Pluripolar hulls and fine analytic structure 8 0 0 Tomas Edlund and Said El Marzguioui 2 n a February 1, 2008 J 0 3 ] V C Abstract . h t a We discuss the relation between pluripolar hulls and fine analytic structure. Our m main result is the following. For each non polar subset S of the complex plane C [ we prove that there exists a pluripolar set E ⊂ (S × C) with the property that the 2 v pluripolar hull of E relative to C2 contains no fineanalytic structure and its projection 2 0 onto the first coordinate plane equals C. 1 2 . 9 0 1 Introduction 7 0 : v Xi Denote by Ω an open subset of Cn and let E ⊂ Ω be a pluripolar subset. It might be r the case that any plurisubharmonic function u(z) defined in Ω that is equal to −∞ on a the set E is necessarily equal to −∞ on a strictly larger set. For instance, if E contains a non polar proper subset of a connected Riemann surface embedded into Cn, then any plurisubharmonic function defined in a neighborhood of the Riemann surface which is equal to −∞ on E is automatically equal to −∞ on the whole Riemann surface. In order to try to understand some aspect of the underlying mechanism of the described ”propagation” property of pluripolar sets, the pluripolar hull of graphs Γ (D) of analytic functions f in a f domain D ⊂ C has been studied in a number of papers. (See for instance [2], [5], [10] and [14].) 2000 Mathematics Subject Classification 31C40, 32U15, 32E20 1 The pluripolar hull E∗ relative to Ω of a pluripolar set E is defined as follows. Ω E∗ = {z ∈ Ω : u(z) = −∞}, Ω \ where the intersection is taken over all plurisubharmonic functions defined in Ω which are equal to −∞ on E. The set E is called complete pluripolar in Ω if there exists a plurisub- harmonic function on Ω which equals −∞ precisely on E. As remarked above a necessary condition for a pluripolar set E to satisfy E∗ = E is that Ω E∩A is polar in A (or E∩A = A) for all one-dimensional complex analytic varieties A ⊂ Ω. The fact that this is not a sufficient condition was proved by Levenberg in [8]. By using a refinement of Wermer’s example of a polynomially convex compact set with no analytic structure (cf. [13]) Levenberg proved that there exists a compact set K ⊂ C2 satisfying K 6= K∗ , and the intersection of K with any one dimensional analytic variety A is polar in C2 A. In this example it is not clear what the pluripolar hull K∗ equals. C2 We will say that a set S ⊂ Cn contains fine analytic structure if there exists a non constant map ϕ : U → S from a fine domain U ⊂ C whose coordinate functions are finely holomorphic in U (see Definition 2.3 below). Such a map ϕ will be called a fine analytic curve. Motivated by recent results of J¨oricke and the first author (cf. [5]), the following result was proved in [3]. Theorem 1.1 Let ϕ : U −→ Cn be a finely holomorphic map on a fine domain U ⊂ C and let E ⊂ Cn be a pluripolar set. Then the following hold (1) ϕ(U) is a pluripolar subset of Cn (2) If ϕ−1(ϕ(U)∩E) is a non polar subset of C then ϕ(U) ⊂ E∗ . Cn In view of this result one may expect to get more information on the pluripolar hull E∗ Cn by examining the intersection of the pluripolar set E with fine analytic curves. Since many curves in Cn are complete pluripolar (see [4]) one cannot expect that E∗ always contains Cn fine analytic