A Stein manifold is called S-Parabolic in case there exits a special

plurisubharmonic exhaustion function that is maximal outside a compact set.

If a continuous special plurisubharmonic exits then we will call the manifold

S*-Parabolic: In one dimensional case these notions are equivalent. However

in several variables the question as to weather these notions coincide seems

open. In this note we establish an interrelation between these two notions

Aytuna, Aydın

Aytuna, Aydin

Sadullaev, Azimbay

English

Sabanci University Research Database

S-Parabolic manifoldsA. Aytuna and A. SadullaevAbstract. A Stein manifold is called S parabolic in case there exits a specialplurisubharmonic exhaustion function that is maximal outside a compact set.If a continuous special plurisubharmonic exits then we will call the manifoldS  parabolic: In one dimensional case these notions are equivalent. Howeverin several variables the question as to weather these notions coincide seemsopen. In this note we establish an interrelation between these two notions.1. IntroductionIn this note we establish an interrelation between two notions of paraboliticityin several complex variables that exit in the literature. We start by giving therelevant denitions.Definition 1. A Stein manifold X of dimension n is called S parabolic in casethere exits a special plurisubharmonic function  2 PSH (X) with the properties:a) The set fz 2 X :  (z)  Cg  X is relatively compact in X, for everyC 2 R: That is  is an exhaustion,b) (ddc)n = 0 o¤ a compact set, i.e.  is maximal outside a compactset.([7])In the previous papers on parabolic manifolds (see for example [9],[10],[4]) au-thors usually required the conditions of continuity or C1  smoothness of  in theabove denition. In this note we distinguish the case of continuity and call a com-plex manifold S   parabolic, in case it possesses a continuous plurisubharmonicexhaustion that is maximal outside a compact subset.We note, that without maximality condition b) an exhaustion function alwaysexist for any Stein manifold X, since in general Stein manifolds can be properlyembedded into CNw for some large N and one take for  the restriction of ln jwj toX.1991 Mathematics Subject Classication. Primary 32U05, 46A61, 32U15 ; Secondary 46A63,32U15.Key words and phrases. Parabolic manifolds, plurisubharmonic exhaustion functions, maxi-mal plurisubharmonic functions.The second author is partially supported by Khorezm Mamun Academy, Grant A-1- 024The rst author is partially supported by a grant from Sabanci University.This paper is in nal form and no version of it will be submitted for publication elsewhere.12 A. AYTUNA AND A. SADULLAEVSpecial plurisubharmonic exhaustion functions on parabolic manifolds X playa key role in Navanlinnas value distribution theory of holomorphic maps from Xinto projective spaces (see for example [4], [6]).Stein manifolds, on which every bounded above plurisubharmonic function re-duces to a constant play a role in the study of the Fréchet spaces of analytic func-tions on Stein manifolds, the bases on them and in nding continuous extensionoperators for analytic functions from complex submanifolds (see for example [1],[2]).Such spaces will be called parabolic in this paper.It is not di¢ cult to see that S parabolic manifolds are parabolic:In particularthere are no bounded non constant analytic functions on such manifolds.The most important example of a S   parabolic manifold is Cnz with the spe-cial plurisubharmonic function ln jzj: A¢ ne algebraic manifolds, parabolic Riemannsurfaces also among the examples of S   parabolic manifoldsS  parabolic manifolds (also S  parabolic Stein spaces) and the structureof certain plurisubharmonic functions and currents on them where studied in detailby Demailly([3]), and Zeriahi ([13]),([14]). Moreover on such manifolds one candene extremal Green functions and apply it to the pluripotential theory on suchmanifolds.Let us x an S  parabolic manifold X and a special exhaustion function  onit. LetL =u 2 PSH (X) : u (z)  + (z) + C, where + = max (; 0) , C = C () R	be the Lelong class of plurisubharmonic functions and for a compact setK  X;and setL (K) = fu 2 L : ujK  0g :Definition 2. Let X and K be as above, the upper regularization V  (z;K) =lim supw!z V (z;K) of V (z;K) = sup (u (z) : u 2 L (K)) is called the Greenfunction of K.We note that the Green function is either +1 (K is pluripolar) or belongs tothe class L: An analytic function f on X will be called a    polynomial in caseln jf jd belongs to L for some integer d. The minimal such d is called the degree ofthe polynomial. The space of all -polynomials of degree less than or equal to d,Pd (X), is a nite dimensional space and dim Pd (X)  n+dnn: This result wasproved in[13] for S  parabolic manifolds but the same proof also works for S parabolic manifolds. . In the special case of a¢ ne algebraic manifolds a detailedanalysis of these generalized polynomials is given in [14].In the one dimensional case the notions of S  parabolicity; S  parabolicityand parabolicity coincide, ([8]) : However in several variables the question as toweather these notions coincide seems open.The aim of this short note is establish an interrelation between S  parabolicityand S  