The Poletsky-Rosay theorem on singular complex spaces
aa r X i v : . [ m a t h . C V ] A ug THE POLETSKY-ROSAY THEOREM ON SINGULARCOMPLEX SPACES
BARBARA DRINOVEC DRNOVˇSEK & FRANC FORSTNERI ˇC
Abstract.
In this paper we extend the Poletsky-Rosay theorem, con-cerning plurisubharmonicity of the Poisson envelope of an upper semi-continuous function, to locally irreducible complex spaces. Introduction
Plurisubharmonic functions were introduced by K. Oka [Oka] and P. Le-long [Lel] in 1942; ever since then they have been playing a major role incomplex analysis. The minimum of two plurisubharmonic functions is notplurisubharmonic in general. There has been a considerable amount of inter-est in studying situations where the infimum actually is plurisubharmonic.The first major result of this kind was Kiselman’s minimum principle [Kis].In the early 1990’s E. Poletsky [Po1, Po2] found a novel way of construct-ing plurisubharmonic functions as pointwise infima of upper semicontinuousfunctions. Set D = { ζ ∈ C : | ζ | < } and T = b D = { ζ ∈ C : | ζ | = 1 } . Let A ( D , X ) denote the set of all analytic discs in a complex space X , that is,continuous maps D → X that are holomorphic in D . Then for x ∈ X let A ( D , X, x ) = { f ∈ A ( D , X ) : f (0) = x } . Our main result is the following. Theorem 1.1.
Let ( X, O X ) be an irreducible and locally irreducible (re-duced, paracompact) complex space, and let u : X → R ∪ {−∞} be an uppersemicontinuous function on X . Then the function (1) b u ( x ) = inf n Z π u ( f (e i t )) d t π : f ∈ A ( D , X, x ) o , x ∈ X is plurisubharmonic on X or identically −∞ ; moreover, b u is the supremumof the plurisubharmonic functions on X which are not greater than u . In the basic case when X = C n this was proved by Poletsky [Po1, Po2] andby Bu and Schachermayer [BS]. The result was extended to some complexmanifolds (and to certain other disc functionals) by L´arusson and Sigurdsson[LS1, LS2, LS3], and to all complex manifolds by Rosay [Ro1, Ro2]. In Date : November 14, 2018.1991
Mathematics Subject Classification.
Primary 32U05; Secondary 32H02, 32E10.
Key words and phrases.
Complex spaces, Stein space, plurisubharmonic function, discfunctional.Research supported by grants P1-0291 and J1-2152, Republic of Slovenia. this paper we use the method of gluing sprays, developed in [DF, For], togive a new proof which applies to any normal complex space X . For alocally irreducible space the result follows easily by using seminormalizationto reduce to the case of a normal space.A reduction to the Poletsky-Rosay theorem on manifolds is also possibleby appealing to the Hironaka desingularization theorem, but this approachseems unreasonable since our proof is no more difficult than the originalproofs given for manifolds. Example 1.2.
The theorem fails in general if X is not irreducible; here isa trivial example. Let X = { ( z, w ) ∈ C : zw = 0 } be the union of twocomplex lines. Let u : X → R be defined by u ( z,
0) = 0 for z ∈ C ∗ and u (0 , w ) = 1 for all w ∈ C . Clearly u is upper semicontinuous. We have b u ( z,
0) = 0 for z ∈ C and b u (0 , w ) = 1 for w ∈ C ∗ ; hence b u fails to be uppersemicontinuos at the point (0 , X is the Riemann sphere withone simple double point, then the above example does not work since theboundaries of analytic discs through any of the two local branches at thedouble point also reach the other branch. (cid:3) The operator P u : A ( D , X ) → R ∪ {−∞} appearing in (1), P u ( f ) = Z π u ( f (e i t )) d t π , f ∈ A ( D , X )is called the Poisson functional . Note that P u ( f ) is the value at 0 ∈ D ofthe harmonic function on D with boundary values u ( f (e i t )). For this reason,the function b u defined by (1) is also called the Poisson envelope of u .Let us first justify the last statement in the theorem. Recall that an uppersemicontinuous function v : X → R ∪ {−∞} on a complex space that is notidentically −∞ on any irreducible component of X is plurisubharmonic ifevery point x ∈ X admits a neighborhood U ⊂ X , embedded as a closedcomplex subvariety in a domain Ω ⊂ C N , such that v | U is the restriction to U of a plurisubharmonic function e v on Ω. By [FN] a function v as above isplurisubharmonic if and only if the composition v ◦ f with any holomorphicdisc f : D → X is subharmonic on D . This holds if and only if v satisfiesthe submeanvalue property on analytic discs, which precisely means that v ( x ) ≤ P v ( f ) for every x ∈ X and f ∈ A ( D , X, x ). Since for the constantdisc f ( ζ ) = x we have P v ( f ) = v ( x ), we conclude that Lemma 1.3.
