Digital Object Identifier (DOI) 10.1007/s002090000126Math. Z. 235, 111–122 (2000)

Certain classes of pluricomplex Green functionsonC

n

Dan Coman

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, USA(e-mail: Dan.F.Coman.2@nd.edu)

Received November 24, 1998; in final form April 19, 1999 /Published online July 3, 2000 –c© Springer-Verlag 2000

Abstract. Weconsider (pluricomplex)Green functions defined onCn, with

logarithmic poles in a finite set and with logarithmic growth at infinity. Forcertain sets, we describe all the corresponding Green functions. The set ofthese functions is large and it carries a certain algebraic structure. We alsoshow that for some sets no such Green functions exist. Our results indicatethe fact that the set of poles should have certain algebro-geometric propertiesin order for these Green functions to exist.

Mathematics Subject Classification (1991):32F05, 32F07, 31C10

1 Introduction

For a bounded domainD ⊂ Cn, let A = p1, . . . , pk be ak-tuple ofpairwise distinct points inD and letW = ν1, . . . , νk be ak-tuple ofpositive real numbers. The pluricomplex Green function ofD with polesin A of weights inW , is defined bygD(z,A,W ) = sup u(z), where thesupremum is taken over the class of negative plurisubharmonic functionsuin D which have a logarithmic pole atpj with weightνj , for j = 1, . . . , k.These functionsexist for anyk-tuplesA,W , and their definition is in analogyto the one dimensional case (see [Kl1], [L]). In [D] and [L] it is shownthat if Ω is hyperconvex thengD(·, A,W ) is the unique solution to thefollowing Dirichlet problem for the complex Monge-Ampere operator:u ∈PSH(D)

⋂C(D\A),u(z)−νj log ‖z−pj‖ = O(1)asz → pj ,(ddcu)n =∑k

j=1 νnj δpj ,u = 0 on∂D. Here, aswell as in the sequel,PSH(D) denotes

the class of plurisubharmonic functions onD, d = ∂+∂, dc = 12πi(∂−∂),

andδpj is the Dirac mass atpj .

112 D. Coman

In the case of entire plurisubharmonic functions,L+ is defined to be theset of functionsu ∈ PSH ∩L∞

loc(Cn) such thatu(z) = log ‖z‖ +O(1) as

‖z‖ → ∞ (see e.g. [B] and [Kl2]). The following Monge-Ampere equationis studied in [Ko1] and [Ko2]:u ∈ L+, (ddcu)n = f dλ, whereλ is theLebesgue measure andf is a non-negative measurable function. In [Ko1] itis shown that this equation has a continuous solution for certain classes offunctionsf , and regularity questions are studied in [Ko2]. The uniquenessof the solution is proved in [BT3]: ifu, v ∈ L+ and(ddcu)n = (ddcv)n

thenu− v is constant.In this paper we study entire plurisubharmonic functions with logarith-

mic growth (in analogy to the classL+), whose Monge-Ampere measureis a finite sum of Dirac masses. LetA = p1, . . . , pk be ak-tuple of pair-wise distinct points inCn andW = ν1, . . . , νk be ak-tuple of positiveweights. Without loss of generality we shall assume throughout the paperthat ν1 ≥ ν2 ≥ . . . ≥ νk > 0. As in the case of a bounded domain, wecan consider the following classesMn(A,W ) of entire pluricomplex Greenfunctions. By definition,Mn(A,W ) is the set of solutions of the followingMonge-Ampere equation:

u ∈ PSH(Cn) ∩ L∞loc(C

n \A) ,

u(z) − νj log ‖z − pj‖ = O(1) asz → pj ,

lim‖z‖→∞

(u(z)/ log ‖z‖) ∈ (0,+∞) exists,

(ddcu)n =k∑

j=1

νnj δpj .

(1.1)

When the weightsνj are equal, hence without loss of generality equal to 1,we denote the corresponding class byMn(A). If A consists of one point,we may assumeA = 0, ν1 = 1, and we denote the corresponding classbyMn(0).

