℂ^{n}.

Proof. First note that we may restrict ourselves to the case n = 1 since Γ_{f} = ⋂_{j=1}^{n} Γ_{j}, where

Γ_{j} := {(x, y_{1}, …, y_{j−1}, f_{j}(x), y_{j+1}, …, y_{n}) | x ∈ RegA, y_{ι} ∈ ℂ}.

We may as well assume that A has pure dimension k (using restrictions to the irreducible components of A) with 0 < k < m (otherwise, since there are no singularities, there is nothing to do — cf. the Analytic Graph Theorem).

If we have (1)⇔(2), the equivalence (2)⇔(3) is quite immediate. Indeed, if \overline{Γ_{f}} is analytic, then Γ_{f} = \overline{Γ_{f}} \(SngA×ℂ) is the difference of two analytic sets, thence is analytically contructible. On the other hand, if Γ_{f} is analytically constructible, then its closure is analytic and so f ∈ O_{w}(A).

We turn now to proving (1)⇔(2). The problem being local we may assume that h ∈ O(Ω) is a global universal denominator for A. Having fixed f ∈ O_{w}(A) we can find g ∈ O(Ω) such that fh = g on RegA. Consider the analytic set X := {(z, t) ∈ A × ℂ | h(z)t = g(z)}. It remains now to observe that the set

Γ_{f} ∩ {(z, t) ∈ Ω × ℂ | h(z) ≠ 0} =
= X ∩ (RegA × ℂ) ∩ {(z, t) ∈ Ω × ℂ | h(z) ≠ 0} =
= X \ [(X ∩ (SngA × ℂ)) ∪ {(z, t) ∈ Ω × ℂ | h(z) = 0}]

is dense in Γ_{f}. Its closure in Ω × ℂ is clearly analytic. □

Note. In the case A = Ω the theorem above asserts that a necessary and sufficient condition for a locally bounded mapping to be holomorphic is the analytic constructibility of its graph.

Finally, we will call simply holomorphic the restrictions to A of holomorphic functions defined in a neighbourhood of A. Their ring will be denoted O(A). Obviously O(A) ⊊ O_{c}(A) ⊊ O_{w}(A) (see [Wh]).

2. Proper projections of analytic sets and irreducibility points

In this section we consider the following situation: X ⊂ ℂ^{N} is a locally analytic set and π: X → A is a proper holomorphic surjection. Without loss of generality we may assume that N = m + n and π is just the restriction to X of the natural projection onto the first m coordinates. Then its multiplicity is well-defined and locally bounded (see e.g. [Ch]). We fix now a neighbourhood Ω of zero in ℂ^{m} and put U := Ω × ℂ^{n} assuming that X and A are closed in U.

The central result of this section is the following theorem. Recall that a function f: Z → ℂ is called analytically constructible (where Z is constructible too) if its graph is analytically constructible.

Theorem 2.1. In the introduced setting the function

μ_{π}: A ∋ a ↦ #π^{−1}(a) ∈ ℤ

is analytically constructible.

Proof. Similarly to [T] we start with observing that the analytic constructibility of μ_{π} is equivalent to the fibres μ_{π}^{−1}(p) being analytically constructible (indeed, the graph coincides with ⋃_{p∈ℤ} μ_{π}^{−1}(p) × {p}). The problem is clearly a local one.

Let d := max{#π^{−1}(a) | a ∈ A}. Consider then the fibred product

X^{\{d\}} := \underbrace{X ×_{π} … ×_{π} X}_{d times} = \{x ∈ X^{d} | π(x_{ι}) = π(x_{j}), ι, j = 1, …, d\}.

It is analytic since it coincides with (π × … × π)^{−1}(Δ_{A^{d}}), where Δ_{A^{d}} is the diagonal in A^{d} (i.e. the set of all points (a, …, a) ∈ A^{d}).

Observe that

f_{ι,j}: X^{{d}} ∋ (x_{1}, …, x_{d}) ↦ (x_{ι} − x_{j}) ∈ ℂ^{n}, 1 ≤ ι < j ≤ d,

is a finite collection of holomorphic functions and the mapping

ρ: X^{{d}} ∋ x ↦ π(x_{1}) ∈ A

is clearly holomorphic and proper (since π is proper).

Now it is obvious that

{μ_{π} = 0} = Ω \ A and {μ_{π} = 1} = A \ ρ (⋃_{ι<j}{f_{ι,j} ≠, 0}) ,

whence both these sets are analytically constructible. Indeed, due to the properness of ρ, the set ρ (⋃_{ι<j}{f_{ι,j} ≠, 0}) = {μ_{π} > 1} is constructible by the Chevalley-Remmert Theorem (see [Ł]).

