A NOTE ON C-ANALYTIC SETS. Alessandro Tancredi

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 36 (2007), A NOTE ON C-ANALYTIC SETS Alessandro Tancredi (Received August 2005) Abstract. It is proved that every C-analytic and C-irreducible set which is not pure dimensional is isomorphic to a real analytic set which is not C-analytic. 1. Introduction and Preliminary Remarks To fix our notations we first recall some well known properties of real and complex analytic sets that we shall freely use in the following (see [2], [3], [4]). By a complex (real) analytic set X in C n (R n ) we mean a locally closed subset of C n (R n ) which locally is the zero set of finitely many complex (real) analytic functions and by complex (real) analytic functions on X we mean functions that locally are restrictions of complex (real) analytic functions of C n (R n ). So an analytic map between analytic sets is locally defined by analytic functions; an isomorphism of analytic sets is a bijective analytic map whose inverse is also an analytic map. The sheaf of analytic functions on X is denoted by O X and by O X,x its stalk at x; the canonical image of a function f in O X,x, the germ of f at x, is denoted by f x. As usual we call holomorphic the complex analytic functions and maps. We recall also that a weakly holomorphic function on an open set U of a complex analytic set X is a holomorphic function f : U S C that is defined outside a thin complex analytic set S in U and locally bounded near S. Two complex (real) analytic sets X, Y are called equivalent at a subset S of X Y if there exists an open neighborhood U of S in C n (R n ) such that X U = Y U. Clearly this gives an equivalence relation; the equivalence classes are called analytic germs at S; if S is a point a the germ of X at a is usually denoted by X a. A real analytic set X is said to be coherent if the ideal sheaf J of real analytic functions vanishing on X is of finite type, i.e. if there exist an open neighborhood of every point x of X and finitely many real analytic functions f 1,..., f p on U such that the germs f 1,x..., f p,x generate the stalk J x as a module over the ring O R n,x. Let X be a real analytic set in R n and X a complex analytic set in C n such that X X and x a point of X: the complex analytic germ X x is said to be the complexification of the real analytic germ X x if every holomorphic function on an open neighborhood of x in C n which vanishes on the germ X x vanishes on the germ X x too; the complex germ X x is uniquely determined, up to isomorphisms, by the real analytic germ X x and we say also that X is the complexification of X at the point x. Moreover X is called the complexification of X if it is the 1991 Mathematics Subject Classification Primary 32C05; Secondary 32B15. Key words and phrases: C-analytic set, real analytic set. The author is member of GNSAGA of INDAM. This work is partially supported by European Contract HPRN-CT

2 36 ALESSANDRO TANCREDI complexification at every point of X; we recall that the complexification X of X, if it exists, is uniquely determined up to isomorphisms as a germ at X. If X is the complexification of X at x, the dimension of the complex analytic germ X x is the same as the dimension of the real analytic germ X x ; moreover X is irreducible (regular) at x if and only if X is irreducible (regular) at x. A real analytic set X has a complexification if and only if it is coherent (see [2], [7]). A real analytic set in R n is called C-analytic if it is determined by global real analytic equations in some open set, i.e. if there exist an open subset Ω R n and finitely many real analytic functions f 1,..., f p on Ω such that X = {x Ω f 1 (x) = = f p (x) = 0}. Every coherent real analytic set in R n is C-analytic, but real analytic sets may fail to be C-analytic and the latter may fail to be coherent. We remark also that the singular points of a C-analytic set X of dimension s are contained in an analytic set of dimension strictly smaller than s (see [2], [4], [7]). A C-analytic set is said to be C-irreducible if it is not the union of two proper closed