We remark thatM1(A,W ) consists precisely of the functions∑k

j=1 νj

log |z − pj | + c, for arbitrary constantsc. Functions inMn(A,W ) canbe viewed as higher dimensional generalizations of these classical Greenfunctions. However, we shall see that whenn > 1 Mn(A,W ) can be insome cases empty. In some other cases we will describe all the elements ofMn(A,W ). In these casesMn(A) has actually many elements, in the sensethat they are not unique up to addition of constants.

In Sect. 2 of the paper we collect a few known results. In Sect. 3 we studythe case whenA = f−1(0), wheref : Cn → Cn is a holomorphic mapsuch that 0 is a regular value forf and the following holds for some integer

Green functions onCn 113

d ≥ 2:

0 < lim inf‖z‖→∞

‖f(z)‖‖z‖d

≤ lim sup‖z‖→∞

‖f(z)‖‖z‖d

< +∞ .(1.2)

We show that the operatorf : Mn(0) → Mn(A), f v = v f , is welldefined and bijective. Using the description of the elements ofMn(0) interms of their Robin functions (given in Sect. 4), we then have a completecharacterization ofMn(A).

In Sect. 4 we deduce some general properties ofMn(A,W ). WhenA consists of one element the corresponding classMn(0) can be easilycharacterized in terms of Robin functions (see also [BT2]). Whenk =cardA ∈ 2, . . . , 2n − 1 we show thatMn(A,W ) = ∅, and whenk = 2n thenMn(A,W ) = ∅ unless the weights are all equal. More-over we remark here thatMn(A,W ) carries a certain algebraic structure:if u, v ∈ Mn(A,W ) thenc1u+ c2v + c3 maxu, v ∈ Mn(A,W ), wherecj ≥ 0 andc1 + c2 + c3 = 1. This is of interest since in general the sum orthe maximum of maximal plurisubharmonic functions is not maximal.

In Sect. 5 we consider the casen = 2. Using results of previous sectionswe obtainM2(A)whencardA = 4. We prove thatM2(A) = ∅ if cardA =5 or cardA = 6, and we give a simple geometric way of constructing setsA for whichM2(A) = ∅ is found explicitly.

Let us recall now that every compact polar setE ⊂ C has an Evanspotential, i.e. there exists a subharmonic function with logarithmic growthonC which is harmonic onC \ E and equal to−∞ onE (see e.g. [R]).Our classesMn(A,W ) can be thought as higher dimensional analogues ofEvans potentials for finite setsA ⊂ Cn. We would like to thank ProfessorJ. Siciak for this observation.

Acknowledgements.Part of this work was done while I was visiting the University of Wup-pertal as a Humboldt Research Fellow. I am grateful to Professor Klas Diederich for theinvitation, and to the Humboldt Foundation for their support.

2 Preliminary results

LetD andD′ be domains inCn andf : D → D′ be a holomorphic map.The operator

f : PSH(D′) → PSH(D) , f v = v f ,(2.1)

is well defined. In addition, iff is proper then (see e.g. [Kl2]) the operator

f : PSH(D) → PSH(D′) , u → f u ,(2.2)

(f u)(w) = maxu(z) : f(z) = w ,

114 D. Coman

is well defined. The following lemma is a direct consequence of these defi-nitions:

Lemma 2.1 We havef f = IdPSH(D′), hencef is surjective andf is

injective.

The next lemma is a particular case of a result of Taylor [T]:

Lemma 2.2 Letu, v ∈ PSH(Cn)∩L∞loc(C

n \K), whereK is a compact

set. Iflim‖z‖→∞ v(z) = +∞ and lim‖z‖→∞u(z)v(z) = 1 then

∫Cn(ddcu)n =∫

Cn(ddcv)n.

The next result is due to Demailly [D]. His proof (of a more generalstatement) is given in the case when the functions involved are continuous,but it can be adapted to the situation below (see also [Kl2]).

