Open Access

On convergence sets of formal power series

Complex Analysis and its Synergies20151:4

https://doi.org/10.1186/s40627-015-0004-4

Received: 10 March 2015

Accepted: 16 March 2015

Published: 9 September 2015

Abstract

The convergence set of a divergent formal power series \(f(x_{0},\dots ,x_{n})\) is the set of all “directions” \(\xi \in \) \(\mathbb {P}^{n}\) along which f is absolutely convergent. We prove that every countable union of closed complete pluripolar sets in \(\mathbb {P}^{n}\) is the convergence set of some divergent series f. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional analogs of the results of Sathaye (J Reine Angew Math 283:86–98, 1976), Lelong (Proc Am Math Soc 2:11–19, 1951), Levenberg and Molzon (Math Z 197:411–420, 1988), and of Ribón (Ann Scuola Norm Sup Pisa Cl Sci (5) 3:657–680 2004) are obtained.

Keywords

Formal power seriesConvergence setsPluripolar setsProjective hulls

Mathematics Subject Classification

Primary: 32A0530C8540A05

1 Background

A formal power series \(f(x_{0},x_{1},\dots ,x_{n})\) with coefficients in \(\mathbb {C}\) is said to be convergent if it converges absolutely in a neighborhood of the origin in \(\mathbb {C}^{n+1}\). A classical result of Hartogs (see [5]) states that a series f converges if and only if \(f_{z}(t):=\) \(f(z_{0}t,z_{1}t,\dots ,\) \(z_{n}t)\) converges, as a series in t, for all \(z\in \mathbb {C}^{n+1}\). This can be interpreted as a formal analog of Hartogs’ theorem on separate analyticity. Because a divergent power series still may converge in certain directions, it is natural and desirable to consider the set of all \(z\in \mathbb {C}^{n+1}\) for which \( f_{z} \) converges. Since \(f_{z}(t)\) converges if and only if \(f_{w}(t)\) converges for all \(w\in \mathbb {C}^{n+1}\) on the affine line through z,  ignoring the trivial case \(z=0,\) the set of directions along which f converges can be identified with a subset of \(\mathbb {P}^{n}.\) The convergence set \({\text {Conv}}(f)\) of a divergent power series f is defined to be the set of all directions \(\xi \in \mathbb {P}^{n}\) such that \(f_{z}(t)\) is convergent for some \(z\in \pi ^{-1}(\xi )\), where \(\pi : \mathbb {C}^{n+1}\backslash \{0\}\rightarrow \mathbb {P}^{n}\) is the natural projection. For the case \(n=1\), Lelong [9] proved that the convergence set of a divergent series \(f(x_{1},x_{2})\) is an \(F_{\sigma }\) polar set (i.e. a countable union of closed sets of vanishing logarithmic capacity) in \(\mathbb {P}^{1}\), and moreover, every \(F_{\sigma }\) polar subset of \(\mathbb {P}^{1}\) is contained in the convergence set of a divergent series \(f(x_{1},x_{2})\). The optimal result was later obtained by Sathaye (see [16]) who showed that the class of convergence sets of divergent power series \(f(x_{1},x_{2})\) is precisely the class of \(F_{\sigma }\) polar sets in \(\mathbb {P}^{1}\). To study the collection \({\text {Conv}}(\mathbb {P}^{n})\) of convergence sets of divergent series in higher dimensions we consider the class \({\text {PSH}}_{\omega }(\mathbb {P} ^{n})\) of \(\omega \)-plurisubharmonic functions on \(\mathbb {P}^{n}\) with respect to the form \(\omega :=dd^{c}\log |Z|\) on \(\mathbb {P}^{n}\). We show that \({\text {Conv}}(\mathbb {P}^{n})\) contains projective hulls of compact pluripolar sets and countable unions of projective varieties. We prove that each convergence set (of divergent power series) is a countable union of projective hulls of compact pluripolar sets. Our main result states that a countable union of closed complete pluripolar sets in \(\mathbb {P}^{n}\) belongs to \({\text {Conv}}(\mathbb {P}^{n})\). This generalizes the results of Lelong [9], Levenberg and Molzon [10], and Sathaye [16]. Our line of approach was inspired by [16], and influenced by the methods developed in [10, 13, 15].

We also consider convergence sets of a formal power series of the type
$$\begin{aligned} f(t,x)=\sum _{j=1}^{\infty }P_{j}(x)t^{j}\in \mathbb {C}[x_{1},x_{2},...,x_{n}][[t]], \end{aligned}$$
where \({{\mathrm{deg}}}\,P_{j}\le j\). The affine convergence set \({\text {Conv}}_{a}(f)\ \) of a divergent power series f(tx) is defined to be the set of all \(x\in \mathbb {C}^{n}\) for which f(tx) is convergent as a series in t. We prove that a countable union of closed complete pluripolar sets is an affine convergence set.

2 Pluripolar sets in \(\mathbb {C}^{n}\)

Let \(\mathbb {C}[[x]]:=\mathbb {C}[[x_{1},\dots ,x_{n}]]\) denote the ring of formal power series with complex coefficients in n indeterminates \(x_1,\dots , x_n\). Let \(\mathbb {C}\{x\}\) be the ring of all power series \(f(x)\in \mathbb {C}[[x]]\ \) that are absolutely convergent in a neighborhood of the origin in \(\mathbb {C}^{n}\).

Let \(\mathcal {H}(\mathbb {C}^{n})\) denote the set of all homogeneous polynomials (including the zero polynomial) in \(x_1,\dots , x_n\) with complex coefficients. For \(k\ge 0\) let \(\mathcal {H}_k(\mathbb {C}^{n})\) denote set of \(p\in \mathcal H(\mathbb {C}^n)\) such that \(p=0\) or p is homogeneous of degree k. So each \(\mathcal H_k(\mathbb {C}^n)\) is a \(\mathbb {C}\)-vector space and \(\mathcal H(\mathbb {C}^n)=\cup _{k=0}^\infty \mathcal H_k(\mathbb {C}^n)\).

For an integer \(k\ge 0,\) let \(\mathcal {P}_{k}(\mathbb {C}^{n})\) denote the set of polynomials of degree at most k in \(x_1,\dots , x_n\) with complex coefficients. For convenience the zero polynomial is considered to have degree \(-1\). So \(\mathcal P_k(\mathbb {C}^n)\) contains 0 and it is a \(\mathbb {C}\)-vector space for \(k\ge 0\). In particular \(\mathcal P_0(\mathbb {C}^n)=\mathbb {C}\).