An upper semicontinuous function v : X → R ∪ {−∞} on acomplex space X that is not identically −∞ on any irreducible componentof X is plurisubharmonic if and only if v = b v , where b v is defined by (1). If v ≤ u then clearly P v ( f ) ≤ P u ( f ) for every f ∈ A ( D , X ), and hence b v ≤ b u . It follows that any plurisubharmonic function v for which v ≤ u satisfies v = b v ≤ b u ≤ u , so b u is indeed the largest such function. HE POLETSKY-ROSAY THEOREM ON SINGULAR COMPLEX SPACES 3
One of the main applications of Theorem 1.1 in the classical case X = C n is the characterization of the polynomially convex hull b K of a compact set K ⊂ C n by analytic discs, due to Poletsky [Po1, Po2] and Bu and Schacher-mayer [BS]. (See also Remark 1.5 below.) In the situation considered inthis paper we obtain a characterization of plurisubharmonic hulls in termsof analytic discs, a fact that was already observed (for complex manifolds)by L´arusson and Sigurdsson and by Rosay. Let Psh( X ) denote the set ofall plurisubharmonic functions on X . Given a compact set K in X , itsplurisubharmonic hull is defined by b K Psh( X ) = (cid:8) x ∈ X : u ( x ) ≤ sup K u, ∀ u ∈ Psh( X ) (cid:9) . Since the modulus | f | of a holomorphic function is a plurisubharmonic func-tion, we always have b K Psh( X ) ⊂ b K O ( X ) . It is a much deeper result of Grauert[Gra] and Narasimhan [Nar] that the two hulls coincide if X is a Stein spaceor, more generally, a 1-convex complex space. (See also the paper [FN] and[GR2].) In the case when X = C n , the equality of the two hulls also followsfrom Poletsky’s theorem as was pointed out in [Po3, Theorem 5.1]. Relatedresults concerning pluripolar hulls of compact sets in C n were obtained in[LP]. Corollary 1.4.
Let K be a compact set in a locally irreducible complex space X such that b K Psh( X ) is compact. Choose an open set V ⋐ X containing b K Psh( X ) . Then a point x ∈ X belongs to b K Psh( X ) if and only if for everyopen set U ⊃ K and every number ǫ > there exist a disc f ∈ A ( D , V ) anda set E f ⊂ [0 , π ] of Lebesgue measure | E f | < ǫ such that f (0) = x and f (e i t ) ∈ U for all t ∈ [0 , π ] \ E f . Proof.
Assume first that a point x ∈ X satisfies the stated conditions; weshall prove that x ∈ b K Psh( X ) . Choose a function ρ ∈ Psh( X ). Set M =sup K ρ and M ′ = sup V ρ . Pick a number ǫ > U with K ⊂ U ⋐ V such that sup U ρ < M + ǫ . Let the disc f ∈ A ( D , V, x ) and theset E f ⊂ [0 , π ] satisfy the hypotheses of the corollary. Then ρ ( x ) ≤ P ρ ( f ) = Z E f ρ ( f (e i t )) d t π + Z [0 , π ] \ E f ρ ( f (e i t )) d t π < M ′ ǫ + M + ǫ. Since this holds for every ǫ >
0, we get ρ ( x ) ≤ M . As ρ ∈ Psh( X ) wasarbitrary, we conclude that x ∈ b K Psh( X ) .Conversely, assume that x ∈ b K Psh( X ) . The function u : V → R whichequals − U ⊃ K and equals 0 on V \ U is upper semicon-tinuous. Let v = b u be the associated plurisubharmonic function defined by(1). Then − ≤ v ≤ V , and v ( x ) = −
1. Theorem 1.1 furnishes a disc f ∈ A ( D , V, x ) with P u ( f ) < − ǫ/ π . By the definition of u this impliesthat the set E f = { t ∈ [0 , π ] : f (e i t ) / ∈ U } has measure at most ǫ . (cid:3) BARBARA DRINOVEC DRNOVˇSEK & FRANC FORSTNERIˇC
Remark 1.5.
Let K be a compact subset of C n . It was shown by Duvaland Sibony [DS1, DS2] that for any point p ∈ b K and Jensen measure σ representing p there exists a positive current T of bidimension (1 ,
1) suchthat dd c T = σ − δ p , where δ p denotes the point evaluation at p . RecentlyWold [Wol] showed that every Duval-Sibony current T is a weak limit ofcurrents T j = ( f j ) ∗ G , where G is the Green current on the unit disc D ,given by G ( ω ) = − Z D log | ζ |· ω, ω ∈ E , ( D ) , and f j : D → C n is a sequence of Poletsky discs. On the other hand, ithas been known since the classical examples of Stolzenberg and Alexanderthat the polynomial hull of K can not be explained in general by analyticvarieties with boundaries in K . (In this direction see the recent paper ofDujardin [Duj].) Hence Poletsky’s characterization of the polynomial hullremains the most universal one that we have at the moment. For more abouthulls we refer the interested reader to Stout’s monographs [St1, St2]. (cid:3) Remark 1.6.