Lemma 2.3 Letu, v ∈ PSH(D) ∩ L∞loc(D \ a) satisfyu(a) = v(a) =

−∞ andu(z)+c < v(z) < u(z)+c′ for z = a in some ballB(a,R) ⊂ D.Then(ddcu)na = (ddcv)na.

3 The caseA = f−1(0)

We consider here the situation when the weights are equal andA = f−1(0),wheref : Cn → Cn is a holomorphic map which satisfies (1.2) and suchthat 0 is a regular value forf . Let us writef = (f1, . . . , fn) and start bystating without proof a few simple properties off .

Lemma 3.1 f is a proper polynomial mapping anddeg(fj) = d for all j.Moreover fori = j, fi andfj have no common factors.

For (z, t) ∈ Cn+1 we define the homogeneous polynomialsfj(z, t) =tdfj

(zt

)and we letf : Pn → Pn, f [z : t] = [f1(z, t) : . . . : fn(z, t) : td],

be the extension off to Pn. We use the standard embeddingCn → Pn,z → [z : 1]. Moreover, we denote byfd

j the homogeneous part of degreed of fj . The next lemma, whose easy proof is omitted, is a consequence ofBezout’s theorem:

Lemma 3.2 The polynomialsfd1 , . . . , f

dn have no common zeros other than

the origin, hence the mapf : Pn → Pn is holomorphic. The proper mapfis dn to 1, hence the setA hasdn elements.

We will use in what follows the description of the classesMn(A) andMn(0) given in Proposition 4.1, respectively in Lemma 4.3 (see Sect. 4).Sincef : Cn → Cn is proper holomorphic the operatorsf andf , givenin (2.1) respectively (2.2), are well defined. In fact we have the following:

Green functions onCn 115

Lemma 3.3 f (Mn(0)) ⊆ Mn(A), f (Mn(A)) ⊆ Mn(0), and f f =

Id : Mn(0) → Mn(0).

Proof. Let v ∈ Mn(0). Since 0 is a regular value forf , the functionf v hasdn logarithmic poles inA. Sincec‖z‖d ≤ ‖f(z)‖ ≤ C‖z‖d for ‖z‖ suffi-ciently large it follows thatlim‖z‖→∞

(fv)(z)log ‖z‖ = d, hencef v ∈ Mn(A) by

Proposition 4.1. Ifu ∈ Mn(A) thenf u ∈ L∞loc(C

n\0) has a logarithmicpole at 0, sinceA = f−1(0). By the above estimates on‖f(z)‖ it followsthat if ‖w‖ is sufficiently large thenc1‖w‖ 1

d ≤ ‖z‖ ≤ C1‖w‖ 1d , for all z

with f(z) = w. This implieslim‖w‖→∞(fu)(w)log ‖w‖ = 1, sof u ∈ Mn(0).

We can now state the main result of this section:

Theorem 3.4 The operatorsf : Mn(0) → Mn(A), f : Mn(A) →Mn(0) are bijective and(f )−1 = f .

Using this theorem and Lemma 4.3 we obtain the following completecharacterization of the classMn(A):

Corollary 3.5 Every functionu ∈ Mn(A) can be uniquely written in theform

u(z) = log ‖f(z)‖ + h(π(f(z))) , z ∈ Cn \A ,

whereh ∈ BP1(Pn−1). Conversely, any suchhdefines by the above formulaan element ofMn(A).

In the proof of Theorem 3.4 we will need the following:

Lemma 3.6 For a = [a1 : . . . : am] ∈ Pm−1 let Ha be the hyperplaneof Cm defined by the equationa1z1 + . . . + amzm = 0. If V is a propersubvariety ofPm−1, the setU =

⋃Ha : a ∈ Pm−1 \ V contains all ofCm except the union of finitely many complex lines through the origin.