A Borel1 subset E of \(\mathbb {C}^{n}\) is said to be pluripolar (polar when \(n=1\)) if for each point \(x\in E\) there is a plurisubharmonic function u, \(u\not \equiv -\infty \), defined in a connected neighborhood U of x in \(\mathbb {C}^{n}\) such that \(u=-\infty \) on \(E\cap U\). A set E is said to be globally pluripolar if there is a nonconstant plurisubharmonic function u defined on \(\mathbb {C}^{n}\) such that \(E\subset \{y:u(y)=-\infty \}\). A theorem of Josefson (see [7]) states that E is pluripolar if and only if E is globally pluripolar.

A set \(E\subset \mathbb {C}^n\) is said to be a complete pluripolar set if there is a non-constant plurisubharmonic function u defined on \(\mathbb {C}^n\) such that \(E= \{y: u(y)=-\infty \}\). So the set \(\{(0, x_2)\in \mathbb {C}^2: |x_2|<1\}\) and its closure are pluripolar, but they are not complete pluripolar sets. A countable union of pluripolar sets is pluripolar. So the set of rationals in the interval [0, 1] is polar. It is not a complete polar set, because each complete pluripolar set is \(G_\delta \). In \(\mathbb {C}\) each \(G_\delta \) polar set is a complete polar set, which is Deny’s Theorem (see [3]).

A subset E of a domain D in \(\mathbb {C}^n\) is said to be a complete pluripolar set in D if there is a non-constant plurisubharmonic function u defined on D such that \(E=\{u=-\infty \}\).

Proposition 2.1

Let w be a nonconstant plurisubharmonic function defined on a Stein manifold \(\Omega \) , and let \(E=\{w=-\infty \}\) . Let v be a continuous, non-negative, plurisubharmonic exhaustion function of \(\Omega \) . Then there is a plurisubharmonic function u on \(\Omega \) such that \(u\le v\) on \(\Omega \) and \(E=\{u=-\infty \}\).

Proof

Let \(V_j=\{x\in \Omega : v(x)<2^j\}\) for \(j\in \mathbb {N}\). Choose an increasing sequence \(\{M_j\}\) of positive numbers such that \(\lim _{j\rightarrow \infty } M_j=\infty \) and \(M_j> \sup _{x\in V_j} w(x)\;\forall j\). For each j, define a function \(u_j\) by
$$\begin{aligned} u_{j}(x)=\left\{ \begin{array}{ll} \max (M_j^{-1}w(x)-1), v(x)-2^j), &{}\quad \text {if} \, x\in V_j, \\ v(x)-2^j, &{}\quad \text {if} \, x\not \in V_j. \end{array} \right. \end{aligned}$$

Then \(u_j\) is plurisubharmonic on \(\Omega \) by the gluing theorem. On each compact subset of \(\Omega \), all but a finite number of \(u_j\) are negative. It follows that the sum \(u(x):=\sum _{j=1}^{\infty }2^{-j}u_{j}(x)\) is plurisubharmonic, since the sequence of the partial sums of the series is eventually decreasing. Since \(u_j(x)\le v(x)\) for each j, we see that \(u(x)\le v(x)\).

Suppose that \(x\in E\). Then \(w(x)=-\infty \), and \(u_j(x)=v(x)-2^j\) for each j. Thus \(u(x)=-\infty \).

Now suppose that \(x\in \Omega {\setminus } E\). There is an m such that \(x\in V_m\). Then
$$\begin{aligned} u(x)\ge & {} \sum _{j=1}^{m-1} 2^{-j}u_j(x)+\sum _{j=m}^\infty 2^{-j}(M_j^{-1}w(x)-1)\\\ge & {} \sum _{j=1}^{m-1} 2^{-j}u_j(x)+\sum _{j=1}^\infty 2^{-j}(-M_1^{-1}|w(x)|-1)\\= & {} \sum _{j=1}^{m-1} 2^{-j}u_j(x)+(-M_1^{-1}|w(x)|-1)>-\infty . \end{aligned}$$
Therefore, \(E=\{x: u(x)=-\infty \}\). \(\square \)

Remark

We got the idea of the proof from Sadullaev (private discussion) and from Bedford and Taylor [2].

For \(x\in \mathbb {C}^{n},\) let \(|x|:=(|x_{1}|^{2}+\cdots +|x_{n}|^{2})^{1/2}\).

Corollary 2.2

Let E be a complete pluripolar set in \(\mathbb {C}^n\) . Then there is a plurisubharmonic function u on \(\mathbb {C}^n\) such that \(u(x)\le (1/2)\log (1+|x|^2)\) on \(\mathbb {C}^n\) and \(E=\{u=-\infty \}\).

Remark

A stronger version of the above corollary appeared in [2].

Let \(B_n\) be the open unit ball in \(\mathbb {C}^n\).

Corollary 2.3

Let E be a complete pluripolar set in \(B_n\) . Then there is a plurisubharmonic function u on \(B_n\) such that \(u(x)\le -\log (1-|x|^2)\) on \(B_n\) and \(E=\{u=-\infty \}\).

Let
$$\begin{aligned} \mathcal {L}(\mathbb {C}^{n})=\{u\in {\text {PSH}}(\mathbb {C}^{n}):\sup _{x\in \mathbb {C}^{n}}(u(x)-(1/2)\log (1+|x|^2))<\infty \} \end{aligned}$$
denote the Lelong class of plurisubharmonic functions. For a polynomial \(P \in \mathcal P_k(\mathbb {C}^n)\), where \(k>0\), the function \((1/k)\log |P(x)|\) is a prototypical member of \(\mathcal {L}(\mathbb {C}^{n}).\)

Corollary 2.2 implies that if E is a complete pluripolar set in \(\mathbb {C}^n\) then there is a \(u\in \mathcal L(\mathbb {C}^n)\) such that \(E=\{u=-\infty \}\).

The pluripolar hull (see [11]) in \(\mathbb {C}^n\) of a pluripolar set E in \(\mathbb {C}^n\) is defined to be
$$\begin{aligned} E^*=\cap \{x\in \mathbb {C}^n: u(x)=-\infty \}, \end{aligned}$$
where the intersection is taken over all plurisubharmonic functions u on \(\mathbb {C}^n\) that are \(-\infty \) on E.
For a polynomial \(P\in \mathcal {P}_{k}( \mathbb {C}^{n}) \) with \(k>0\) and a subset K of \(\mathbb {C}^n\) we set
$$\begin{aligned} \langle P(x)\rangle _{k}=\frac{|P(x)|^{1/k}}{ \sqrt{1+|x|^{2}}},\quad \langle P\rangle _{k, K}=\sup _{x\in K}\langle P(x)\rangle _k,\quad \langle P\rangle _{k}=\langle P\rangle _{k,\mathbb {C}^n}. \end{aligned}$$
Note that if \(P\in \mathcal {P}_{k}\left( \mathbb {C}^{n}\right) \) and m is a positive integer, then \(\left\langle P^m(x)\right\rangle _{km}=\left\langle P(x)\right\rangle _{k}\).