parabolicity:2. ResultsLet X be a S  parabolic manifold and choose a special plurisubharmonic ex-haustion function : If  is not continuous it is of interest to examine the jumps atS-PARABOLIC MANIFOLDS 3its discontinuities. Note that for any plurisubharmonic function  and any point zin its domain of denition ;  (z) = lim supw!z  (w) =  (z) : For a given pointz in the domain of denition of the function we set  (z) = lim infw!z  (w) : If (z) <  (z) then we have a jump at z:Definition 3. Let  be a plurisubharmonic function exhaustion of a complexmanifold. We say that  is strongly continuous at the point at innity in caselim(z)!1 (z) (z)= 1:Lemma 1. Let X be a S  parabolic manifold and  its special exhaustionfunction. Then the following are equivalent:a) The function  is strongly continuous at the point at innityb) The Green function V  (z;K) corresponding to  is strongly continuous atthe point at innity for any nonpluripolar compact set K:Proof. Fix a nonpluripolar compact set K  X . There are positive constantsC1 and C2 such that (z)  C1  V  (z;K)   (z) + C2 .The rst inequality is by denition of the Green function and the second follows fromthe remarks given in section 1. The Lemma follows easily from these inequalities.Now we can state our result.Theorem 1. Let X be a S   parabolic manifold. Then X is S   parabolic ifand only if there is an plurisubharmonic exhaustion function on X that is maximaloutside a compact set and strongly continuous at the point at innity.Proof. Lets assume that there exits a plurisubharmonic exhaustion functionon X that is maximal outside a compact set and is strongly continuous at the pointat innity. We x a big pluriregular compact set K  X: Let us denote Greenfunction V  (z;K) corresponding to this compact set by v (z) : Then in view of thelemma v (z) is strongly continuous at the point at innity. Using the approximationtheorem given in [11] (this fact was also proved independently, but later by thesecond author see [12]) we can nd a sequence of plurisubharmonic functions vj (z)2 PSH (X) \ C1 (X) ; vj (z) # v (z) 8z 2 X:Since K is pluriregular, vjK = 0: In view of Hartogs theorem for an arbitrary > 0; we can nd a j0 such that vj <  uniformly on K; j  j0:Since v is strongly continuous at innity, there is an R > 0 such thatv (z)  v (z) + v (z) for z =2 BR where BR = fz "X : v (z) < Rg :In particular we have vj@BR  (1 + ) R: Applying again Hartogs theorem wecan nd an j1 so thatvj (z)  (1 + 2)R; j > j1  j0, z "@BR:For j > j1 we put:w (z) =max fvj (z) ; (1 + 3) v (z)  Rg if z 2 BR(1 + 3) v (z)  R if z =2 BR :4 A. AYTUNA AND A. SADULLAEVSince for z 2 BR we have w (z) = (1 + 3) v (z)  R  (1 + 2)R  vj (z) thefunction w (z) is plurisubharmonic onX: It follows that the function 11+3 [w (z)  ]belongs to the Lelong class L (K) : So we have11 + 3[w (z)  ]  V  (z;K) = v (z) :In particular vj (z)  (1 + 3)V  (z;K) + : Since vj (z)  V  (z;K) the conti-nuity of V  (z;K) follows. Corollary 1. If the "jumps" of the special exhaustion function  of an S  parabolic manifold X satisfy (z)   (z) = o ( (z)) ;then X is S   parabolic:References[1] Aytuna A., Stein Spaces M for which O(M) is isomorphic to a power series space,Advancesin the theory of Fréchet spaces, Istanbul,1988,115-154[2] Aytuna A., Linear tame extension operator from closed subvarieties of, Proc. Amer. Math.Soc., 123:3 (1995),759-763[3] Demailly J.P., Mesures de Monge - Ampère et caractèrisation gèomètrique des variétés al-gébriques a¢ nes, Memoires de la societe mathematique de France.V.19(1985),1-125[4] Gri¢ ts P. and King J.,Nevanlinna theory and holomorphic mappings between algebraic va-rieties,Acta mathematica,V.130 (1973), 145-220[5][6] Sadullaev A., Valirons defect divisors for holomorphic map,Mathem. Sbornic,V 108:4 (1979),567-580.=Math. USSR -Sb.V.36(1980),535-547[7] Sadullaev A., Plurisubharmonic measure and capacity on complex manifolds, UspehiMath.Nauk, Moscow, V.36 N4, (220) (1981) , 53-105 = Russian Mathem.Surveys V.36 (1981),61-119[8] Sario,L-M.Nakai, Classication theory of Riemann surfaces, Springer,Berlin-Heidelberg-NewYork-1970, pp.216.[9] Stoll W., Value distribution of holomorphic maps into compact complex manifolds, Lecturenotes,N 135,Springer,Berlin-Heidelberg-New York-1970[10] Stoll W.,Value distribution on parabolic spaces, Lecture notes, 600, Springer,Berlin-Heidelberg-New York-1977[11] Fornaess J.E. and Narasimhan R., The Levi problem on complex spaces with singularities,Math.Ann., 248 (1980), 47-72[12] Sadullaev A.,Continuation plurisubharmonic functions from submanifold,Dokl.AN Uzbek-istan,N 5(1982), 3[13] Zeriahi A., Fonction de Green pluricomplex a pole a linni sur un espace de Stein parabolique,Math.Scand., V.69 (1991), 89-126[14] Zeriahi A., Approximation polynomial et extension holomorphe avec croissance sur une varietealgebrique, Ann.Polon.Math. V.63:1, (1996), 35-50S-PARABOLIC MANIFOLDS 5FENS, SABANCI UNIVERSITY, 34956 TUZLA, ISTANBUL, TURKEYE-mail address : aytuna@sabanciuniv.eduCurrent address : MATHEMATICS DEPARTMENT, NATIONAL UNIVERSITY OF UZBEK-ISTAN VUZ GORODOK, 700174, TASHKENT, UZBEKISTANE-mail address : sadullaev@mail.ru

S-parabolic manifolds

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S-parabolic manifolds

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