Observe that in the case d = 1, 2 there is nothing more to prove (for d = 1 leads to A = {μ_{π} = 1} while d = 2 yields {μ_{π} = 2} = A \ {μ_{π} = 1}). We might suppose thus that d ≥ 3.

It is enough to prove that each set {μ_{π} > p} is analytically constructible, since {μ_{π} = p} = {μ_{π} > p − 1} \ {μ_{π} > p}.

Clearly enough, we have in general for p ∈ {2, …, d},

{μ_{π} > p − 1} = ρ ( ⋃_{1≤ι_{1}<...<ι_{p}≤r} ⋂_{1≤j<κ≤p}{f_{ι_{j},ι_{κ}} ≠ 0}) .

The Chevalley-Remmert Theorem ends the proof. □

This theorem has many interesting consequences. The first one concerns the irreducibility points of A, i.e. points at which A induces an irreducible germ.

Corollary 2.2. In the introduced setting let

A^{≺} := {a ∈ SngA | the germ A_{a} is irreducible}.

Then A^{≺} is analytically constructible.

Before we prove this corollary, observe that one cannot possibly hope to obtain analycity of A^{≺} in general, unless dimA = 1:

Example 2.3. Let X := {x^{2}y = z^{2}} be Whitney’s Umbrella and consider Y := {z = 0} ∪ {y^{2} = x^{3}}, both sets being taken in ℂ^{3}. Then

X^{≺} = {(0, 0)}, but Y^{≺} = {x = y = 0, z ≠ 0}.

The difference in structure of these sets is due to the fact that Y is globally reducible while X is not and in both cases the singularities are along onedimensional sets.

The corollary above is actually itself a corollary to the following one.

Corollary 2.4. The function

μ: SngA ∋ a ↦ #{irreducible components of A_{a}} ∈ ℤ

is analytically constructible.

Proof of corollary 2.2. The problem being local we may suppose that zero belongs to A^{≺}, A is a closed analytic subset of some neighbourhood of zero and π: N → A is its normalization, where N ⊂ ℂ^{m} × ℂ^{k} for some k ≥ 1 (see e.g. [Ł] Local Normalization Theorem). We are exactly in the situation of theorem 2.1 and μ = μ_{π}|_{SngA}, whence A^{≺} = {μ = 1} is constructible. □

Proof of corollary 2.4. Once again it suffices to consider the local normalizing mapping π and apply theorem 2.1 to μ = μ_{π}|_{SngA}. □

Note. The matter being local, the corollary above has clearly a counterpart on complex analytic manifolds (assumed to be second-countable by definition) since the local maps are biholomorphisms and so they preserve analytic constructibility (i.e. we consider A as an analytic subset of some analytic manifold Ω).

Corollary 2.5. If A is analytic in the open set (or more generally in the analytic manifold) Ω, then the function μ̂: Ω → ℤ defined by

μ̂(x) = \begin{cases}μ(x), when x ∈ SngA; \\ 1, when x ∈ RegA; \\ 0, when x ∉ A, \end{cases}

is analytically constructible and locally bounded.

Proof. Clearly μ^{−1}(0) = Ω \ A is analytically constructible. As to μ^{−1}(1), it is the union of μ^{−1}(1) and RegA, and both are analytically constructible. □

Recall now that an analytic cycle on some manifold Ω is the formal sum Z = ∑ α_{ι}Z_{ι}, where α_{ι} ∈ ℤ and {Z_{ι}} is a locally finite family of pairwise distinct, analytic, irreducible subsets of Ω (see [T]). The analytic set |Z| := ⋃ Z_{ι} is called the support of the cycle Z. We define also the degree of Z by

ν(Z, x) := ∑ α_{ι}ν(Z_{ι}, x),

where ν(Z_{ι}, x) is the classical degree or Lelong number of Z_{ι} at x ∈ Ω (if x ∉ Z_{ι}, then ν(Z_{ι}, x) = 0).

By [T] (2.1), there exists exactly one analytic cycle Z in Ω such that μ̂ coincides with ν(Z), where

ν(Z): Ω ∋ x ↦ ν(Z, x) ∈ ℤ

is the degree function of Z. A natural question arises here: what kind of information on A does Z bring with, and, besides, what actually is Z as a cycle?

Note that in general A ⊊ |Z| (otherwise, if A = |Z|, we obtain α_{ι} = 1 and so μ̂(x) = ν(A, x) which is not true in general as one can see for instance by considering Whitney’s Umbrella).

3. Singular points of weakly holomorphic functions

Fix a function f ∈ O_{w}(A) and put

𝒩 (f) := {a ∈ SngA | f is not continuous at a}.

Theorem 3.1. The set 𝒩 (f) is analytically constructible.