Definition 2.4

Let \(F\subset \mathbb {C}^{n}\), \(F\not =\emptyset \), \(x\in \mathbb {C}^{n}\), and \(0\le r\le 1\). Define
$$\begin{aligned} \tau (x,F,r) =\inf \{\langle P\rangle _{k,F}: k\in \mathbb {N}, P\in \mathcal P_k(\mathbb {C}^n), \langle P(x)\rangle _k\ge r, \langle P\rangle _k\le 1\} \end{aligned}$$
and
$$\begin{aligned} T(x,F) =\sup \{r:0\le r\le 1, \tau (x,F,r)=0\}. \end{aligned}$$

For the empty set, we put \(\tau (x,\emptyset ,r)=0\) and \(T(x,\emptyset )=1\). It follows directly from the definition that if \(E\subset F\), then \(\tau (x,E,r)\le \tau (x,F,r)\) and \(T(x,E)\ge T(x,F)\).

Lemma 2.5

Let \(u\in \mathcal L(\mathbb {C}^n)\) . Suppose that the set \(E:=\{u=-\infty \}\) is closed. Then for each \(x\in \mathbb {C}^{n}{\setminus } E\) , and each non-empty compact set \(K\subset E\) , we have
$$\begin{aligned} T(x,K)\ge (1+|x|^{2})^{-1/2}e^{u(x)-b}, \end{aligned}$$
where
$$\begin{aligned} b:=\sup _{z\in \mathbb {C}^n} (u(z)-\frac{1}{2}\log (1+|z|^{2})). \end{aligned}$$

Proof

Without loss of generality, we assume that \(b=0\). Let \(g(x)=e^{u(x)}\). Then \((1+|x|^{2})^{-1/2}g(x)\le 1\) for \(x\in \mathbb {C}^n\). Fix a \(x\in \mathbb {C}^{n}{\setminus } E\) and a non-empty compact set \(K\subset E\). Let \(r>0\) be such that \(r<(1+|x|^{2})^{-1/2}g(x)\). Let \(\eta \) be a positive number with \(\eta <r\). Let \(\lambda \) be a positive number that is less than the distance between the closed set \(\{y:g(y)\ge \eta \}\) and the compact set K, and that is so small that
$$\begin{aligned} (\lambda +\eta )^{1-\lambda }<\sqrt{\eta }. \end{aligned}$$
(1)
Let
$$\begin{aligned} \omega (y)=\left\{ \begin{array}{ll} c_{n}\exp (-1/(1-|y|^{2})), &{} \text{ if }\,|y|<1, \\ 0, &{} \text{ if }\,|y|\ge 1, \end{array} \right. \;\;\int \omega (y)\,\mathrm{d}y=1, \end{aligned}$$
where \(c_n\) is so chosen that \(\int \omega (t)\,\mathrm{d}t=1\). For \(\mu >0\), let \(g_{\mu }(y)=\int g(y+\mu z)\omega (z)\,\mathrm{d}z\). Then \(g_{\mu }\) is \(C^{\infty }\), positive, and \(g_{\mu }\downarrow g\) as \(\mu \downarrow 0\). We now show that \(\log g_\mu \in \mathcal L(\mathbb {C}^n)\). Let f be a polynomial in \(y\in \mathbb {C}^n\) and let \(a, \zeta \in \mathbb {C}^n\) with \(\zeta \ne 0\). Assume that \(\log g_\mu \le \mathfrak {R}f\) on the boundary \(\partial D:=\{a+w\zeta : w\in \mathbb {C}, |w|=1\}\) of the disc \(D:=\{a+w\zeta : w\in \mathbb {C}, |w|\le 1\}\), that is, \(g_\mu \le |e^f|\) on \(\partial D\). Since \(u(y+\mu z)- \mathfrak {R}f(y)\) is plurisubharmonic in y, its exponential \(g(y+\mu z)|e^{-f(y)}|\) is plurisubharmonic in y. Hence \(g_\mu |e^{-f}|\) is plurisubharmonic and therefore \(\le 1\) on D, that is, \(\log g_\mu \le \mathfrak {R}f\) on D. This proves that \(\log g_\mu \) is plurisubharmonic. For \(y\in \mathbb {C}^n\),
$$\begin{aligned} \begin{aligned} g_{\mu }(y)&\le \int (1+|y+\mu z|^{2})^{1/2}\omega (z)\, \mathrm{d}z\nonumber \\&\le (1+(|y|+\mu )^{2})^{1/2}\nonumber \\&\le (1+\mu )(1+|y|^{2})^{1/2}. \end{aligned} \end{aligned}$$
(2)
Therefore, \(\log g_\mu \in \mathcal L(\mathbb {C}^n)\).
If \(y\in K\), and if \(|z|<1\), then \(y+\lambda z\not \in \{u:g(u)\ge \eta \}\), and hence \(g(y+\lambda z)<\eta \). It follows that \(g_{\lambda }(y)=\int g(y+\lambda z)\omega (z)\,\mathrm{d}z<\int \eta \omega (z)\,\mathrm{d}z=\eta \). By (2) we also have \(g_\lambda (y)\le (1+\lambda )(1+|y|^2)^{1/2}\) for \(y\in \mathbb {C}^n\). As in ([15], p. 17) (we tried to find a more accessible reference without success), we define a function \(\phi _{\lambda }\) on \(\mathbb {C}\times \mathbb {C}^n\) by
$$\begin{aligned} \phi _{\lambda }(y_{0},y)=\left\{ \begin{array}{ll} |y_{0}|(\lambda +g_{\lambda }(y/y_{0}))^{1-\lambda }+\lambda (|y_{0}|^{2}+|y|^{2})^{1/2}, &{} \text{ if }\,y_{0}\not =0, \\ \lambda |y|, &{} \text{ if }\,y_{0}=0. \end{array} \right. \end{aligned}$$
Then \(\phi _\lambda \) is continuous and plurisubharmonic. Moreover, it satisfies \(\phi _\lambda (cw)=|c|\phi _\lambda (w)\) for \(c\in \mathbb {C}\) and \(w\in \mathbb {C}^{n+1}\). We then define \(\psi _{\lambda }\) by
$$\begin{aligned} \psi _{\lambda }(y)=\phi _{\lambda }(1,y)=(\lambda +g_{\lambda }(y))^{1-\lambda }+\lambda (1+|y|^{2})^{1/2}. \end{aligned}$$
Then \(\psi _{\lambda }\) is \(C^{\infty }\) and \(\log \psi _\lambda \in \mathcal L(\mathbb {C}^n)\) by [15, Prop. 2.7].
By [15, Prop. 2.10],
$$\begin{aligned} \phi _{\lambda }(y_{0},y)=\sup \{|h(y_{0},y)|^{1/{{\mathrm{deg}}}\, h}\}, \end{aligned}$$
where the supremum is taken over all homogeneous polynomials h of \(n+1\) variables such that \(|h(z_{0},z)|^{1/{{\mathrm{deg}}}\, h}\le \phi _{\lambda }(z_{0},z)\;\forall (z_{0},z)\in \mathbb {C}\times \mathbb {C}^{n}\). Since \( \psi _{\lambda }(z)/\sqrt{1+|z|^{2}}\) extends to a continuous function on \( \mathbb {P}^{n}\), it follows that
$$\begin{aligned} \psi _{\lambda }(x)=\sup \{|P(x)|^{1/k}: k\in \mathbb {N}, P\in \mathcal {P}_{k}(\mathbb {C}^{n}),\;|P(y)|^{1/k}\le\, \psi _{\lambda }(y)\;\forall y\in \mathbb {C}^{n}\}. \end{aligned}$$
(3)
For all \(y\in \mathbb {C}^{n}\),
$$\begin{aligned} \psi _{\lambda }(y)\,= \,& {} (\lambda +g_{\lambda }(y))^{1-\lambda }+\lambda (1+|y|^{2})^{1/2} \\\le & \,{} (\lambda +(1+\lambda )(1+|y|^{2})^{1/2})^{1-\lambda }+\lambda (1+|y|^{2})^{1/2} \\< & \,{} (1+3\lambda )(1+|y|^{2})^{1/2}. \end{aligned}$$
If \(y\in K\), then
$$\begin{aligned} \psi _{\lambda }(y)\,=\, & {} (\lambda +g_{\lambda }(y))^{1-\lambda }+\lambda (1+|y|^{2})^{1/2} \\\le &\, {} (\lambda +\eta )^{1-\lambda }+\lambda (1+|y|^{2})^{1/2} \\< &\, {} \sqrt{\eta }+\lambda (1+|y|^{2})^{1/2} \\\le &\, {} (\sqrt{\eta }+\lambda )(1+|y|^{2})^{1/2}. \end{aligned}$$
If \(P\in \mathcal {P}_{k}(\mathbb {C}^{n})\) and if \(|P(z)|^{1/k}\le \psi _{\lambda }(z)\,\forall z\in \mathbb {C}^{n}\), then
$$\begin{aligned} \langle P\rangle _k \le 1+3\lambda \quad \text {and } \quad \langle P\rangle _{k, K}\le \sqrt{\eta }+\lambda . \end{aligned}$$
(4)
For sufficiently small \(\lambda \),
$$\begin{aligned} (\lambda +g_{\lambda }(x))^{1-\lambda }+\lambda (1+|x|^{2})^{1/2}>(1+3\lambda )r(1+|x|^{2})^{1/2}, \end{aligned}$$
since as \(\lambda \) approaches 0, the difference of the left side minus the right side tends to \(g(x)-r(1+|x|^{2})^{1/2}>0\). It follows that for sufficiently small \(\lambda \),
$$\begin{aligned} \psi _{\lambda }(x)>(1+3\lambda )r(1+|x|^{2})^{1/2}. \end{aligned}$$
(5)
By (2), (3) and (4), we have \(\tau (x, K,r)\le (1+3\lambda )^{-1}(\sqrt{\eta }+\lambda )\). Letting \(\lambda \rightarrow 0\), and then \( \eta \rightarrow 0\), yields that \(\tau (x, K,r)=0\). Since this holds for every \( r<g(x)(1+|x|^2)^{-1/2}\), it follows that \(T(x, K)\ge g(x)(1+|x|^2)^{-1/2}\). \(\square \)