C-analytic sets; however a C-irreducible set may not be irreducible (see [1], [4]). An old problem (see e.g. [8]) is to check if the C-analycity of a real analytic set depends on the imbedding in R n. It is well known that every real analytic set isomorphic to a coherent real analytic set is coherent too and then it is a C-analytic set (see [2], [7]). On the other hand there exist real analytic sets X such that any real analytic set containing the singular points of X has the same dimension as X (see [4], [8]); clearly such a real analytic set can not be isomorphic to a C-analytic set. In some particular cases (see [5], [6]) it was pointed out by examples that a C-analytic set may be isomorphic to a real analytic set which is not C-analytic. In this note we prove that every C-analytic and C-irreducible set X which is not pure dimensional is isomorphic to a real analytic set which is not C-analytic. The key ingredient of the proof is the construction of a suitable real analytic function on X that is induced by a meromorphic function having an indeterminate point on X. We remark that not pure dimensional C-analytic and C-irreducible sets are not coherent and so their dimension is strictly bigger than one; as a matter of fact coherent irreducible real analytic sets are pure dimensional and every real analytic set of dimension one is coherent (see [2], [7]). In the following theorem we collect some properties of the smallest complex analytic set which contains a C-analytic set that we will use many times in the following. Theorem 1.1. Let X be a C-irreducible and C-analytic set in an open set Ω R n of dimension s and let σ be the antiinvolution on C n induced by the conjugation. There exist a Stein open neighborhood Ω of Ω in C n, such that σ( Ω) = Ω and Ω R n = Ω, and a closed irreducible complex analytic set X in Ω of dimension s that satisfy the following properties. i) σ( X) = X and X = {x X σ(x) = x} = X Ω. ii) X is the complexification of X at every point x X such that X x is irreducible and X x has dimension s. iii) If X is regular at a point x X, then X is regular at x of dimension s.

3 A NOTE ON C-ANALYTIC SETS 37 iv) If x is a point of X of dimension s, there exist points y arbitrarily near to x such that X and X are regular of dimension s at y and then X is the complexification of X at y. v) If Ω is an open neighborhood of X in Ω such that σ( Ω ) = Ω, then there exists one and only one irreducible component of X Ω which satisfies i), ii), iii) and iv). Proof. These results are well known (see [1], [2], [4], [7]) and we recall only the construction of X. The set X is the zero set of finitely many real analytic functions on Ω that extend to holomorphic functions on some Stein open set Ω, which may be assumed invariant with respect to σ: their zero set is a complex analytic set in Ω which satisfies i). By taking the intersection of all such complex analytic sets in Ω we get X. The complex set X is uniquely defined as a germ at X and in the following we refer to X as the weak complexification of X. It is easy to prove that if X is coherent, then its complexification and its weak complexification coincide as a germs at X. 2. Results and Proofs Theorem 2.1. Let X be a C-analytic and C-irreducible set in an open set Ω of R n of dimension s 2. If X is not pure dimensional, then there exist real analytic functions on X that do not extend to any open neighborhood of X in Ω. Proof. If X is not pure dimensional there exist a connected open set M of X, a point a in the closure of M and t N, with t < s, such that dim R X a = s and dim R X x = t for every x M. The smallest C-analytic set T in Ω that contains M is, by [1], the intersection of all C-analytic sets in Ω that contains T ; trivially M T X and then a T and M x = T x = X x for every x M. Moreover, by the minimality of T and the irreducibility of M, T is C-irreducible. Let σ be the antiinvolution induced by the conjugation of C n ; by Theorem 1.1 there exists a Stein open subset Ω of C n such that Ω R n = Ω and σ( Ω) = Ω, and there exist irreducible complex analytic set T X, which are the weak