Another immediate implication of Theorem 1.1 is the fol-lowing result which was observed in the smooth case by J.-P. Rosay [Ro1,Corollary 0.2]: If X is as in Theorem 1.1 and if every bounded plurisub-harmonic function on X is constant, then for every point p ∈ X , nonemptyopen set U ⊂ X and number ǫ > f : D → X such that f (0) = p and the set { t ∈ [0 , π ) : f (e i t ) ∈ U } has measure atleast 2 π − ǫ . This follows by observing that the envelope b u defined by (1) ofthe negative characteristic function u = − χ U of the set U is bounded fromabove by 0, and hence it is constantly equal to − X does not admit any nonconstant bounded plurisubharmonic function (suchspace is said to be Liouville) if and only if every closed loop in X can beapproximated on a set of almost full linear measure in the circle by theboundary values of holomorphic discs in X . We hope to return to thesequestions in a future publication. (cid:3) A nonlinear Cousin-I problem
We recall from [DF, For] the relevant results concerning holomorphicsprays, adjusting them to the applications in this paper.
Definition 2.1.
Let ℓ ≥ r ∈ { , . . . , ℓ } be integers. Assume that X is a complex space, D is a relatively compact domain with C ℓ boundary in C , and σ is a finite set of points in D . A spray of maps of class A r ( D ) withthe exceptional set σ and with values in X is a map f : P × ¯ D → X , where P (the parameter set of the spray) is an open subset of a Euclidean space C m containing the origin, such that the following hold:(i) f is holomorphic on P × D and of class C r on P × ¯ D , HE POLETSKY-ROSAY THEOREM ON SINGULAR COMPLEX SPACES 5 (ii) the maps f (0 , · ) and f ( t, · ) agree on σ for all t ∈ P , and(iii) if z ∈ ¯ D \ σ and t ∈ P then f ( t, z ) ∈ X reg and ∂ t f ( t, z ) : T t C m → T f ( t,z ) X is surjective (the domination property ).We call f = f (0 , · ) the core (or central ) map of the spray f .The following lemma is a special case of [DF, Lemma 4.2]. Lemma 2.2. (Existence of sprays)
Assume that ℓ , r , D , σ and X are inaccordance with Definition 2.1. Given a map f : ¯ D → X of class A r ( D ) such that the set { z ∈ ¯ D : f ( z ) ∈ X sing } is contained in σ , there exists aspray f : P × ¯ D → X of class A r ( D ) , with the exceptional set σ , such that f (0 , · ) = f . Definition 2.3.
Let ℓ ≥ D , D ⋐ C is said to be a Cartan pair of class C ℓ if(i) D , D , D = D ∪ D and D , = D ∩ D are domains with C ℓ smooth boundaries, and(ii) D \ D ∩ D \ D = ∅ (the separation property).The following is the main result on gluing sprays in the particular situationthat we are considering (see [DF, Proposition 4.3] or [For, Lemma 3.2]). Thisis in fact a solution of a nonlinear Cousin-I problem. Proposition 2.4. (Gluing sprays)
Let ( D , D ) be a Cartan pair of class C ℓ ( ℓ ≥ in C (Def. 2.3). Set D = D ∪ D , D , = D ∩ D . Let X be acomplex space. Given an integer r ∈ { , , . . . , ℓ } and a spray f : P × ¯ D → X of class A r ( D ) with the exceptional set σ such that σ ∩ ¯ D , = ∅ , there isan open set P ⋐ P ⊂ C m containing ∈ C m and satisfying the following.For every spray f ′ : P × ¯ D → X of class A r ( D ) , with the exceptionalset σ ′ such that f ′ is sufficiently C r close to f on P × ¯ D , and σ ′ ∩ ¯ D , = ∅ ,there exists a spray F : P × ¯ D → X of class A r ( D ) , with the exceptional set σ ∪ σ ′ , enjoying the following properties: (i) the restriction F : P × ¯ D → X is close to f : P × ¯ D → X in the C r -topology (depending on the C r -distance of f and f ′ on P × ¯ D , ), (ii) the core map F = F (0 , · ) is homotopic to f = f (0 , · ) on ¯ D , and F is homotopic to f ′ = f ′ (0 , · ) on ¯ D , (iii) F agrees with f on σ , and it agrees with f ′ on σ ′ , and (iv) F ( t, z ) ∈ { f ′ ( s, z ) : s ∈ P } for each t ∈ P and z ∈ ¯ D . Here is a brief outline of the proof. The first step is to find a domain P ′ ⊂ C m such that 0 ∈ P ′ ⋐ P , and a transition map between the sprays f and f ′ , that is, a C r map γ : P ′ × ¯ D , → P × ¯ D , , γ ( t, z ) = (cid:0) c ( t, z ) , z (cid:1) BARBARA DRINOVEC DRNOVˇSEK & FRANC FORSTNERIˇC that is holomorphic in P ′ × D , and is C r close to the identity map (de-pending on the C r -distance between f and f ′ over P × ¯ D , ), such that f = f ′ ◦ γ on P ′ × ¯ D , . This is an application of both the implicit function theorem and the factthat Cartan’s Theorem B holds for holomorphic vector bundles on domainsin C that are smooth of class C r up to the boundary. The key step is to splitthe map γ in the form γ = β ◦ α − where α ( t, z ) = (cid:0) a ( t, z ) , z (cid:1) and β ( t, z ) = (cid:0) b ( t, z ) , z (cid:1) are maps with similarproperties over P × ¯ D and P × ¯ D , respectively, for some slightly smallerparameter set 0 ∈ P ⋐ P ′ . This splitting is accomplished by nonlinearoperators whose linearization involves a solution operator for the ¯ ∂ -equationwith C r estimates on D = D ∪ D . The final step is to observe that over P × ¯ D , we have f = f ′ ◦ γ = f ′ ◦ β ◦ α − = ⇒ f ◦ α = f ′ ◦ β. Hence the two sides amalgamate into a spray F over ¯ D , and it is easilyverified that F satisfies Proposition 2.4.3. A Riemann-Hilbert problem
In this section we explain how to find an approximate solution of aRiemann-Hilbert problem with the control of the average of a given functionon a boundary arc. Results of this kind have been used by several authors;see e.g. [Po1, Po2, BS, FG1, FG2].Recall that T = b D = { ζ ∈ C : | ζ | = 1 } . Given a measurable subset I ⊂ T and a measurable function v : I → R , R I v (e i t ) d t will denote the integral overthe set of points t ∈ [0 , π ] for which e i t ∈ I . Lemma 3.1.