Proof. Let z ∈ Cm \ U . Then the hyperplaneHz = a ∈ Pm−1 : z1a1 +. . .+ zmam = 0 is contained inV . But V contains at most finitely manyhyperplanes ofPm−1, which we denote byHwj , wj ∈ Cm \ 0, j =1, . . . , l. If Lj = λwj : λ ∈ C, it follows thatz ∈ L1 ∪ . . . ∪ Ll. Proof of Theorem 3.4.It suffices to show thatf f = Id : Mn(A) →Mn(A). We letu ∈ Mn(A) andu = f f u ∈ Mn(A). Thenu is given by

u(z) = maxu(z′) : f(z′) = f(z) .(3.1)

Clearlyu ≤ u onCn, and we have to proveu = u. This will be done byconsidering inverse imagesf−1(L) ⊂ Cn of complex linesL in Cn whichpass through the origin. Thenf−1(L) determines an algebraic curve inPn

116 D. Coman

andA ⊂ f−1(L). Sinceu = f v for v = f u ∈ Mn(0), we see by Lemma4.3 thatu(z) = log ‖f(z)‖ + c for z ∈ f−1(L), wherec is some constant.Without loss of generality we may assume thatL is given by the equationswj = βjw1, j = 2, . . . , n. LetG(z) = log |f1(z)| + 1

2 log(1 + ‖β‖2) + c,whereβ = (β2, . . . , βn). ThenG = u on f−1(L). Sincef1 = 0 ∩f−1(L) = A, u − G is plurisubharmonic on an open neighborhood off−1(L)\A inCn. It follows thatu− u ≤ 0 is subharmonic onf−1(L)\A.For a fixedz ∈ f−1(L)\Awe have thatf−1(f(z)) ⊂ f−1(L)\A is a finiteset, hence by (3.1) there existsz′ ∈ f−1(f(z)) such thatu(z′) = u(z′). Itthen follows by the maximum principle thatu = u alongf−1(L), providedthatf−1(L) is connected.

In the remainder of the proof we show thatf−1(L) is connected for ”suf-ficiently many” linesL in order to conclude thatu = u onCn. For this wewill use two classical theorems of Bertini about linear systems on projec-tive manifolds (see e.g. [N], p.255), and the following result from [FH]: ifX, Y ⊂ Pn are irreducible varieties such thatdim(X)+dim(Y ) > n thenX ∩ Y is connected. We adopt the following notation: ifH is a (complex)vector subspace ofCn thenf−1(H) ⊂ Cn determines a subvariety ofPn

which we still denote byf−1(H).We consider the linear system of divisors onPn Λ = (a1f1 + . . . +

anfn) determined by the rational mapf = [f1 : . . . : fn] : Pn → Pn−1.By Lemma3.1, this linear system iswithout fixed components. Note thatf isnon-degenerate, in the sense thatf(Pn) is not contained in any hyperplaneof Pn−1. In fact f is onto, sodim(f(Pn)) = n − 1. The base locus ofΛ is Bs(Λ) = A ⊂ Cn. Indeed, if[z : 1] ∈ Bs(Λ) then z ∈ A, andBs(Λ) ∩ t = 0 = [z : 0] : fd

1 (z) = . . . = fdn(z) = 0 = ∅ by Lemma

3.2. Moreover, since 0 is a regular value off we see that every element ofΛ is smooth at each point ofBs(Λ) = A. Bertini’s theorems imply thatthe generic element ofΛ is a smooth hypersurface ofPn, irreducible ifn − 1 > 1. These facts hold also whenn = 2, since any algebraic curvein P2 is connected. We denote byFn−1 = a1w1 + . . . + anwn = 0 thefamily of (n− 1)-dimensional vector subspaces ofCn which correspond tothe smooth and connected elements ofΛ. ThenFn−1 contains the generic(n − 1)-dimensional vector subspace ofCn, and forV ∈ Fn−1, f−1(V )determines a smooth connected hypersurface ofPn.