Proof. We know by theorem 1.4 that \overline{Γ_{f}} is analytic. Moreover, for each a ∈ SngA there is #(\{a\} × ℂ) ∩ \overline{Γ_{f}} < +∞. Indeed, for any such a point we have a finite decomposition of the germ A_{a} into irreducible germs, say r of them, and the restriction of f to each of them is continuous at a (see e.g. [Wh]). Thus at a we have at most r possible values for f(a).

Now let π be the restriction to \overline{Γ_{f}} of the natural projection ℂ^{m} × ℂ → ℂ^{m}. Clearly it is a proper mapping (since f is locally bounded) and 𝒩 (f) = {μ_{π} > 1} according to the notation introduced in the previous section. Thus, by theorem 2.1, the set 𝒩 (f) is analytically constructible. □

Recall that the Zariski tangent space of an analytic germ X at a point a ∈ ℂ^{m} is defined as

T_{a}^{Zar}X := ⋂{Ker d_{a}ϕ | ϕ ∈ ℐ_{a}(X)},

where ℐ_{a}(X) ⊂ O_{m} is the ideal of germs of holomorphic functions vanishing on the germ X (see [Wh]).

There is a nice holomorphicity criterion for c-holomorphic functions, whose short proof one can find in [D1]. The point is that this proof works also in the weakly holomorphic case and we obtain the following

Theorem 3.2. Let a ∈ A \ 𝒩 (f). Then f is holomorphic at a iff

T_{(a,f(a))}^{Zar}\overline{Γ_{f}} ∩ (\{0\} × ℂ) = \{0\}.

Proof. For simplicity sake assume that a = f(a) = 0. If F ⊃ f is a holomorphic extension of f to a neighbourhood of zero in ℂ^{m}, then φ(z, t) = t − F (z) (here (z, t) ∈ ℂ^{m} × ℂ) belongs to ℐ_{0}(\overline{Γ_{f}}). Since φ is a submersion, then Ker d_{0}φ = T_{0}Γ_{F}. Hence

T_{0}^{Zar}\overline{Γ_{f}} ∩ (\{0\} × ℂ) ⊂ T_{0}Γ_{F} ∩ (\{0\} × ℂ) = \{0\}.

To prove the ‘only if’ part first note that the assumption leads to m ≥ dimT_{0}^{Zar}\overline{Γ_{f}}. We know (cf. [Wh]) that there exists a submanifold Γ such that Γ_{0} ⊃ (\overline{Γ_{f}})_{0} and T_{0}Γ = T_{0}^{Zar}\overline{Γ_{f}}. We apply now Lemma (1.4) from [D1] to find an m-dimensional submanifold Γ̃ whose germ at zero contains the germ Γ_{0} and whose tangent space at zero meets {0} × ℂ only at zero.

By the Implicit Function Theorem it is now clear that Γ̃ is the germ of the graph of a holomorphic function over ℂ^{m}. □

Now let us introduce the set

𝒪_{w}(f) := {a ∈ A | f is not holomorphic at a}.

It is clearly a closed subset of SngA.

Theorem 3.3. The set 𝒪_{w}(f) is an analytic subset of SngA.

Proof. First observe that 𝒩 (f) ⊂ 𝒪_{w}(f).

Since 𝒪_{w}(f) is closed, we only need to show its constructibility at any point a ∈ 𝒪_{w}(f). Fix such a point a. There is a finite set {b_{1}, …, b_{r}} ⊂ ℂ such that \overline{Γ_{f}} ∩ (\{a\} × ℂ) = \{(a, b_{1}), …, (a, b_{r})\}. Take pairwise disjoint discs D_{j} ∋ b_{j}. Let π(z) = (z_{1}, …, z_{m}) denote the natural projection from ℂ^{m} × ℂ onto ℂ^{m}. It is proper on \overline{Γ_{f}}.

By Cartan’s Coherence Theorem (see [Ł]), we can find pairwise disjoint neighbourhoods U_{j} = V × W_{j} ⊂ Ω × D_{j} of (a, b_{j}) and some holomorphic functions g_{j,1}, …, g_{j,r_{j}} ∈ O(U_{j}) whose germs at z ∈ U_{j} generate the ideal ℐ_{z}(\overline{Γ_{f}}). Then, for z ∈ U_{j} ∩ \overline{Γ_{f}},

T_{z}^{Zar}\overline{Γ_{f}} = ⋂_{ι=1}^{r_{j}} Ker d_{z}g_{j,ι}.

Indeed, the inclusion ‘⊂’ is quite obvious from the definition of the Zariski tangent space. To prove that the converse one holds too, observe that any g ∈ ℐ_{z}(\overline{Γ_{f}}) can be written as the sum ∑_{1}^{r_{j}} h_{ι}g_{j,ι} with some holomorphic germs h_{ι}. Then d_{z}g = ∑_{1}^{r_{j}}(h_{ι}(z)d_{z}g_{j,ι} + g_{j,ι}(z)d_{z}h_{ι}) and since g_{j,ι}(z) = 0, there is Ker d_{z}g ⊃ ⋂_{1}^{r_{j}} Ker d_{z}g_{j,ι}.