Remark

Some comments about (2) may be in order. By [8], Theorem 5.1.6 (iii)]
$$\begin{aligned} \psi _{\lambda }=(\limsup _{j\rightarrow \infty }|P_j|^{1/j})^* \end{aligned}$$
for some sequence \(\{P_j\}\) of polynomials on \(\mathbb {C}^n\) such that \(\deg P_j\le j\). Here \(w^*\) denotes the upper semicontinuous regularization of w. The stronger equality (2) depends on the fact that \(\psi _\lambda (z)/\sqrt{1+|z|^2}\) extends to a continuous function on \(\mathbb {P}^n\).

Definition 2.6

A subset E of \(\mathbb {C}^n\) is said to have Property J (in \(\mathbb {C}^n\)) if for each \(x\in \mathbb {C}^n{\setminus } E\) there is a positive number \(r_x\) such that \(T(x, K\cap E)\ge r_x\) for each compact subset K of \(\mathbb {C}^n\).

The inequality \(T(x, K\cap E)\ge r_x\) means that there is a sequence \(\{k_j\}\) of positive integers, and a sequence \(\{P_j\}\) of polynomials, with \({{\mathrm{deg}}}P_j\le k_j\), such that
$$\begin{aligned} { \langle P_j(x)\rangle _{k_j}\ge r_x,\;\;\langle P_j\rangle _{k_j}\le 1,\;\;\lim _{j\rightarrow \infty }\langle P_j\rangle _{k_j, \, K\cap E}=0.} \end{aligned}$$
(6)
Suppose that E has Property J and \(x\not \in E\). Let \(K=\{y\in \mathbb {C}^n: |y|\le |x|+1\}\). By (5), \( \langle P_j(x)\rangle _{k_j}>\langle P_j\rangle _{k_j, K\cap E}\) for sufficiently large j, which implies that x does not belong to the closure \(\overline{E}\) of E. It follows that \(\overline{E}{\setminus } E=\emptyset \), and E is closed. Therefore, each set that has Property J must be closed.

Theorem 2.7

A subset of \(\mathbb {C}^n\) has Property J if and only if it is a closed complete pluripolar set.