We fix nowV ∈ Fn−1 and letM be the corresponding connected sub-manifold ofPn, given bya1f1 + . . .+anfn = 0. Without loss of generalitywe assumean = 1, and we consider the rational mapf : M → Pn−2,f = [f1 : . . . : fn−1]. Sincef : M ∩ Cn → V is onto, so is the mapf , hence it determines the linear systemΛ = (b1f1 + . . . + bn−1fn−1)onM . If [z : 1] ∈ M ∩ Bs(Λ) thenf1(z) = . . . = fn−1(z) = 0, hencefn(z) = 0 by using the defining equation ofM , so z ∈ A. Moreover,

Green functions onCn 117

points [z : 0] ∈ M cannot be inBs(Λ) (otherwise it would follow thatfd1 (z) = . . . = fd

n−1(z) = 0, sofdn(z) = 0). ThusBs(Λ) = A, and every

divisor inΛ is smooth at each base point ofΛ. By Bertini’s theorems thegeneric element ofΛ is a smoothhypersurfaceofM , irreducible ifn−2 > 1.WedenotebyF(V ) theset of(n−2)-dimensional subspacesofV whichcor-respond to these elements ofΛ, andwe letFn−2 =

⋃F(V ) : V ∈ Fn−1.Proceeding in this manner we construct for eachj ∈ 2, . . . , n − 1

a family Fj of j-dimensional vector subspaces ofCn with the followingproperties:

– For everyVj+1 ∈ Fj+1,Fj contains the genericj-dimensional subspaceof Vj+1.

– If Vj ∈ Fj thenf−1(Vj) determines an irreducible (hence connected)submanifold ofPn of dimensionj.

In order to construct a familyF1 with these properties, we fixV ∈ F2.In view of the above construction,M = f−1(V ) is cut by a triangularsystem of equations, without loss of generality of the formaj

1f1 + . . . +aj

n−j+1fn−j+1 = 0, whereajn−j+1 = 0 andj = 1, . . . , n − 2. ThenΛ =

(b1f1 +b2f2) is a one dimensional linear system onM determined by therational mapf : M → P1, f = [f1 : f2]. As beforeBs(Λ) = A, hence thegeneric element ofΛ is a smooth curve inM . The generic element ofFn−1intersectsV in a complex line. It follows by Lemma 3.6 that all but finitelymany lines L (passing through 0) inV can be written asL = V ∩ Vn−1, forVn−1 ∈ Fn−1. Since the varietiesf−1(V ) andf−1(Vn−1) are irreducible ofdimension 2 respectivelyn− 1, the result of [FH] mentioned above impliesthatf−1(L) = f−1(V ) ∩ f−1(Vn−1) is a connected algebraic curve inPn.We conclude that the generic element ofΛ is a smooth connected (henceirreducible) curve, and we letF(V ) be the family of lines corresponding tothese elements ofΛ. Then the family of linesF1 =

⋃F(V ) : V ∈ F2verifies the above two properties of the familiesFj .

For any lineL ∈ F1 we havef−1(L) ⊂ Cn is connected, henceu = ualongf−1(L), by the considerations at the beginning of the proof. Next,for a fixedV ∈ F2, F1 contains the generic line ofV . By Lemma 3.6 andsincef is proper it follows thatu = u almost everywhere with respect to theLebesguemeasure onf−1(V ). Henceu = u onf−1(V ), for everyV ∈ F2.Continuing like this we conclude thatu = u everywhere onCn, and theproof is complete.

118 D. Coman

4 The classesMn(A, W )

Recall the definition (1.1) of the classMn(A,W ), whereA = p1, . . . , pk,W = ν1, . . . , νk, ν1 ≥ ν2 ≥ . . . ≥ νk > 0, and let

α = α(n,W ) = (νn1 + . . .+ νn

k )1n .(4.1)

We begin with the following simple characterization ofMn(A,W ):

Proposition 4.1 Let u ∈ PSH(Cn) ∩ L∞loc(C

n \ A) such thatu has alogarithmic pole at eachpj ∈ A with weight νj ∈ W , and the limit

γ = lim‖z‖→∞u(z)

log ‖z‖ ∈ (0,+∞) exists. Thenγ ≥ α. Moreover,u ∈Mn(A,W ) if and only ifγ = α.