Let f ∈ A ( D , C n ) , and let g : T × D → C n be a continuous mapsuch that for each ζ ∈ T we have g ( ζ, · ) ∈ A ( D , X, f ( ζ )) . Given numbers ǫ > and < r < , an arc I ⊂ T , and a continuous function u : C n → R ,there are a number r ′ ∈ [ r, and a disc h ∈ A ( D , C n , f (0)) satisfying (2) Z I u (cid:0) h (e i t ) (cid:1) d t π < Z π Z I u (cid:0) g (e i t , e i θ ) (cid:1) d t π d θ π + ǫ and also the following properties: (i) for any ζ ∈ T we have dist (cid:0) h ( ζ ) , g ( ζ, T ) (cid:1) < ǫ , (ii) for any ζ ∈ T and ρ ∈ [ r ′ , we have dist (cid:0) h ( ρζ ) , g ( ζ, D ) (cid:1) < ǫ , (iii) for any | ζ | ≤ r ′ we have | h ( ζ ) − f ( ζ ) | < ǫ , and (iv) if g ( ζ, · ) = f ( ζ ) is the constant disc for all ζ ∈ T \ J , where J ⊂ T is an arc containing I , then we can choose h such that | h − f | < ǫ holds outside any given neighborhood of J in D . HE POLETSKY-ROSAY THEOREM ON SINGULAR COMPLEX SPACES 7
Proof.
Write g ( ζ, z ) = f ( ζ ) + λ ( ζ, z ) , ζ ∈ T , z ∈ D , where λ ( ζ, z ) is continuous on ( ζ, z ) ∈ T × D , and for every fixed ζ ∈ T thefunction D ∋ z λ ( ζ, z ) is holomorphic on D and satisfies λ ( ζ,
0) = 0. Wecan approximate λ uniformly on T × D by Laurent polynomials of the form e λ ( ζ, z ) = 1 ζ m N X j =1 A j ( ζ ) z j = zζ m N X j =1 A j ( ζ ) z j − with polynomial coefficients A j ( ζ ). Hence we can choose a map e λ as aboveand a number r ′ ∈ [ r,
1) such that(3) (cid:12)(cid:12)e λ ( ρ e i t , z ) − λ (e i t , z ) (cid:12)(cid:12) < ǫ , t ∈ R , r ′ ≤ ρ ≤ , | z | ≤ (cid:12)(cid:12) f ( ρ e i t ) − f (e i t ) (cid:12)(cid:12) < ǫ , t ∈ R , r ′ ≤ ρ ≤ . Choose an integer k > m and a number c = e i φ ∈ T and set h k ( ζ, c ) = f ( ζ ) + e λ ( ζ, cζ k ) = f ( ζ ) + c ζ k − m N X j =1 A j ( ζ ) (cid:16) cζ k (cid:17) j − , | ζ | ≤ . This is an analytic disc in C n satisfying h k (0 , c ) = f (0), so it belongs to A ( D , C n , f (0)). For ζ = e i t ∈ T we have(5) h k (cid:0) e i t , c (cid:1) = f (cid:0) e i t (cid:1) + e λ (cid:0) e i t , e i φ e k i t (cid:1) ≈ g (cid:0) e i t , e i( φ + kt ) (cid:1) , and hence property (i) holds in view of (3). Similarly, if r ′ ≤ ρ ≤ (cid:12)(cid:12)(cid:12) h k (cid:0) ρ e i t , c (cid:1) − g (cid:0) e i t , cρ k e i kt (cid:1)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)e λ (cid:0) ρ e i t , c ρ k e i kt (cid:1) − λ (cid:0) e i t , c ρ k e i kt (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) f ( ρ e i t ) − f (e i t ) (cid:12)(cid:12) < ǫ (6)by (3) and (4), so (ii) holds as well.If k → + ∞ then h k ( ζ, · ) → f ( ζ ) uniformly on the set {| ζ | ≤ r ′ }× T , so weget condition (iii) if k is chosen big enough. Property (iv) is a consequenceof (ii) and (iii).It remains to show that the inequality (2) can be achieved by a suitablechoice of the number c = e i φ ∈ T . We clearly have Z π Z I u (cid:0) g (e i t , e i θ ) (cid:1) d t π d θ π = Z π Z I u (cid:0) g (e i t , e i( θ + kt ) ) (cid:1) d t π d θ π . By the mean value theorem there exists φ ∈ [0 , π ) such that this equals Z I u (cid:0) g (e i t , e i( φ + kt ) ) (cid:1) d t π . BARBARA DRINOVEC DRNOVˇSEK & FRANC FORSTNERIˇC
By (5) this number differs by at most ǫ from R π u (cid:0) h k (e i t , e i φ ) (cid:1) d t π if k ischosen big enough. This completes the proof. (cid:3) Proof of Theorem 1.1
Step 1.