Proof

Suppose that E is a closed complete pluripolar set in \(\mathbb {C}^n\). By Corollary 2.2, there is a \(u\in \mathcal L(\mathbb {C}^n)\) with \(E=\{u=-\infty \}\). Without loss of generality, we assume that \(\sup (u(z)-\frac{1}{2}\log (1+|z|^{2}))=0\). For each point \(x\in \mathbb {C}^n\backslash E\) and each compact set K, one has, by Lemma 2.5, that
$$\begin{aligned} T(x,K\cap E)\ge (1+|x|^{2})^{-1/2}e^{u(x)}. \end{aligned}$$
Thus E has Property J.
Conversely, suppose that \(E\subset \mathbb {C}^n\) has Property J. Then E is closed. Set \(E_j=E\cap \{z\in \mathbb {C}^n: |z|\le j\}\). Let \(x\in \mathbb {C}^n{\setminus } E\). Then there is a positive number \(r_x\) such that \(T(x, E_j)\ge r_x\) for each positive integer j. Thus there is a sequence \(\{k_j\}\) of positive integers, and a sequence \(\{P_j\}\) of polynomials, with \({{\mathrm{deg}}}P_j\le k_j\), such that
$$\begin{aligned} \langle P_j(x)\rangle _{k_j}\ge r_x,\;\;\langle P_j\rangle _{k_j}\le 1,\;\;\langle P_j\rangle _{k_j, \, E_j}\le \exp (-2^j). \end{aligned}$$
Let
$$\begin{aligned} u(z):=\sum _{j=1}^\infty 2^{-j}\log |P_j(z)|^{1/k_j}=\frac{1}{2}\log (1+|z|^2)+\sum _{j=1}^\infty 2^{-j}\log \langle P_j(z)\rangle _{k_j}. \end{aligned}$$
Then u is plurisubharmonic with \(u(z)\le (1/2)\log (1+|z|^2)\) on \(\mathbb {C}^n\), and \(u(x)\ge \log r_x+(1/2)\log (1+|x|^2)\). Let \(y\in E\). Then there is a positive integer m such that \(y\in E_j\) for \(j\ge m\), hence
$$\begin{aligned} u(y)&=\frac{1}{2}\log (1+|y|^2)+\sum _{j=1}^\infty 2^{-j}\log \langle P_j(y)\rangle _{k_j}\\&\le \frac{1}{2}\log (1+|y|^2)+\sum _{j=m}^\infty 2^{-j}(-2^j)=-\infty . \end{aligned}$$
It follows that \(u=-\infty \) on E. Thus x does not belong to the pluripolar hull of E. Therefore, the pluripolar hull of E is E. A Theorem of Zeriahi [17] states that if a pluripolar set F is both \(F_\sigma \) and \(G_\delta \), and if the pluripolar hull of F equals F, then F is a complete pluripolar set. Since E, being a closed set, is both \(G_\delta \) and \(F_\sigma \), it follows that E is a complete pluripolar set. \(\square \)

3 Pluripolar sets in \(\mathbb {P}^{n}\)

Let \(\pi :\mathbb {C}^{n+1}\backslash \{0\}\rightarrow \mathbb {P}^{n}\) denote the standard projection mapping that maps a point \(z=(z_{0},z_{1},...,z_{n})\in \mathbb {C}^{n+1}\) to its corresponding homogeneous coordinates \(\pi (z)=[z]=[z_{0}:z_{1}:...:z_{n}]\in \mathbb {P}^{n}\). Suppose that \(z=(z_{0},z_{1},...,z_{n})\in \mathbb {C}^{n+1}{\setminus } \{0\}\) and \(Z=[Z_0:Z_1:\dots :Z_n]\in \mathbb {P}^n\). Then \(Z=\pi (z)\) if and only if
$$\begin{aligned}{}[Z_0:Z_1:\dots :Z_n]=[z_{0}:z_{1}:...:z_{n}], \end{aligned}$$
or, equivalently,
$$\begin{aligned} z_jZ_k=z_kZ_j,\quad \text{for } j, k = 0, \dots , n. \end{aligned}$$
Suppose that \(p\in \mathcal H_k(\mathbb {C}^{n+1})\) with \(k>0\) and that \(Z=[z]\in \mathbb {P}^n\), where \(z\in \mathbb {C}^{n+1}{\setminus } \{0\}\). Set
$$\begin{aligned} { \langle p(Z)\rangle :=\frac{|p(Z)|^{1/k}}{|Z|}=\frac{|p(z)|^{1/k}}{|z|}.} \end{aligned}$$
(7)
When \(p\equiv 0\), let \(\langle p(Z)\rangle =0\), which is consistent with (6). For \(p\in \mathcal H_0(\mathbb {C}^n)\), let \(\langle p(Z)\rangle =|p(Z)|\). Note that \(\langle p(Z)\rangle \) is independent of the choice of the representative z and is a well-defined function on \(\mathbb {P}^n\). Furthermore, if \(m, k>0\) and \(p\in \mathcal H_k(\mathbb {C}^n)\), then \(\langle p^m(Z)\rangle =\langle p(Z)\rangle \). For a set \(K\subset \mathbb {P}^{n}\), put
$$\begin{aligned} \langle p\rangle _{K}=\sup _{Z\in K}\langle p(Z)\rangle ,\;\;\;\langle p\rangle =\langle p\rangle _{\mathbb {P}^n}. \end{aligned}$$

Definition 3.1

The projective hull \(\hat{K}\) of a compact set \(K\subset \mathbb {P}^{n}\) is the set of all points \(Z \in \mathbb {P}^{n}\) for which there exists a constant \(C=C_Z>0\) such that
$$\begin{aligned} \langle p(Z )\rangle \le C\langle p\rangle _{K} \end{aligned}$$
(8)
for all homogeneous polynomials \(p\in \mathcal {H}(\mathbb {C}^{n+1})\). A compact set \(K\subset \mathbb {P}^{n}\) is said to be projectively convex if \(\hat{K}=K\).

Since, for \(k\ge 1\), the set \(H^0(\mathbb {P}^n, \mathcal O(k))\) of global holomorphic sections of the line bundle \(\mathcal O(k)\) is canonically identified with the set \(\mathcal H_k(\mathbb {C}^{n+1})\) of homogeneous polynomials of degree k, it follows that the above definition of projective hulls is equivalent to that in [6, p. 607].

Consider an algebraic variety \(K:=\{p=0\}\subset \mathbb {P}^n\), where \(p\in \mathcal H(\mathbb {C}^{n+1})\). Assume that \(Z\not \in K\). Then (7) does not hold since the right side equals 0, which implies that \(Z\not \in \hat{K}\). Thus, \(\hat{K}{\setminus } K\) is empty, and hence K is projectively convex. Therefore, each algebraic variety is projectively convex. In particular, each finite set is projectively convex.