Proof. By Lemmas 2.2 and 2.3 we have

γn =∫

Cn

(ddcu)n =∫

Cn\A(ddcu)n +

k∑j=1

νnj ≥ αn .

Sinceu ∈ Mn(A,W ) if and only if∫

Cn\A(ddcu)n = 0 the propositionfollows.

As a consequence we see thatMn(A,W ) has the following algebraicproperty:

Corollary 4.2 Let u, v ∈ Mn(A,W ) and c1, c2, c3 ≥ 0 be such thatc1 + c2 + c3 = 1. Thenρ = c1u+ c2v + c3 maxu, v ∈ Mn(A,W ).

Proof. Note thatρ has logarithmic poles in(A,W ) andlim‖z‖→∞ρ(z)

log ‖z‖ =α.

We now consider the case whenA has one element, sayA = 0 andα = ν1 = 1. Foru ∈ Mn(0) and for a fixedz ∈ Cn \ 0, the functiont ∈ C → u(tz)−log ‖tz‖extends to a subharmonic function onCwhich iso(log |t|) as|t| → ∞, hence it is constant. Sowe canwriteu(z) = log ‖z‖+h(z), for z ∈ Cn\0, whereh is upper semicontinuous and constant alonglines through the origin, hence it is bounded. Ifπ : Cn \ 0 → Pn−1 is thestandard projection map thenh induces a bounded upper semicontinuousfunctionh onPn−1 such thath = hπ. The functionh is theRobin functionof u (see [BT2]). Sinceu is plurisubharmonic it follows thatddch + ω ≥0, whereω is the standard Kahler form onPn−1 (corresponding to theFubini-Study metric). Following [BT2] we writeBP1(Pn−1) for the classof bounded upper semicontinuous functionsh onPn−1 with ddch+ω ≥ 0.We summarize this facts in the following lemma:

Lemma 4.3 The operatorBP1(Pn−1) → Mn(0), h → u, whereu(z) =log ‖z‖ + h(π(z)), is well defined and bijective.

Green functions onCn 119

We remark that the classBP1(Pn−1) has many elements. Indeed, by[Ko1], if F ∈ Lp(Pn−1), p > 1, is non-negative and is normalized by∫

Pn−1 Fωn−1 = 1, then there existsh ∈ C(Pn−1) such thatddch+ ω ≥ 0

and(ddch+ ω)n−1 = Fωn−1.If n ≥ 2 thenMn(A,W ) = ∅ in the case when the number of poles is

not large enough:

Proposition 4.4 i) Letu andγ be as in Proposition 4.1. Thenγ ≥ ν1 + ν2.ii) If k ∈ 2, . . . , 2n − 1 thenMn(A,W ) = ∅ for every choice of

k-tuplesA and W . If k = 2n and the weightsνj are not equal thenMn(A,W ) = ∅.

Proof. i) Let v(t) = u(p1 + t(p2 − p1)), t ∈ C. By Lemmas 2.2 and 2.3we have

γ =12π

∫C

∆v =12π

∫C\0,1

∆v + ν1 + ν2 ≥ ν1 + ν2 .

ii) We assumeMn(A,W ) = ∅ andk ≥ 2. Then by Proposition 4.1 andpart i) we haveα ≥ ν1 + ν2, whereα is defined in (4.1). By the orderingassumption on the weights we have

νn1 + . . .+ νn

k ≥ (ν1 + ν2)n ≥ νn1 + (2n − 1)νn

2 .(4.2)

Henceνn1 + (k − 1)νn

2 ≥ νn1 + . . . + νn

k ≥ νn1 + (2n − 1)νn

2 , whichshowsk ≥ 2n. If k = 2n this impliesν2 = ν3 = . . . = νk, so by (4.2)νn1 + (2n − 1)νn

2 = (ν1 + ν2)n. Using the binomial expansion andν1 ≥ ν2this showsν1 = ν2.