We reduce to the case when X is a normal complex space.Recall that a function on a complex space X is weakly holomorphic if itis holomorphic on the regular part X reg and is locally bounded near eachsingular point. A reduced complex space X is said to be normal if everyweakly holomorphic function is in fact holomorphic; it is seminormal if everycontinuous weakly holomorphic function on X is holomorphic. A holomor-phic map π : e X → X of complex spaces is called a seminormalization (ora maximalization ) of X if e X is a seminormal complex space and π is ahomeomorphism. We will use the following facts (see [Rem]): • every reduced complex space admits a seminormalization, • every locally irreducible seminormal complex space is normal, and • seminormalization is a functor; in particular, the lift π − ◦ f of anyholomorphic disc f : D → X is a holomorphic disc in e X .This implies that if π : e X → X is a seminormalization of X then the com-position u u ◦ π induces an isomorphism from P SH ( X ) onto P SH ( e X ).Assume now that Theorem 1.1 holds for normal complex spaces. Let X be a locally irreducible complex space and u : X → R ∪ {−∞} an uppersemicontinuous function. Let π : e X → X be a seminormalization of X . Thefunction e u = u ◦ π is upper semicontinuous on e X . Since e X is a normalcomplex space, the Poisson envelope e v = be u of e u is plurisubharmonic on e X . Each holomorphic disc f ∈ A ( D , X ) admits a holomorphic lifting e f ∈A ( D , e X ) with π ◦ e f = f , and we clearly have P e u ( e f ) = P u ( f ). This impliesthat e v ( π − ( x )) = ˆ( x ) and hence ˆ u is plurisubharmonic. Step 2.
We reduce to the case when the function u : X → R is continuousand bounded from below.Since u : X → R ∪ {−∞} is upper semicontinuous, there is a decreasingsequence of continuous functions u ≥ u ≥ . . . ≥ u such that u = lim k →∞ u k pointwise in X . Replacing u k by max { u k , − k } we may assume in additionthat u k ≥ − k on X ; hence b u k ≥ − k as well. Assuming that the resultholds for each u k , the Poisson envelopes b u ≥ b u ≥ . . . form a decreasingsequence of plurisubharmonic functions, and hence v = lim k →∞ b u k is alsoplurisubharmonic or identically −∞ . Since b u k ≤ u k , it follows that v ≤ u .For any point x ∈ X and disc f ∈ A ( D , X, x ) we have v ( x ) ≤ P v ( f ) ≤ P u ( f ) ≤ P u k ( f ) by monotonicity. Taking the infimum over all such f showsthat v ( x ) ≤ b u ( x ) ≤ b u k ( x ). Letting k → ∞ gives v ( x ) = b u ( x ), so b u isplurisubharmonic or −∞ . HE POLETSKY-ROSAY THEOREM ON SINGULAR COMPLEX SPACES 9
Step 3.
We show that the Poisson envelope v = b u given by (1) is uppersemicontinuous on the regular locus X reg . (We assume that X is a reducedcomplex space. The reductions in Steps 1 and 2 will not be used here.)Pick a point x ∈ X reg and a number ǫ >
0. Assume first that v ( x ) > −∞ . By the definition of v there exists a disc f ∈ A ( D , X, x ) such that v ( x ) ≤ P u ( f ) < v ( x ) + ǫ . By shrinking D slightly we may assume that f ( T ) ⊂ X reg ; hence the set σ = { ζ ∈ D : f ( ζ ) ∈ X sing } is finite and 0 / ∈ σ .We embed f as the central map f = f (0 , · ) in a spray of holomorphic discs f : P × D → X with the exceptional set σ , where P is an open set in C m containing the origin. (See Def. 2.1 and Lemma 2.2; on X = C n we cansimply use the family of translates f y = f + ( y − x )). If P ′ ⋐ P is a smallopen neighborhood of 0 ∈ C m then P u ( f ( t, · )) < P u ( f ) + ǫ for each t ∈ P ′ and hence v ( f ( t, ≤ P u ( f ( t, · )) ≤ P u ( f ) + ǫ < v ( x ) + 2 ǫ. By the domination property the set { f ( t,
0) : t ∈ P ′ } fills a neighborhood ofthe point x = f (0) in X , so we see that v is upper semicontinuous at x . Asimilar argument works at points where v ( x ) = −∞ . Step 4.