complexifications of T and X respectively. Let r = dim R T = dim C T : clearly r t. If r > t, T would be contained in the singular set of T : indeed, by Theorem 1.1, a point of T which is regular of dimension r for T would be regular of dimension r for T ; by the minimality of T, we can conclude that r = t. By the same argument there exists a point c M which is regular for T and then T is the complexification of T at c. Let g be the function z n j=1 (z j c j ) 2 on C n and C = {z Ω g(z) = 0}. The dimension at every point of the complex analytic set C X is bigger or equal to s 1, but if an irreducible component S of C X were of dimension s it would be S = X and then X Ω {c}, that is absurd. It follows that C X is a complex analytic set of pure dimension s 1. In the same way one can see that T is not contained in any irreducible component of C X. On the other hand T does not contain any irreducible component S of C X: if t < s 1, this is trivial and if

4 38 ALESSANDRO TANCREDI t = s 1 we would have again the absurd T Ω {c}. In particular T C is a thin analytic subset of T. Since T, X and C are invariant with respect to σ, we can choose a point ζ l on every irreducible component S l of C X in such a way that the set {ζ l, σ(ζ l ) l N} is closed, discrete and does not meet T. By Cartan Theorem B, there exists a holomorphic function f on Ω such that f T = 0 and f(ζ l ) = f(σ(ζ l )) = 1 for every l N. Moreover, since σ( Ω) = Ω, by replacing f with the function z 1 2 ( f(z) + σ( f(σ(z))), we may suppose that f(ω) R. Now let us consider the meromorphic function m = f g on Ω, which is holomorphic on the dense open set D = Ω C. By construction, m X is a not identically zero meromorphic function on X, holomorphic on the dense open set X D = X X C and vanishing on T D. We observe that the same conclusion holds if g is replaced by any power g d, d N. In order to conclude the proof of the Theorem we start by stating the following lemma, which will also be used further ahead. Lemma 2.2. Let Ω be an open neighborhood of X in Ω such that σ( Ω ) = Ω. There exists an irreducible component X of Ω X such that X X and, for every ξ C, there exists a sequence (x ν ) ν N of points in X D such that x ν c and m(x ν ) ξ. Proof. Since T c is irreducible there exists an irreducible component of Xc which contains T c ; it follows that there exist an open neighborhood Ũ of c in Ω and an irreducible component Z of X Ũ such that Z Ũ is irreducible, T Ũ Z Ũ and the germ Z c is irreducible. Moreover we can find a countable neighborhood basis (Ũν) ν N of c in Ω such that Z Ũν is irreducible and T Ũν Z Ũν for every ν N, with Ũ1 = Ũ. For every ν N, C Z Ũ ν is a proper analytic set in Z Ũν; moreover no irreducible component of C Z Ũν contains any irreducible component of T Ũν and no irreducible component of T Ũ ν contains any irreducible component of C Z Ũν. It follows that, for every ν N, m Z Ũ ν is a not identically zero meromorphic function on Z Ũν, holomorphic on Z Ũν D and vanishing on T D Ũν. Let X be the irreducible component of X Ω containing Z and T the irreducible component of T Ω containing T. Since T c = T c Z c X c it follows that T X, and so T X. Moreover, by the C-irreducibility of X, X must contains X too. In order to get the conclusion it is enough to prove that for every ν N the set m( Z Ũν D) is a dense subset of C. By contradiction, let us suppose that there exist ζ C and r R + such that m(x) ζ > r for every x Z Ũν D: then the function µ = 1 m ζ is meromorphic on Z Ũν and bounded holomorphic on Z Ũν D. It follows that µ is weakly holomorphic and then, since Z c is irreducible, µ is continuous at c (see [3]). If µ(c) = 0, since T D is a dense open subset of T, we can choose a sequence of points x l T D Ũν converging to c, and that implies lim ν m(x l ) = 0. If µ(c) 0, by replacing g with a suitable power g d, d N, if necessary, it is not difficult to find a sequence of points x l Z Ũν D converging to c such that f(x l ) 0 for every l N and lim ν m(x l ) =. In