structure. However if we consider the non trivial part E∗ r E the situation Cn is up to now slightly different. In fact, all examples we have seen so far have the property that if E∗ rE is nonempty then for each w ∈ E∗ rE there exists a finely analytic curve Cn Cn ϕ contained in E∗ which passes through the point w. (i.e. ϕ : U → E∗ is a finely analytic Cn C2 2 curve and ϕ(z) = w for some z ∈ U). In this paper we prove that no such conclusion holds in general. We have the following main result. Theorem 1.2 For each proper non polar subset S ⊂ C there exists a pluripolar set E ⊂ (S ×C) with the property that E∗ contains no fine analytic structure and the projection of C2 E∗ onto the first coordinate plane equals C. C2 The set E will be a subset of a complete pluripolar set X which is constructed in the same spirit as Wermer’s polynomially convex compact set without analytic structure. Let us describe more precisely the content of the paper. In Section 2 we briefly recall the construction of Wermer’s set and prove that it contains no fine analytic structure. This leads to Theorem 2.4 which slightly generalizes a result in [8]. The main result is proved in Section 3. Subsection 3.1 is devoted to construct the above mentioned set X and in Subsection 3.2 we show that X contains no fine analytic structure. In Subsection 3.3 we define the set E and describe E∗ . Finally, in Section 4 we make some remarks and pose two open questions. C2 Readers who are not familiar with basic results on finely holomorphic functions and fine potential theory are referred to [6] and [7]. Acknowledgments. Part of this work was completed while the first author was visiting Korteweg-de Vries Institute for Mathematics, University of Amsterdam. He would like to thank this institution for its hospitality. The authors thank Professor Jan Wiegerinck for very helpful discussions. 2 Wermer’s example In this Section we sketch the details of Wermer’s construction given in [13]. Denote by D r the open disk with center zero and radius r and by C the opencylinder D ×C. Let a ,a ,.... r r 1 2 denote the points in the disk D with rational real and imaginary part. For each j we denote 1 2 by B (z) the algebraic (2-valued) function j B (z) = (z −a )(z −a )...(z −a ) (z −a ). j 1 2 j−1 q j To each n-tuple of positive constants c ,c ,...,c we associate the algebraic (2n-valued) func- 1 2 n tion g (z) = n c B (z). Let (c ,...,c ), n = 1,2,... be the subset of the Riemann n Pj=1 j j P 1 n surface of g (z) which lies in C . n 1 2 3 Lemma 2.1 [[13], lemma 1] There exist positive constants c ,c ,..., with c = 1 and c ≤ 1 2 1 10 n+1 ( 1 )c , n = 1,2,... and a sequence of polynomials {p (z,w)} such that: 10 n n (1) {p = 0}∩{|z| ≤ 1} = (c ,...,c ), n = 1,2,... n 2 P 1 n (2) {|p | ≤ ε }∩{|z| ≤ 1} ⊂ {|p | ≤ ε }∩{|z| ≤ 1}, n = 1,2,... n+1 n+1 2 n n 2 (3) If |a| ≤ 1 and |p (a,w)| ≤ ε , then there is a w with p (a,w ) = 0 and |w−w | ≤ 1, 2 n n n n n n n n = 1,2,.... With p , ε , n = 1,2,... chosen as in Lemma 2.1, we put n n ∞ 1 Y = [{|p | ≤ ε }∩{|z| ≤ }]. \ n n 2 n=1 Clearly, Y is a compact polynomially convex subset of C2. It was shown by Wermer that Y has no analytic structure i.e. Y contains no non-constant analytic disk. In fact he proves something stronger. The set Y defined above contains no graph