In this main step of the proof we assume that X is an irreduciblenormal complex space and that u : X → R is a continuous function which isbounded from below (see Steps 1 and 2). We shall prove that the Poissonenvelope v = b u given by (1) is plurisubharmonic on X reg . (Plurisubhar-monicity on X sing will be proved in Step 5.)We need to show that for every point x ∈ X reg and for every analytic disc f ∈ A ( D , X, x ) we have the submeanvalue property(7) v ( x ) = v ( f (0)) ≤ Z π v ( f (e i t )) d t π . Since plurisubharmonicity is a local property, it is enough to consider smalldiscs; hence we may assume that f ( D ) ⊂ X reg and that f is holomorphicon a larger disc r D for some r > ǫ >
0. Fix a point e i t ∈ T . By the definition of v thereexists an analytic disc g t = g (e i t , · ) ∈ A ( D , X, f (e i t )) such that(8) v ( f (e i t )) ≤ Z π u (cid:0) g (e i t , e i θ ) (cid:1) d θ π < v ( f (e i t )) + ǫ. Since the set { ζ ∈ D : g t ( ζ ) ∈ X sing } is discrete, we can replace g t by themap ζ g t ( rζ ) for some r < g t ( T ) ⊂ X reg . By Lemma 2.2 there is a domain P ⊂ C m containing the origin and a holomorphic spray of discs G : P × D → X with the central map G (0 , · ) = g t . (We use sprays of class A ( D ).) Since G (0 ,
0) = g t (0) = f (e i t ) ∈ X reg and the spray G is dominating, the set G ( P,
0) = { G ( w,
0) : w ∈ P } is a neighborhood of the point f (e i t ) in X . By the implicit mapping theorem there are a disc D ⊂ r D centered at thepoint e i t ∈ T and a holomorphic map ϕ : D → P such that ϕ (e i t ) = 0 and G ( ϕ ( ζ ) ,
0) = f ( ζ ) , ζ ∈ D. Consider the continuous map g : D × D → X defined by g ( ζ, z ) = G ( ϕ ( ζ ) , z ) , ζ ∈ D, z ∈ D . Note that g is holomorphic on D × D , and g (e i t , · ) = G (0 , · ) = g t ; g ( ζ,
0) = f ( ζ ) , ζ ∈ D. Since g ( ζ, · ) is uniformly close to g t when ζ is close to e i t , it follows from(8) that there is a small arc I ⋐ T ∩ D around the point e i t such that Z π Z I u (cid:0) g (e i η , e i θ ) (cid:1) d η π d θ π ≤ Z I v (cid:0) f (e i η ) (cid:1) d η π + | I | ǫ π . By repeating this construction at other points of T we find finitely manypairs of arcs I j ⋐ I ′ j ⋐ T ∩ D j ( j = 1 , . . . , l ), where D j is a disc contained in r D , such that I ′ j ∩ I ′ k = ∅ if j = k and the set E = T \ ∪ mj =1 I j has arbitrarilysmall measure | E | , and holomorphic families of discs g j ( ζ, z ) for ζ ∈ D j and z ∈ D such that(9) Z π Z I j u (cid:0) g j (e i t , e i θ ) (cid:1) d t π d θ π < Z I j v (cid:0) f (e i t ) (cid:1) d t π + | I j | ǫ π . For each j = 1 , . . . , l we choose a smoothly bounded simply connecteddomain ∆ j ⊂ D j ∩ D such that ∆ j ⊂ D , ∆ j ∩ T = I ′ j , and ∆ j ∩ ∆ k = ∅ when1 ≤ j = k ≤ l . (The situation with l = 3 is illustrated in Fig. 1. The readershould keep in mind that the gaps between the segments I j have very smalltotal length. The role of the sets D , D ⊂ D with D ∪ D = D is explainedin the proof of Lemma 4.1 below.)Let χ : C → [0 ,
1] be a smooth function such that χ = 1 on ∪ lj =1 I j ⊂ T and χ = 0 on a neighborhood of the set D \ ∪ lj =1 (∆ j ∪ I ′ j ). Consider themap ξ : D × D → X defined by(10) ξ ( ζ, z ) = ( g j (cid:0) ζ, χ ( ζ ) z (cid:1) , ζ ∈ ∆ j , z ∈ D , j = 1 , . . . , l ; f ( ζ ) , χ ( ζ ) = 0 , z ∈ D . The latter condition holds in particular if ζ ∈ T \ ∪ mj =1 I ′ j . Note that ξ iscontinuous and is holomorphic in the second variable. Then(11) Z π Z π u (cid:0) ξ (e i t , e i θ ) (cid:1) d t π d θ π < Z π v (cid:0) f (e i t ) (cid:1) d t π + 2 ǫ. Indeed, the integral over ∪ lj =1 I j is estimated by adding up the inequalities(9), while the integral over the complementary set E = T \ ∪ lj =1 I j gives atmost ǫ if the measure | E | is small enough. (When estimating the integralover E it is important to observe that u ≥ − M by the assumption, and u HE POLETSKY-ROSAY THEOREM ON SINGULAR COMPLEX SPACES 11 D ∆ ∆ ∆ I I I b D bD D = ∪ j ∆ j Figure 1.