We note thatMn(A) can be non-empty whenk = 2n. For instance,assume thatA consists of the2n points ofCn whose coordinates takevalues in the set−1, 1, and apply Corollary 3.5 tof(z1, . . . , zn) =(z2

1 − 1, . . . , z2n − 1).

5 The casen = 2

Wediscussnow theclassesM2(A)whencardA ≤ 6. In thecaseof onepole,M2(0) is identified by Lemma 4.3 withBP1(P1). We also mention here thefollowing well-known fact. Ifu ∈ M2(0) is such that the functionv = e2u

is of classC2 near 0, thenu(z) = 12 log(a|z1|2 + 2(bz1z2) + c|z2|2),

where the expression inside the logarithm is positive definite. Indeed, fort ∈ C we havev(tz) = |t|2v(z). Differentiating this with respect tot andtand lettingt = 0 yields the above formula.

By Proposition 4.4,M2(A,W ) = ∅ if cardA ∈ 2, 3, or if cardA = 4and the weights are not equal. We assume now thatA = p1, p2, p3, p4. If

120 D. Coman

there exists a complex line passing through three of the poles thenM2(A) =∅. Indeed, we assume for a contradiction thatu ∈ M2(A) and the linep1p2containsp3. Then the subharmonic functionv(t) = u(p1 + t(p2 − p1)),t ∈ C, has three logarithmic poles, so12π

∫C∆v ≥ 3, but growth at infinity

like 2 log |t|, so 12π

∫C∆v = 2.

We consider now the case when no three poles lie on the same complexline. By an affine change of coordinates wemay assumeA = (0, 0), (1, 0),(0, 1), (t1, t2),wheret1 = 0 = t2, t1+t2 = 1. Then themapf : C2 → C2,

f(z1, z2) =(z1

(z1 +

1 − t1t2

z2 − 1), z2

(z2 +

1 − t2t1

z1 − 1))

,

(5.1)

satisfies the condition (1.2) withd = 2, A = f−1(0), and 0 is a regularvalue forf . In fact, it is easy to see that any non-degenerate holomorphicmaph : C2 → C2 such thath(A) = 0and‖h(z)‖ ≤ O(‖z‖2)as‖z‖ → ∞,is of the formh = L f for someL ∈ GL2(C). Hence the elements ofM2(A) are described by Corollary 3.5 with the mapf of (5.1).

In the case of 5 or 6 poles we have the following:

Theorem 5.1 If cardA ∈ 5, 6 thenM2(A) = ∅.

Proof. We assume for a contradiction that there existsu ∈ M2(A). Letα bethenumber defined in (4.1). Thenα ∈ √

5,√

6. Sinceα < 3 it follows thatno three poles inA can lie on the same complex line. By an affine change ofcoordinates we may assume thatA′ = (0, 0), (1, 0), (0, 1), (t1, t2) ⊂ A,wheret1 = 0 = t2, t1 + t2 = 1. We fix w ∈ A \ A′ and for the setA′we consider the mapf in (5.1). Thenf(w) = 0 and we denote byL thecomplex line passing through 0 andf(w). The conicC ⊂ P2 determined byf−1(L) containsA′ ∪w and it is irreducible, hence non-singular. Indeed,otherwiseC would be the union of two lines so we would have at least threepoles on the same line. There exists a holomorphic embeddingF : P1 → P2

such thatF (P1) = C (see e.g. [Ki]).The conicC intersects the hyperplanet = 0 at infinity in P2 in two

distinct points or in a double point. In the first case we may assume thatthese points correspond to0,∞ ⊂ P1, henceF : C → C2 is a holo-morphic embedding withF (C ) = f−1(L). In the second case we obtaina holomorphic embeddingF : C → C2 with F (C) = f−1(L). In orderto discuss these cases together we writeF : D → C2, F (D) = f−1(L),whereD = C orD = C , and∂D = ∞ respectively∂D = 0,∞. Byabove, whenζ → ∂D thenF (ζ)will approach the point(s) at infinity onC.