The complement of a hyperplane in \(\mathbb {P}^n\) is called an affine open set. An affine open set in \(\mathbb {P}^n\) is biholomorphically equivalent to \(\mathbb {C}^n\). The sets
$$\begin{aligned} U_j:=\{Z=[Z_0:Z_1:\dots :Z_n]\in \mathbb {P}^n: Z_j\not =0\},\quad j=0,\dots ,n,\end{aligned}$$
(9)
are canonical affine open sets in \(\mathbb {P}^n\).

Definition 3.2

A subset F of \(\mathbb {P}^n\) is said to be a pluripolar set if for each \(Z\in F\) there is a neighborhood V of Z in \(\mathbb {P}^n\) and a nonconstant plurisubharmonic function u defined on V such that u is identically \(-\infty \) on \(V\cap F\). A subset E of \(\mathbb {P}^n\) is said to be a complete pluripolar set in \(\mathbb {P}^n\) if for each affine open set U in \(\mathbb {P}^n\) the set \(U\cap E\) is a complete pluripolar set in U.

We emphasize that a compact set K in \(\mathbb {P}^n\) is pluripolar if and only if \(\hat{K}\ne \mathbb {P}^n\) (see [6, Corollary 4.4]).

In the above definition, the pluripolar sets and the complete pluripolar sets are defined “locally”, because there are no globally defined nonconstant plurisubharmonic functions on \(\mathbb {P}^n\). It is desirable to have an equivalent definition of (complete) pluripolar sets in terms of some kind of substitutes of plurisubharmonic functions that are globally defined on \(\mathbb {P}^n\). We could use the \(\omega \)-plurisubharmonic functions described below to define pluripolar sets and complete pluripolar sets. Definition 3.2 is to emphasize that the notion of (complete) pluripolar sets is independent of any differential forms.

We fix a Kähler form
$$\begin{aligned} \omega :=dd^c\log |Z|=i\partial \overline{\partial }\log (|Z_0|^2+\cdots +|Z_n|^2) \end{aligned}$$
on \(\mathbb {P}^n\), where \(d^c=i(\overline{\partial }-\partial )\). Note that \((2\pi )^{-1}\omega \) is the Fubini-Study form on \(\mathbb {P}^{n}\). An upper semicontinuous function u from an open subset of \(\mathbb {P}^n\) to \(\mathbb {R}\cup \{-\infty \}\) is said to be \(\omega \)-plurisubharmonic if \(dd^c u+\omega \ge 0\) (see, e.g., [4]). Let \({\text {PSH}}_{\omega }(\mathbb {P}^{n})\) denote the family of \(\omega \)-plurisubharmonic functions on \(\mathbb {P}^n\). For a homogeneous polynomial p, the function \(Z\mapsto \log \langle p(Z)\rangle \) is a prototypical function in \({\text {PSH}}_{\omega }(\mathbb {P}^{n})\). Suppose that \(\ell (Z):=a_0Z_0+\cdots +a_nZ_n\) is a linear form and Q is an open subset of \(U_\ell :=\{Z\in \mathbb {P}^n:\ell (Z)\ne 0\}\). Then a function u on Q is \(\omega \)-plurisubharmonic if and only if the function \(u(Z)+\log (|Z|/|\ell (Z)|)\) is plurisubharmonic.
There is a one to one correspondence between \({\text {PSH}}_{\omega }(\mathbb {P}^{n})\) and the Lelong class \(\mathcal {L}(\mathbb {C}^{n})\). This can be seen by identifying \(\mathbb {C}^{n}\) with the affine open set \(U_0\):
$$\begin{aligned} \mathbb {C}^{n}\simeq \left\{ [1:\xi _{1}:\xi _{2}:...:\xi _{n}]\in \mathbb {P}^{n}:(\xi _{1},...,\xi _{n})\in \mathbb {C}^{n}\right\} =U_0. \end{aligned}$$
(10)
Given \(\varphi \in {\text {PSH}}_{\omega }(\mathbb {P}^{n}),\) the function
$$\begin{aligned} \hat{\varphi }(z_{1},...,z_{n}):=\frac{1}{2}\log (1+|z|^{2})+\varphi (1:z_1:\dots :z_n)\end{aligned}$$
belongs to \(\mathcal {L}(\mathbb {C}^{n})\), and the map \(\varphi \mapsto \hat{\varphi }\) is a bijection from \({\text {PSH}}_\omega (\mathbb {P}^n)\) onto \(\mathcal L(\mathbb {C}^n)\) (see [4]).

The following proposition can be found in [4] or [6, Theorem 4.3].

Proposition 3.3

Let E be a subset of \(\mathbb {P}^n\) . Then E is pluripolar if and only if there is a function \(u\in {\text {PSH}}_\omega (\mathbb {P}^n)\) , \(u\not \equiv -\infty \) , such that \(E\subset \{u=-\infty \}\).

Lemma 3.4

Suppose that \(n\ge 2\) and \(H\cong \mathbb {P}^{n-1}\) is a hyperplane in \(\mathbb {P}^n\) . Let u be an \(\omega \) -plurisubharmonic function on H . Then there is an \(\omega \) -plurisubharmonic function v on \(\mathbb {P}^n\) such that
$$\begin{aligned} \{Z\in \mathbb {P}^n: v(Z)=-\infty \}=\{Z\in H: u(Z)=-\infty \}.\end{aligned}$$
(11)

Proof

Without loss of generality we assume that \(u\le 0\) and \(H=\{Z=[Z_0:\dots :Z_n]: Z_0=0\}\). Let \(O=[1:0:\dots :0]\). The function
$$\begin{aligned} w(Z):=u(0:Z_1:\dots :Z_n)+\frac{1}{2}\log \frac{|Z_1|^2+\cdots +|Z_n|^2}{|Z_0|^2+|Z_1|^2+\cdots +|Z_n|^2} \end{aligned}$$
is an \(\omega \)-plurisubharmonic function on \(\mathbb {P}^n{\setminus } O\) such that \(w|_H=u\). Let
$$\begin{aligned} v(Z)=\left\{ \begin{array}{ll} \max (w(Z), 1+\log (|Z_0|/|Z|)),&{}\quad \text { if}\, Z\in \mathbb {P}^n{\setminus } O,\\ 1,&\quad {} \text {if} \, Z=O. \end{array} \right. \end{aligned}$$
Since the function \(1+\log (|Z_0|/|Z|)\) is \(>0\ge w\) on \(U{\setminus } O\) for some neighborhood U of O, we see that \(v(Z)=1+\log (|Z_0|/|Z|)\) on \(U{\setminus } O\), hence \(v(Z)=1+\log (|Z_0|/|Z|)\) on U. It follows that \(v\in {\text {PSH}}_\omega (\mathbb {P}^n)\). Since \(v=w\) on H, and since \(v>-\infty \) on U, it follows that (10) holds. \(\square \)

Proposition 3.5

Let \(E\subset \mathbb {P}^n\) . Then E is a complete pluripolar set in \(\mathbb {P}^n\) if and only if there is a non-constant \(u\in {\text {PSH}}_\omega (\mathbb {P}^n)\) such that \(E=\{Z: u(Z)=-\infty \}\).