The Cartan pair ( D , D )is bounded from above on each compact set. Since − M ≤ v = b u ≤ u , thesame holds for v .)To conclude the proof we apply the following lemma to the data that wehave just constructed. We state it in a more general form since we shallneed it again in Step 5 below. Lemma 4.1.
Let X be a reduced complex space and let u : X → R be acontinuous function on X . Assume that f ∈ A ( D , X ) is a holomorphic discsuch that f ( T ) ⊂ X reg , and ξ : T × D → X is a continuous map such that ξ ( ζ, · ) ∈ A ( D , X, f ( ζ )) for every ζ ∈ T . Given ǫ > there exists an analyticdisc h ∈ A ( D , X, f (0)) such that Z π u ( h (e i t )) d t π < Z π Z π u (cid:0) ξ (e i t , e i θ ) (cid:1) d t π d θ π + ǫ. (12)Note that the estimate (12) is the same as (2) in Lemma 3.1 which per-tains to the case X = C n , but we do not get (and do not need) the otherapproximation statements in that lemma.Combining the inequalities (11) and (12) we obtain v ( x ) ≤ Z π u ( h (e i t )) d t π < Z π v ( f (e i t )) d t π + 3 ǫ. Since this holds for every ǫ >
0, the property (7) follows. This proves that v is plurisubharmonic on X reg provided that Lemma 4.1 holds. Proof of Lemma 4.1.
In the case when X is a complex manifold, the firstproof of Rosay [Ro1] uses a rather delicate construction of Stein neighbor-hoods; this approach was developed further by L´arusson and Sigurdsson [LS2]. Later Rosay [Ro2] gave another proof using an initial approximationby non-holomorphic discs with small ¯ ∂ -derivatives, approximating these byholomorphic discs, and finally patching the partial solutions together by solv-ing a nonlinear Cousin-I problem. None of these methods seems to extendto complex spaces with singularities without major technical difficulties. Onthe other hand, the method that we use here works in essentially the sameway as in the nonsingular case.Since the function u is bounded on compacts, it is a trivial matter toreduce the proof to the special situation considered above so that the doubleintegral in (11) changes by less than ǫ ; in the sequel we consider this case.Let ∆ j be the discs chosen in the paragraph following (9) (see Fig. 1). Fix anindex j ∈ { , . . . , l } . We shall apply Lemma 3.1 over ∆ j to find an analyticdisc f ′ j : ∆ j → X that approximates f uniformly as close as desired outsideof a small neighborhood of the arc I j in ∆ j and satisfies the estimate(13) Z I j u (cid:0) f ′ j (e i t ) (cid:1) d t π < Z π Z I j u (cid:0) ξ (e i t , e i θ ) (cid:1) d t π d θ π + | I j | ǫ π . Recall from (10) that ξ ( ζ, z ) = g j (cid:0) ζ, χ ( ζ ) z (cid:1) for ζ ∈ ∆ j and z ∈ D . Considerthe function u ′ j ( ζ, z ) = u (cid:0) g j (cid:0) ζ, z ) (cid:1) , ζ ∈ ∆ j , z ∈ D and the smooth family of analytic discs in C ζ,z ) given by g ′ j ( ζ, z ) = (cid:0) ζ, χ ( ζ ) z (cid:1) , ζ ∈ b ∆ j , z ∈ D . Then(14) ξ = g j ◦ g ′ j and u ′ j = u ◦ g j on b ∆ j × D . Applying Lemma 3.1 with D replaced by ∆ j , u replaced by u ′ j and g replacedby g ′ j furnishes an analytic disc h ′ j ∈ A (∆ j , C ) which approximates the disc ζ ( ζ,
0) outside of a small neighborhood of the arc I j and such that(15) Z I j u ′ j (cid:0) h ′ j (e i t ) (cid:1) d t π < Z π Z I j u ′ (cid:0) g ′ j (e i t , e i θ ) (cid:1) d t π d θ π + | I j | ǫ π . Since g j : ∆ j × D → X is holomorphic in the interior ∆ j × D , the map(16) f ′ j := g j ◦ h ′ j : ∆ j → X is an analytic disc in X such that f ′ j ( ζ ) ≈ g j ( ζ,