We consider now the subharmonic functions onD ρ(ζ) = u(F (ζ))andv(ζ) = α

2 log ‖f(F (ζ))‖. The functionv is actually harmonic onD \F−1(A′), and it has four logarithmic poles with weightα/2 at the points of

Green functions onCn 121

F−1(A′). The functionρ has five logarithmic poles (with weight 1) at thepoints ofF−1(A′) ∪ F−1(w). Hence

12π

∫D∆v = 2α ,

12π

∫D∆ρ ≥ 5 .

We note that

limζ→∂D

v(ζ)ρ(ζ)

= lim‖z‖→∞

α log ‖f(z)‖2u(z)

= 1 ,

hence∫D∆ρ =

∫D∆v. If D = C this follows from Lemma 2.2, and when

D = C we use Lemma 5.2 below. We conclude that2α ≥ 5, which isimpossible sinceα ∈ √

5,√

6, and the proof is complete. Lemma 5.2 LetD be a domain inC andu, v be subharmonic functions onD such thatlimζ→∂D u(ζ) = +∞ and limζ→∂D

u(ζ)v(ζ) = 1. Then

∫D∆u =∫

D∆v.

Proof. The proof is similar to the one of Lemma 2.2–see [T]. By takingthe maximum of each of the functionsu andv with 1, we may assume thatu, v ≥ 1 onD. Forε, C > 0 the setS = S(ε, C) = (1+ ε)u < v+C isrelatively compact inD, andS(ε, C) D asC +∞. By the comparisontheorem

∫S ∆v ≤ (1 + ε)

∫S ∆u. LettingC +∞ and thenε 0 we

conclude that∫D∆v ≤ ∫

D∆u. WesawbyTheorem3.4 thatM2(A)canbenon-emptywhencardA = 9.

Given a subsetA ofC2 with 9 elements the following geometric conditionsare necessary in order forM2(A) to be non-empty. First, no complex lineshould pass through 4 poles inA. Moreover, iff is the map of (5.1) cor-responding to any 4 poles inA which are in general position and ifL isany complex line through the origin, thenf−1(L) cannot contain 7 of theelements ofA. This follows by the same argument as the one in the proofof Theorem 5.1.

We conclude with a sufficient geometric condition forM2(A) to be non-empty.

Proposition 5.3 Let C1, C2 be algebraic curves inC2 of degreed ≥ 2,defined by the polynomialsf1 respectivelyf2, such that the setA = C1 ∩C2hasd2 elements. Then the conclusion of Theorem 3.4 holds for the mapf = (f1, f2).

Proof. We have to check thatf satisfies (1.2) and 0 is a regular valuefor f . Consider the homogeneous polynomials onC3 fj(z, t) = tdfj

(zt

),

j = 1, 2, and the setS = [z : t] : f1(z, t) = f2(z, t) = 0 ⊂ P2. AsS ∩ t = 0 is finite, it follows by Bezout’s theorem thatS = A ⊂ C2. We

122 D. Coman

assumelim inf‖z‖→∞‖f(z)‖‖z‖d = 0. This implieslim inf‖z‖→∞

‖fd(z)‖‖z‖d = 0,

wherefd = (fd1 , f

d2 ) andfd

j is the homogeneous part of degreed of fj .Hence there existsz0 ∈ C2 with ‖z0‖ = 1 andfd(z0) = 0. Then [z0 :0] ∈ S, which contradictsS = A, so we conclude thatf verifies (1.2).This showsf is proper, hence 0 is a regular value off sincef−1(0) hasd2

elements.

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