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Plurisubharmonic functions with singularities and affine invariants for finite sets in ℂn

Abstract : This paper studies the problem of determining the minimal growth of plurisubharmonic (psh) functions on Cn that have logarthmic poles on a given finite set S, and particularly those that are maximal on Cn∖S. There always exist such psh functions of logarithmic growth, for example, u(z)=∑p∈Slog|z−p|, which grows like γlog|z|+O(1) as |z|→∞ for γ=|S|, the number of elements in S. The main results are stated in terms of two affine invariants γ(S) and γ˜(S) defined as follows. Let E˜(S) denote the class of all psh functions of logarithmic growth with poles on S, that is, u(z)=O(log|z|) as |z|→∞ and u(z)=log|z−p|+O(1) as z→p∈S. Let E(S) denote the subclass consisting of those functions in E˜(S) that are (locally) maximal outside of S. For u∈E˜(S), let γu=limsup|z|→∞u(z)/log|z|. The numbers γ˜(S)=inf{γu:u∈E˜(S)} and γ(S)=inf{γu:u∈E(S)} then provide a measure of this minimal growth. One of the nice results of this paper is that |S|1/n≤γ˜(S)≤γ(S)≤|S| and that the left and right hand inequalities cannot be improved in general. They further show that equality in the left hand inequality, γ˜(S)=|S|1/n, occurs if and only if S is contained in a complex line. The authors also state that they do not know any examples where γ˜(S) is strictly smaller than γ(S). Several other interesting results appear in the paper, including comparison with the affine invariants ω(S), the singular degree of M. Waldschmidt [in Séminaire Pierre Lelong (Analyse) année 1975/76, 108–135. Lecture Notes in Math., 578, Springer, Berlin, 1977; MR0453659], and m(S)=maxjmj(S)/j where mj(S) denotes the maximal number of points in S that can lie on an algebraic curve of degree j. They prove ω(S)≤γ˜(S)n−1 and m(S)≤γ˜(S). The paper also continues work of the first author [Math. Z. 235 (2000), no. 1, 111–122; MR1785074] who studied the problem of determining when there exists a function in E˜(S) such that the limit lim|z|→∞u(z)/log|z| exists. He showed that such functions do not exist for some sets S, and gave sufficient conditions for their existence. The authors show here that in some cases when n=2, this sufficient condition is also necessary. They also handle all examples when S has a small number of points (including |S|≤8).
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Coman Dan, Stephanie Nivoche. Plurisubharmonic functions with singularities and affine invariants for finite sets in ℂn . Mathematische Annalen, Springer Verlag, 2002, 322 (2), pp.317-332. ⟨hal-01293626⟩

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