Observe that \{0\}×ℂ ⊂ T_{z}^{Zar}\overline{Γ_{f}} (where z ∈ U_{j} ∩\overline{Γ_{f}}) is equivalent to \frac{∂g_{j,ι}}{∂z_{m+1}} (z) = 0 for all ι = 1, …, r_{j}. Now,

𝒪_{w}(f)∩V = (𝒩 (f)∩V)∪ ⋃_{j=1}^{r} π (\{z ∈ \overline{Γ_{f}} ∩ U_{j} | \frac{∂g_{j,ι}}{∂z_{m+1}} (z) = 0, ι = 1, …, r_{j}\}) .

Indeed, to see that ‘⊃’ holds note that if x is a point from the set on the righthand side and x ∉ 𝒩 (f) ∩ V, then f(x) is well-defined as the only point over x in \overline{Γ_{f}} and so (x, f(x)) belongs to exactly one of the sets U_{j}. By Theorem 3.2, f cannot be holomorphic at x. On the other hand, if x ∈ 𝒪_{w}(f) ∩ V \ 𝒩 (f), then once again x is a continuity point of f at which f is not holomorphic and since \{(x, f(x))\} = \overline{Γ_{f}} ∩ (\{x\} × ℂ), there is exactly one j such that (x, f(x)) ∈ U_{j}. Then we apply Theorem 3.2.

Finally, Remmert’s Proper Mapping Theorem implies that the projections under consideration ara analytic in V, whence 𝒪_{w}(f)∩V is analytically constructible in V. □

Corollary 3.4. The set A \ ⋂{𝒪_{w}(g) | g ∈ O_{w}(A)} coincides with the set of normal points of A and its complement in A is an analytic set.

Corollary 3.5. If f ∈ O_{c}(A), then the set

𝒪_{c}(f) := {a ∈ A | f is not holomorphic at a}

is analytic in SngA. In particular, A \ ⋂{𝒪_{c}(g) | g ∈ O_{c}(A)} is the set of weakly normal points of A and its complement in A is analytic.

Obviously, 𝒪_{c}(f) = 𝒪_{w}(f), but we would like to distinguish the c-holomorphic case from the weak holomorphic one. In doing so we are aiming at the following proposition.

First fix an open set Ω in which A is closed and put U := Ω \ \overline{𝒩 (f)} . Then U is open (we are substracting an analytic subset of SngA). Let X := A ∩ U. Notice that X ≠ ∅ but there may be SngX = ∅.

Proposition 3.6. In the introduced setting f|_{X} ∈ O_{c}(X) and the following holds: (1) The set 𝒪_{c}(f|_{X}) is analytically constructible in Ω;

(2) There is 𝒪_{w}(f) = \overline{𝒩 (f)} ∪ 𝒪_{c}(f|_{X}), the union being disjoint.

Proof. Observe first that

\overline{𝒩 (f)} ⊂ 𝒪_{w}(f).

Indeed, f cannot have a holomorphic extension at a point a ∈ ∂𝒩 (f), since otherwise this extension would be discontinuous in a sequence of points converging to a.

Clearly enough f|_{X} is c-holomorphic. Therefore, by theorem 3.5, 𝒪_{c}(f|_{X}) is analytic in U. But it is also easy to see that

𝒪_{w}(f) = \overline{𝒩 (f)} ∪ 𝒪_{c}(f|_{X})

the union being disjoint by definition. Since 𝒪_{w}(f) is analytic, then there is 𝒪_{c}(f|_{X}) = 𝒪_{w}(f) \ \overline{𝒩 (f)} is the difference of two analytic sets and so is constructible. □

Acknowledgements I wish to acknowledge here my indebtedness to Prof. Piotr Tworzewski for suggesting to me the use of fibred products in the proof of theorem 2.2.

References

[Ch] E. M. Chirka, Complex Analytic Sets, Kluwer Acad. Publ. 1989;

[D1] M. P. Denkowski, The Łojasiewicz exponent of c-holomorphic mappings, Ann. Polon. Math. LXXXVII.1 (2005), pp. 63-81;

[D2] M. P. Denkowski, A note on the Nullstellensatz for c-holomorphic functions, (2006) submitted to Ann. Polon. Math.;

[Ł] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel

1991;

[T] P. Tworzewski, Intersection theory in complex analytic geometry, Ann. Polon. Math. LXII.2 (1995), pp. 177-191;

[Wh] H. Whitney, Complex Analytic Varieties, Addison-Wesley Publ. Co. 1972.

Institute of Mathematics Jagiellonian University Reymonta 4, 30-059 Kraków Poland

e-mail: denkowsk@im.uj.edu.pl