0) = f ( ζ ) for ζ outside asmall neighborhood of I j in ∆ j . From (14) and (16) it follows that u ◦ f ′ j = u ◦ g j ◦ h ′ j = u ′ j ◦ h ′ j , u ◦ ξ = u ◦ g j ◦ g ′ j = u ′ j ◦ g ′ j hold on b ∆ j × D . Hence the integrals in (13) equal the corresponding inte-grals in (15), and so the disc f ′ j satisfies the desired properties.If the approximation of f by f ′ j is close enough for each j = 1 , . . . , l , wecan use Proposition 2.4 to glue this collection of discs into a single analytic HE POLETSKY-ROSAY THEOREM ON SINGULAR COMPLEX SPACES 13 disc h : D → X which approximates f away from the union of arcs ∪ lj =1 I j ,and which approximates the disc f ′ j over a neighborhood of I j for each j .In particular we can insure that for each j = 1 , . . . , l we have Z I j u (cid:0) h (e i t ) (cid:1) d t π ≈ Z π Z I j u (cid:0) g (e i t , e i θ ) (cid:1) d t π d θ π . By adding up these terms and estimating the difference over the remainingset E = T \ ∪ j I j with small length | E | we obtain (12).Let us explain the gluing. We apply Proposition 2.2 to embed f as thecentral map f in a dominating spray of discs f p ∈ A ( D , X ) dependingholomorphically on a parameter p in an open ball P in some C N . Nextwe apply the above approximation procedure simultaneously to all discs f p in the spray, with a holomorphic dependence on the parameter. (Theobvious details need not be repeated.) This gives for every j a holomorphicspray of discs f ′ p,j ∈ A (∆ j , X ) ( p ∈ P ) approximating the spray f p over thecomplement of a small neigborhood of I j in ∆ j . If the approximations areclose enough, we can glue these sprays into a spray of discs F p ∈ A ( D , X ) byusing Proposition 2.4. (Here we allow the parameter set P to shrink.) Forgluing we use a Cartan pair ( D , D ) with D ∪ D = D obtained as follows(see Fig. 1). We get D by denting the boundary circle T = b D slightlyinward along each of the arcs I j ⊂ T . The set D equals ∪ lj =1 ∆ j . Then thetwo sprays are close to each other over ¯ D ∩ ¯ D , so Proposition 2.4 applies.The core disc h = F obtained in this way is close to the initial disc f over the complement of a small neighborhood of ∪ j I j in D , while over aneighborhood of I j it is close to f ′ j . If the approximations are close enoughthen h also satisfies the estimate (12) in Lemma 4.1. (cid:3) Step 5.
We now prove that v = ˆ u is plurisubharmonic on all of X . Let w : X → R ∪ {−∞} be the upper regularization of v | X reg : w ( p ) = (cid:26) v ( p ) , p ∈ X reg ;lim sup q ∈ X reg ,q → p v ( q ) , p ∈ X sing . It is easy to see that w ≤ u . Since X is normal, w is plurisubharmonic on X according to a result of Grauert and Remmert [GR1, Satz 4].To complete the proof of the theorem we show that v = w on X sing .Fix a point p ∈ X sing and a disc f ∈ A ( D , X, p ). The fact that w isplurisubharmonic implies that w ( p ) ≤ P w ( f ). Since w ≤ u , we also have P w ( f ) ≤ P u ( f ). Therefore w ( p ) ≤ P u ( f ) for every f ∈ A ( D , X, p ) whichimplies that w ( p ) ≤ v ( p ).Suppose now that w ( p ) < v ( p ); we shall reach a contradiction. Choose ǫ > w ( p ) + 3 ǫ < v ( p ). Since w is upper semicontinuous, there is aneighborhood U of p such that(17) w ( q ) ≤ w ( p ) + ǫ for each q ∈ U. We can choose an analytic disc f ∈ A ( D , U, p ) such that f ( T ) ⊂ U ∩ X reg .Since w = v on X reg , we have using (17)(18) P v ( f ) = P w ( f ) ≤ sup { w ◦ f ( ζ ) : ζ ∈ T } ≤ w ( p ) + ǫ. For each point e i t ∈ T we choose a disc g t = g (e i t , · ) ∈ A ( D , X, f (e i t )) suchthat P u ( g t ) < v ( f (e i t )) + ǫ . As in Step 3 we can deform g t to a continuousfamily g : T × D → X of analytic discs such that(19) (cid:12)(cid:12)(cid:12)(cid:12)Z π v (cid:0) f (e i t ) (cid:1) d t π − Z π Z π u (cid:0) g (e i t , e i θ ) (cid:1) d t π d θ π (cid:12)(cid:12)(cid:12)(cid:12) < ǫ. Lemma 4.1 furnishes an analytic disc h ∈ A ( D , X, p ) such that P u ( h ) < Z π Z π u (cid:0) g (e i t , e i θ ) (cid:1) d t π d θ π + ǫ. By (19) and (18) we get P u ( h ) ≤ Z π v (cid:0) f (e i t ) (cid:1) d t π + 2 ǫ ≤ w ( p ) + 3 ǫ < v ( p )which contradicts the definition of v . This concludes the proof. Acknowledgement.
The authors would like to thank the referee for hisremarks.Added to page proofs: Since the submission of this paper, the authorshave used the techniques developed in the present paper to obtain similarresults for some other classes of disc functionals; see [DF2].
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