of a continuous function defined on a circle in D which avoids all the branch points {a }. Using this observation the 1 i 2 following lemma follows. Lemma 2.2 There is no fine analytic curve contained in Y. Before we prove Lemma 2.2 we recall the following definition (cf. [7], page 75): Definition 2.3 Let U be a finely open set in C. A function f : U −→ C is said to be finely holomorphic if every point of U has a compact (in the usual topology) fine neighbourhood K ⊂ U such that the restriction f | belongs to R(K). K Here R(K) denotes the uniform closure of the algebra of all restrictions to K of rational functions on C with poles off K. Proof of Lemma 2.2. Let ϕ : U → Y, z 7→ (ϕ (z),ϕ (z)) be a fine analytic curve con- 1 2 tained in Y. If ϕ (z) is constant on U then ϕ (z) must also be a constant since non constant 1 2 finely holomorphic functions are finely open maps and by the construction of the set Y the fibre Y ∩({z}×C) is a Cantor set or a finite set for any point z ∈ D . Assume therefore 1/2 that ϕ (z) is non-constant. In particular, there is a point z ∈ U where the fine deriva- 1 0 tive of ϕ (z) does not vanish. Hence ϕ (z) is one-to-one on some finely open neighborhood 1 1 V ⊂ U of the point z . By considering the map z 7→ (ϕ ◦ ϕ−1(z),ϕ ◦ ϕ−1(z)), defined 0 1 1 2 1 on the finely open set ϕ (V) we may assume that ϕ is of the form z 7→ (z,g(z)) where 1 4 g(z) = ϕ ◦ ϕ−1(z) is finely holomorphic in the finely open set V′ = ϕ (V) ⊂ D . By 2 1 1 1/2 Definition 2.3 there exists a compact subset K ⊂ V′ with non-empty fine interior such that g(z) is a continuous function on K (with respect to the Euclidean topology). Shrinking K if necessary we may assume that K ∩{a ,a ,....} = ∅. Let p be a point in the fine interior 1 2 of K. It is well known that there exists a sequence of circles {C(p,r )} contained in K j with centers p and radii r → 0 as j → ∞. Clearly, the circle C(p,r ) avoids the branch j j points {a ,a ,....} and its image under the continuous map z 7→ (z,g(z)) is contained in Y. 1 2 By the above observation this is not possible. Hence Y contains no fine analytic structure. (cid:3) Denote by d the degree of the one variable polynomial w 7→ p (z,w) where p (z,w) n n n is the polynomial given in Lemma 2.1. Assume that the set Y is constructed using the parameters ǫ satisfying the following condition n lim(ǫ )1/dn = 0. (1) n n→∞ It is shown in [9] that with this choice the set Y ∩C is complete pluripolar in C . Using 1/2 1/2 this result and Lemma 2.2 we are able to generalize a result in [8]. Theorem 2.4 Fix δ ∈ (0,1/2) and let Y = ∞ [{|p | ≤ ε } ∩ {|z| ≤ δ}] be constructed δ Tn=1 n n using the parameters ε satisfying (1). Then n (a) ϕ−1(ϕ(U)∩Y ) is a polar subset of U for all fine analytic curves ϕ : U → C2. δ (b) Y 6= (Y )∗ . δ δ C2 Proof of Theorem 2.4. In order to prove (a) we argue by contradiction. Assume therefore that ϕ : U → C2 is a fine analytic curve and ϕ−1(ϕ(U)∩Y ) is a non polar subset of U. Then δ there is a fine domain U ⊆ U such that ϕ(U ) ⊂ C and ϕ−1(ϕ(U )∩Y ) is non polar. k0 k0 1/2 k0 δ Indeed, the set ϕ−1(ϕ(U)∩C ) is a finely open subset of U and hence has at most countably 1/2 many finely connected components {U }∞ . Moreover, ϕ−1(ϕ(U) ∩Y ) ∩ U is non polar k k=1 δ k0 for some natural number k , since otherwise ∞ {ϕ−1(ϕ(U)∩Y )∩U } = ϕ−1(ϕ(U)∩Y ) 0 Sk=1 δ k δ would be polar contrary to our assumption. Since Y ∩ C is complete pluripolar in C 1/2 1/2 there exists a plurisubharmonic function u defined in C which is equal to −∞ exactly on 1/2 Y ∩C . The function u◦ϕ is either finely subharmonic on U or identically equal to −∞ 1/2 k0 (cf. [3], Lemma 3.1). Since u equals −∞ on the non polar subset ϕ−1(ϕ(U) ∩ Y ) ∩ U , δ k0 it must be identically equal to −∞ on U . Therefore ϕ(U ) ⊂ {u = −∞} = Y ∩ C k0 k0 1/2 contradicting Lemma 2.2 and (a) follows. 5 The proof of assertion (b) follows immediately from the proof of Proposition 3.1 in [8]. Indeed, if u is a plurisubharmonic function defined in C2 which equals −∞ on Y then the δ function z 7→ max{u(z,w) : (z,w) ∈ Y} is subharmonic in D and since it equals −∞ on 1/2 D it equals −∞ on D . Consequently Y ∩C ⊂ (Y )∗ and hence Y 6= (Y )∗ . (cid:3) δ 1/2 1/2 δ C2 δ δ C2 Remark. It follows from the argument used in the proof of assertion (b) in Theorem 2.4 that Y ∩C ⊂ (Y )∗ . Since the first set is complete pluripolar in C it follows that 1/2 δ C 1/2 1/2 (Y )∗ = Y ∩C . Consequently, (Y )∗ contains no fine analytic structure. It would be δ C 1/2 δ C 1/2 1/2 nice to determine what the set (Y )∗ equals and to figure out whether this set contains fine δ C2 analytic structure. We are unable to do this. But by modifying Wermer’s construction, we will in the next Section construct a complete pluripolar Wermer-like set X ⊂ C2 with the property that (X ∩ (S × C))∗ contains no fine analytic structure for all non polar subset C2 S ⊂ C. 3 Proof of Theorem 1.2 3.1 Construction of the set X In this Subsection we construct the set X. Denote by {a }∞ the points in the complex k k=1 plane both of whose coordinates are rational numbers. Without loss of generality we may assume that a ∈ D . For any sequence of points {a }j we denote by B (z) the algebraic k k l l=1 j function B (z) = (z −a )...(z −a ) (z −a ). j 1 j−1 q j Denote by γ a simple smooth curve with endpoints a and ∞. For each j B (z) has two j j j single-valued analytic branches on C rγ . Following the notation in [13] we choose one of j the branches B (z) arbitrarily and denote it by β (z). Then |β (z)| = |B (z)| is continuous j j j j on C. Foreachn+1-tupleofpositiveconstants(c ,c ,...,c )wedenotebyg (z)thealgebraic 1 2 n+1 n function defined recursively in the following way. Put g (z) = c B (z) and g (z) = c B (z)+ 1 1 1 2 1 1 c B (z)andifg (z)hasbeenchosenwewillchooseg (z)asdescribedbelow. PutZ (z) = 1 2 2 n n+1 1 and for n = 2,3,... define the function Z (z) as follows. Denote by z ,z ,...,z all the zeros n 1 2 l 6 ofallpossible different differences h (z)−h (z) (i 6= j) ofbranches h (z),h (z) ofthefunction j i i j g (z). Suppose z is a zero of h (z) − h (z) of order m and put Z (z) = Πl (z − z )mi. n k j i k n i=1 i Note that the zeros of Z (z) are also zeros of the function Z (z) of the same or greater n n+1 multiplicity. Define g (z) = g (z)+c Z (z)B (z). n+1 n n+1 n n+1 By Σ(c ,...,c ) we mean the Riemann surface of g (z) which lies in C2. In other words, 1 n n Σ(c ,...,c ) = {(z,w) : z ∈ C,w = w ,j = 1,2,...,2n}, where w , j = 1,2,...,2n are the 1 n j j values of g (z) at z. n We will choose positive constants c , ǫ and polynomials p (z,w) recursively so that n n n {p (z,w) = 0}∩C = Σ(c ,c ,...,c )∩C and (2) n n+1 1 2 n n+1 {|p (z,w)| ≤ ǫ }∩C ⊂ {|p (z,w)| < ǫ }∩C (3) n+1 n+1 n+1 n n n+1 hold for n = 1,2,.... The set X will be of the form ∞ ∞ X = {|p (z,w)| ≤ ǫ }∩C . (4) [ (cid:16) \ j j n+1(cid:17) n=1 j=n Put c = 1andletp (z,w) = w2−(z−a ). ItisclearthatΣ(c )∩C = {p (z,w) = 0}∩C . 