Proof

Suppose that there is a \(u\in {\text {PSH}}_\omega (\mathbb {P}^n)\) such that \(E=\{u=-\infty \}\). Let U be an affine open set, and let \(H=\mathbb {P}^n{\setminus } U=\{Z: \Lambda (Z)=0\}\), where \(\Lambda (Z):=a_0Z_0+a_1Z_1+\cdots +a_nZ_n\) is a linear form. Then \(v(Z):=u(Z)+\log (|Z|/|\Lambda (Z)|)\) is plurisubharmonic on U and
$$\begin{aligned} U\cap E=\{Z\in U: u(Z)=-\infty \}=\{Z\in U: v(Z)=-\infty \}. \end{aligned}$$
It follows that \(U\cap E\) is a complete pluripolar set in U for each affine open set U. Therefore, E is a complete pluripolar set in \(\mathbb {P}^n\).

Conversely, suppose that E is a complete pluripolar set in \(\mathbb {P}^n\). For \(j=0,\dots , n\), let \(H_j=\{Z_j=0\}\) and \(U_j=\mathbb {P}^n{\setminus } H_j\). Since \(E\cap U_j\) is a complete pluripolar set in \(U_j\), there is a plurisubharmonic function \(h_j\) on \(U_j\) with \(E\cap U_j=\{Z\in U_j: h_j(Z)=-\infty \}\) and \(h_j(Z)\le \log (|Z|/|Z_j|)\), by Corollary 2.2. The function \(h_j(Z) - \log (|Z|/|Z_j|)\) is a non-positive \(\omega \)-plurisubharmonic function on \(U_j\), which extends uniquely to an \(\omega \)-plurisubharmonic function \(w_j\) on \(\mathbb {P}^n\).

Let \(w=\max w_j\). For each j,
$$\begin{aligned} \{Z\in U_j: w(Z)=-\infty \}\subset \{Z\in U_j: w_j(Z)=-\infty \}=E\cap U_j. \end{aligned}$$
It follows that \(\{w=-\infty \}\subset E\). Let \(\Omega =U_0\cap \dots \cap U_n\). Then
$$\begin{aligned} \{Z\in \Omega : w_j(Z)=-\infty \}=\Omega \cap \{Z\in U_j: w_j(Z)=-\infty \}=\Omega \cap (E\cap U_j)=E\cap \Omega , \end{aligned}$$
for each j, hence
$$\begin{aligned} \{w=-\infty \}\cap \Omega =\cap _j (\{w_j=-\infty \}\cap \Omega )=E\cap \Omega . \end{aligned}$$
To summarize, we have \(\{w=-\infty \}\subset E\) and \(\{w=-\infty \}\cap \Omega = E\cap \Omega \).

To prove that there is a \(u\in {\text {PSH}}_\omega (\mathbb {P}^n)\) with \(E=\{u=-\infty \}\), we proceed by induction on n. Suppose that \(n=1\). If \([0:1]\in E\), let \(v_0=\log (|Z_0|/|Z|)\); if \([0:1]\not \in E\), let \(v_0=0\). Then \(v_0\in {\text {PSH}}_\omega (\mathbb {P}^1)\) and \(\{v_0=-\infty \}=E\cap H_0\). We similarly define \(v_1\in {\text {PSH}}_\omega (\mathbb {P}^1)\) so that \(\{v_1=-\infty \}=E\cap H_1\). Then \(u:=(w+v_0+v_1)/3\) belongs to \({\text {PSH}}_\omega (\mathbb {P}^1)\) and \(\{u=-\infty \}=E\). The statement holds for \(n=1\).

Suppose that \(n\ge 2\) and that the statement holds for \(n-1\). For each j, either \(H_j\subset E\), or \(H_j\cap E\) is a complete pluripolar set in \(H_j\). If \(H_0\subset E\), let \(v_0=\log (|Z_0|/|Z|)\). If \(H_0\not \subset E\), then, by the induction hypothesis, there is a \(u_0\in {\text {PSH}}_\omega (H_0)\) with \(\{Z\in H_0: u_0(Z)=-\infty \}=E\cap H_0\), and hence, by Lemma 3.4, there is a \(v_0\in {\text {PSH}}_\omega (\mathbb {P}^n)\) with \(\{Z\in \mathbb {P}^n: v_0(Z)=-\infty \}=E\cap H_0\). Either case, we have \(v_0\in {\text {PSH}}_\omega (\mathbb {P}^n)\) and \(\{v_0=-\infty \}=E\cap H_0\). We similarly obtain \(v_j\) so that \(v_j\in {\text {PSH}}_\omega (\mathbb {P}^n)\) and \(\{v_j=-\infty \}=E\cap H_j\), for each j. Then \(u:=(w+v_0+\cdots +v_n)/(n+2)\) belongs to \({\text {PSH}}_\omega (\mathbb {P}^n)\) and \(\{u=-\infty \}=E\), which completes the proof. \(\square \)

Propositions 3.3 and 3.5 can be considered equivalent definitions of pluripolar sets and complete pluripolar sets respectively. Since Definition 3.2 is local, the union of a countable collection of pluripolar sets in \(\mathbb {P}^n\) is pluripolar for the same reason that the corresponding statement is true in affine space. It is a consequence of Propositions 3.3 and 3.5 that each pluripolar set in \(\mathbb {P}^n\) is contained in a complete pluripolar set in \(\mathbb {P}^n\).

Let E be a pluripolar set in \(\mathbb {P}^n\). The intersection of all complete pluripolar sets in \(\mathbb {P}^n\) that contain E is called the pluripolar hull (see [11]) of E (in \(\mathbb {P}^n\)), and is denoted by \(E^*\).

Proposition 3.6

Let K be a compact pluripolar set in \(\mathbb {P}^n\) . Then \(\hat{K}\subset K^*\).

Proof

Let
$$\begin{aligned} L_K(Z)=\sup \{\varphi (Z):\varphi \in {\text {PSH}}_\omega (\mathbb {P}^n) \quad \text { and} \quad \varphi |_K\le 0\} \end{aligned}$$
and
$$\begin{aligned} \Lambda _K(Z)=\sup \{\log \langle p(Z)\rangle : p\in \mathcal H(\mathbb {C}^{n+1}),\; \langle p\rangle _K\le 1\}. \end{aligned}$$
Then \(\Lambda _K(Z)=L_K(Z)\), by [6, Preposition 4.2].