5 A NOTE ON C-ANALYTIC SETS 39 both cases we get a contradiction and then we can conclude that m( Z Ũν D) is a dense subset of C. Now we return to the proof of the Theorem 2.1. Let f = f Ω and g = g Ω. The restriction to Ω {c} of m is the real analytic functions f g that we denote by m. Let λ be the function on X defined by { m(x) if x c λ(x) = 0 if x = c. Since m vanishes on M {c} it is straightforward to see that λ is a real analytic function on X. We want to prove that λ does not extend to any open neighborhood of X in Ω. Let us assume by contradiction that there exist an open neighborhood Ω of X in Ω and a real analytic function h : Ω R such that h X = λ. Now h extends to a holomorphic function on an open set Ω of C n such that σ( Ω ) = Ω and Ω R n = Ω ; moreover we may assume that Ω Ω. Let X be the irreducible component of X Ω as in Lemma 2.2. By Theorem 1.1 there exists a regular point b of X of dimension s, arbitrarily close to a, such that X is the complexification of X at b and m is holomorphic at b, since m is holomorphic in a neighborhood of a. The function m h vanishes on X b and then it vanishes on X b too. It follows that m X = h X which is impossible by Lemma 2.2. Theorem 2.3. Let X be a C-analytic and C-irreducible set in an open set Ω of R n. If X is not pure dimensional, then it is isomorphic to a real analytic set in R n+1 which is not C-analytic. Proof. We use the notations of the previous theorem. Let φ : X R n+1 be the real analytic map x (x, λ(x)) and let Y = φ(x); Y is a closed analytic set in Ω R isomorphic to X. Let Z = {(x 1,..., x n+1 ) X R g(x 1,..., x n )x n+1 = f(x 1,..., x n )} and L = {(x 1,..., x n+1 ) Z x 1 = c 1,..., x n = c n }. Clearly Z is a C-analytic set in Ω R and it is easy to check that Z = Y L and Y L = {(c, 0)}. Let A be an open neighborhood of Y in R n+1 and h : A R a real analytic function such that h X = 0. We want to prove that h vanishes on L too and then that Y is not C-analytic. Let π : R n+1 R n and π : C n+1 C n be the canonical projections. We may assume that A π 1 (Ω). Then there exist an open set à of Cn+1 such that à π 1 (Ω) = A and a holomorphic function h : à C such that h A = h. We may assume that à is invariant with respect to the conjugation of C n+1 and that à π 1 ( Ω); then π(ã) is invariant with respect to the conjugation of C n and π(ã) Ω. Let Ω = π(ã); clearly π(a) Ω Ω and X {c} Ω D. Let X be an irreducible component of Ω X, as in Lemma 2.2. For every (c, ξ) L there exists a sequence of points (x ν ) ν N in X D such that x ν c and m(x ν ) ξ. If ξ is small enough we may suppose that (c, ξ) A and then that (x ν, ξ) is in à for every ν N. Since the function h vanishes on a neighborhood of the point (a, λ(a)) = (a, m(a)) in Y, then the function x h(x, m(x)) vanishes on the real

6 40 ALESSANDRO TANCREDI analytic germ X a. Since m is holomorphic in a neighbourhood of a, by Theorem 1.1 it follows that the function z h(z, m(z)) vanishes on some complex analytic germ X b, where b is a point of X, arbitrarily close to a, which is regular for X. By the irreducibility of X we have h(x ν, m(x ν )) = 0 for every ν N. Therefore, by the choice of ξ, we can conclude that h vanishes on L and then that Y is not C-analytic. References 1. F. Bruhat, H. Whitney, Quelques propriétés fondamentales des ensembles analytiques réels, Comm. Math. Helv. 33 (1959), H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), L. Kaup, B. Kaup, Holomorphic Functions of Several Variables, Berlin-New York: de Gruyter R. Narashiman, Introduction to the Theory of Analytic Spaces, Lecture Notes in Mathematics 25. Berlin-Heidelberg-New York: Springer G. Nardelli, A. Tancredi, A note on the extension of analytic functions off real analytic subsets, Revista Matemática de la Universidad Complutense de Madrid, 9 (1996), A. Tancredi, Un osservazione sui sottoinsiemi C-analitici, Ann. Univ. Ferrara 29 (1983), A. Tognoli, Proprietà globali degli spazi analitici reali, Ann. Mat. pura e Appl. 75 (1967), A. Tognoli, Pathology and Imbedding Problems for Real Analytic Spaces, Singularities of analytic spaces, Roma: Cremonese 1975 Alessandro Tancredi Dipartimento di Matematica e Informatica Università di Perugia Via Vanvitelli Perugia (PG) Italy altan@unipg.it