1 1 1 1 2 1 2 Choose ǫ > 0 so that if z ∈ D and |p (z ,w)| ≤ ǫ then there exists (z ,w ) ∈ Σ(c )∩C 1 0 2 1 0 1 0 1 1 2 with |w−w | ≤ 1. Let B = D ×D be a bidisk where ρ is chosen so that 1 2 2 ρ1 1 {|p (z,w)| ≤ ǫ }∩C = {|p (z,w)| ≤ ǫ }∩B . 1 1 2 1 1 2 Assume that c ,ǫ and p (z,w) have been chosen so that (2) and (3) hold. We will now n n n choose c and p (z,w). We denote by w (z), j = 1,2,...,2n the roots of p (z,·) = 0 n+1 n+1 j n and to each positive constant c we assign a polynomial p (z,w) by putting c p (z,w) = Π2n (w−w (z))2 −c2(Z (z)B (z))2 . (5) c j=1(cid:16) j n n+1 (cid:17) Then p (z,·) = 0 has the roots w (z)±cZ (z)B (z), j = 1,2,...,2n and so c j n n+1 {p (z,w) = 0} = Σ(c ,c ,...,c ,c). c 1 2 n Note that from (5) p = p2 +c2q +...+(c2)2nq , c n 1 2n where the q are polynomials in z and w, not depending on c. Choose c > 0 so that j 7 Σ(c ,c ,...,c ,c)∩C ⊂ {|p (z,w)| < ǫ /2}∩C and (6) 1 2 n n+1 n n n+1 c·|Z (z)B (z)| ≤ (1/10)c |Z (z)B (z)| holds for all z ∈ D . (7) n n+1 n n−1 n n+1 Decreasing c if necessary we may assume thatif h (z) andh (z) areany different branches i j of the function g (z) the estimate n |h (z)−h (z)| ≥ 2c|Z (z)B (z)| (8) j i n n+1 holds in D with equality exactly at the zeros of Z (z) which are contained in D and n+1 n n+1 at the points a ,...a . This estimate will be needed later when we prove that X contains 1 n no fine analytic structure. Choose c = c. n+1 Let B = D × D be a bidisk where ρ is chosen so that {|p (z,w)| ≤ ǫ } ∩ n+2 n+2 ρn+2 n+2 n n C = {|p (z,w)| ≤ ǫ } ∩ B and ρ > ρ + 1. Let δ > 0 be a constant such that n+2 n n n+2 n+2 n+1 |δ ·p (z,w)| < 1 in B and choose p (z,w) = δ ·p (z,w). c n+2 n+1 c We now turn to the choice of ǫ . Since the part of the zero set of p (z,w) which is n+1 n+1 contained in B is a subset of {|p (z,w)| < ǫ /2} ∩ B it is possible to find a natural n+1 n n n+1 number m so that n+1 1 1 log|p (z,w)| ≥ − for all (z,w) ∈ B r{|p (z,w)| ≤ ǫ }. (9) m n+1 2n n+1 n n n+1 Choose ǫ < ǫ so that n+1 n 1 log|p (z,w)| ≤ −1 for all (z,w) ∈ {|p (z,w)| ≤ ǫ }∩C . (10) n+1 n+1 n+1 n+2 m n+1 By decreasing ǫ we may assume that (3) and the following assumption hold. n+1 If (z ,w) ∈ C and |p (z ,w)| ≤ ǫ , then there exists 0 n+2 n+1 0 n+1 (11) (z ,w ) ∈ C such that |p (z ,w )| = 0 and |w−w | ≤ 1/n. 0 n n+2 n+1 0 n n This ends the recursion. Lemma 3.1 The set X defined by (4) is complete pluripolar in C2. Proof. Define for n ≥ 2 the plurisubharmonic function 1 u (z,w) = max log|p (z,w)|,−1 n n (cid:8)m (cid:9) n 8 and put u(z,w) = u (z,w). Then u(z,w) is plurisubharmonic in C2. Indeed, since the Pn≥2 n bidisks B exhaust C2 and |p (z,w)| < 1 in B the series u (z,w) will be decreasing n n n+1 Pn≥2 n on each fixed bidisk B after a finite number of terms and hence plurisubharmonic there. N Since plurisubharmonicity is a local property u(z,w) is plurisubharmonic in C2. If (z ,w ) ∈ 0 0 ∞ X, then for some natural number N, (z ,w ) ∈ {|p (z,w)| ≤ ǫ }∩C . Condition (10) 0 0 Tj=N j j N+1 above implies that u(z ,w ) = Const+ u (z ,w ) = −∞. Finally if (z ,w ) ∈/ X then 0 0 Pn>N n 0 0 0 0 thereexistsanaturalnumberN suchthat(z ,w ) ∈ B and(z ,w ) ∈/ {|p (z,w)| ≤ ǫ }∩B 0 0 N 0 0 n n N for all n ≥ N. By (9) 1 1 u(z,w) = Const+ max log|p (z,w)|,−1 ≥ Const+ − > −∞. X (cid:8)m n (cid:9) X 2n n n>N n>N (cid:3) The Lemma follows. 3.2 X contains no fine analytic structure In this Section we show that X contains no fine analytic structure. Suppose that z 7→ (ϕ (z),ϕ (z)) is a fine analytic curve whose image is contained in X. If ϕ (z) is constant 1 2 1 then ϕ (z) must be constant since X∩({z }×C) is a Cantor set or a finite set for any point 2 0 z ∈ C. On the other hand, if ϕ (z) is non-constant, then using the arguments given in 0 1 the proof of Lemma 2.2 we may assume that the fine analytic curve contained in X is given by z 7→ (z,m(z)) where m(z) is a finely holomorphic function defined in U where U ⊂ D n for some natural number n. Fix a point z′ ∈ U r{a ,...,a } . By the definition of finely 1 n holomorphic functions we can find a compact (in the usual topology) fine neighborhood K ⊂ U of z′ where m(z) is continuous. Shrinking K if necessary we may assume that (K r {z′}) ∩ ({a }∞ ∪ {Z (z) = 0}∞ ) = ∅. Since the complement of K is thin at z′, j j=1 k−1 k=2 one can find a sequence of circles {C(z′,r )} ⊂ K with r → 0 as i → ∞. Choose one of the i i circles C(z′,r ) so that none of the points a ,...,a are contained in {|z−z′| ≤ r }. Let a j 1 n j k be the first point in the sequence {a }∞ which is contained in {|z−z′| ≤ r }. Note that j j=n+1 j a ∈ {|z−z′| < r }, m(z) is continuous on C(z′,r ) and the function Z (z)β (z) 6= 0 when k j j k−1 k z ∈ C(z′,r ). The fact that the image of C(z′,r ) under the map z 7→ (z,m(z)) is a subset of j j X will lead us to a contradiction and hence X contains no fine analytic structure. In order to prove this fix a point z ∈ C(z′,r ) and denote by ℜ the 2k branches of the algebraic 1 j function g (z) defined on C(z′,r )r{z }. k j 1 9 Lemma 3.2 If h (z) and h (z) are any different functions from ℜ then i j |h (z)−h (z)| > (3/2)c |Z (z)β (z)| (12) i j k k−1 k holds for all z ∈ C(z′,r )r{z }. j 1 Proof. This is follows directly from (8) since C(z′,r ) ⊂ D and C(z′,r ) does not j n j intersect any of the branch points a ,...,a or the zeros of Z (z). (cid:3) 1 k k−1 From now on the proof that X contains no fine analytic structure follows the arguments given in [13]. Lemma 3.3 Fix z in C(z′,r ) r {z }. There exists a function h (z) ∈ ℜ, where h (z) 0 j 1 i i depends on z such that 0 |m(z )−h (z )| < (1/4)c |Z (z )β (z )| (13) 0 i 0 k k−1 0 k 0 Proof. By (11) there exists N ≥ k and w such that (z ,w ) lies on Σ(c ,...,c ) and N 0 N 1 N m(z ) = w +R(z ) where |R(z )| ≤ (1/10)c |Z (z )β (z )|. Thus 0 N 0 0 k k−1 0 k 0 N m(z ) = ±c β (z )+ ±c Z (z )β (z )+R(z ) = 0 1 1 0 X ν ν−1 0 ν 0 0 ν=2 N def = h (z )+ c Z (z )β (z )+R(z ). i 0 X ν ν−1 0 ν 0 0 ν=k+1 Since C(z′,r ) ⊂ D and the constants c are chosen so that (7) holds, j n+1 ν N |m(z )−h (z )| ≤ c |Z (z )β (z )|+|R(z )| ≤ 0 i 0 X ν ν−1 0 ν 0 0 ν=k+1 1 1 ≤ c |Z (z )β (z )|( + +...)+|R(z )| = k k−1 0 k 0 10 102 0 1 1 = c |Z (z )β (z )|+ c |Z (z )β (z )| < k k−1 0 k 0 k k−1 0 k 0 9 10 < (1/4)c |Z (z )β (z )|. k k−1 0 k 0 (cid:3) Hence (13) holds and the Lemma is proved. 10