Let \(X\in {K^*}^c\), the complement of \(K^*\). Then there is a \(u\in {\text {PSH}}_\omega (\mathbb {P}^n)\) with \(u(X)>-\infty \) and \(u|_K\equiv -\infty \). Thus \(L_K(X)=\infty \). It follows that \(\Lambda _K(X)=\infty \), which implies that \(X\in {\hat{K}}^c\), the complement of \(\hat{K}\). Therefore, \({K^*}^c\subset {\hat{K}}^c\), which is equivalent to \(\hat{K}\subset K^*\). \(\square \)

Corollary 3.7

Each compact complete pluripolar set in \(\mathbb {P}^n\) is projectively convex.

Remark

The converse to Corollary 3.7 is false. The set \(E:=\{[1:z: e^z]\in \mathbb {P}^2: z\in \mathbb {C}, |z|\le 1\}\) is projectively convex in \(\mathbb {P}^2\) (see [6]), but it is not a complete pluripolar set.

Proposition 3.8

Let U be an affine open set in \(\mathbb {P}^n\) , and let E be a compact complete pluripolar set in U. Then E is a complete pluripolar set in \(\mathbb {P}^n\).

Proof

Without loss of generality, we assume that \(U=\{Z_0\ne 0\}\) and \(E\subset \{Z\in U: |Z/Z_0|<\sqrt{2}\}\). By the proof of ([10, Lemma 5.4) there is a number \(a>1\) and a plurisubharmonic function u defined on U such that \(E=\{X\in U: u(X)=-\infty \}\) and \(u(Z)=\log (\sqrt{|Z_1|^2+\cdots +|Z_n|^2}/|Z_0|)\) on the set \(\{Z\in U: \sqrt{|Z_1|^2+\cdots +|Z_n|^2}/|Z_0|\ge a\}\). Let
$$\begin{aligned} v(Z)=\left\{ \begin{array}{ll} u(Z)-\log (|Z|/|Z_0|),\quad &{}\hbox {if} \, Z\in U,\\ 0,&{}\hbox {if} \, Z\not \in U.\end{array} \right. \end{aligned}$$
Then \(v\in {\text {PSH}}_\omega (\mathbb {P}^n)\) and \(E=\{Z\in \mathbb {P}^n: v(Z)=-\infty \}\). Therefore, E is a complete pluripolar set in \(\mathbb {P}^n\). \(\square \)

4 Convergence sets in \(\mathbb {P}^{n}\)

For a given \(f(x)\in \mathbb {C}[[x]]=\mathbb {C}[[x_0,x_1,\dots ,x_n]]\), we are interested in the set of all \( z\in \mathbb {C}^{n+1}\) for which the restriction \(f_{z}(t):=f(z_{0}t,z_{1}t, \dots ,z_{n}t)\in \mathbb {C}\{t\}\). Since for \(\lambda \in \mathbb {C} \backslash \{0\},\) the series \(f_{z}(t)\) and \(f_{\lambda z}\) either both converge or both diverge, it is more appropriate to consider the set of affine lines along which f converges. Thus the set of affine lines along which f converges can be identified as a subset of the projective space \(\mathbb {P}^{n}\). Since a \(f\in \mathbb {C}[[x]]\) converges if and only if \(f_{z}(t)\in \mathbb {C}\{t\},\forall z\in \mathbb {C}^{n+1}\), our focus will be on the divergent power series.

The convergence set of a power series \(f\in \mathbb {C}[[x]]\), denoted by \({\text {Conv}}(f)\), is the set of all \(Z\in \mathbb {P}^n\) such that \(f_z(t)\in \mathbb {C}\{t\}\) for some (and hence all) \(z\in \pi ^{-1}(Z)\). Then f converges if and only if \({\text {Conv}}(f)=\mathbb {P}^n\).

Definition 4.1

A subset E of \(\mathbb {P}^n\) is said to be a convergence set (in \(\mathbb {P}^n\)) if \(E={\text {Conv}}(f)\) for some divergent power series f. Let \({\text {Conv}}(\mathbb {P}^n)\) denote the collection of all convergence sets in \(\mathbb {P}^n\):
$$\begin{aligned} {\text {Conv}}(\mathbb {P}^n):=\{{\text {Conv}}(f): f\in \mathbb {C}[[x_0,x_1,\dots ,x_n]],\; f\text {diverges}\}. \end{aligned}$$

Convergence sets were first studied in [1].

Consider a divergent series \(f\in \mathbb {C}[[x_0,x_1,\dots , x_n]]\). Since
$$\begin{aligned} f_{z}(t):=f(z_{0}t,z_{1}t,\dots ,z_{n}t)=\sum _{j=1}^{\infty }p_{j}(z)t^{j},\quad \text { }p_{j}\in \mathcal {H}_j(\mathbb {C}^{n+1}), \end{aligned}$$
we see that
$$\begin{aligned} {\text {Conv}}(f)=\{ Z \in \mathbb {P}^{n}:\sup _{j}\langle p_j(Z)\rangle <\infty \}. \end{aligned}$$
In fact, we have the following lemma (see [16]).

Lemma 4.2

Suppose that \(E\subsetneqq \mathbb {P}^{n}\) . Then \(E\in {\text {Conv}}(\mathbb {P}^n)\) if and only if there exists a countable family \(\mathscr {F}\) of non-constant homogeneous polynomials in \(\mathcal H(\mathbb {C}^{n+1})\) such that
$$\begin{aligned} E=\{Z\in \mathbb {P}^n: \sup _{p\in \mathscr {F}}\langle p(Z)\rangle <\infty \}. \end{aligned}$$
(12)
Note that there is no need to require the degrees of the polynomials in \(\mathscr {F}\) to form a strictly increasing sequence. Suppose that \(E\subsetneqq \mathbb {P}^{n}\), and that there exists a countable family \(\mathscr {F}\subset \mathcal H(\mathbb {C}^{n+1})\) of non-constant homogeneous polynomials such that (11) holds. Let \(\{ p_j\}\) be an enumeration of \(\mathscr {F}\). Raise \(p_j\) to suitable powers to obtain \(q_j:=p_j^{k_j}\) so that the sequence \(\{{{\mathrm{deg}}}q_j\}\) is strictly increasing. Since \(\langle p_j(Z)\rangle =\langle q_j(Z)\rangle \), we see that
$$\begin{aligned} E=\{ Z \in \mathbb {P}^{n}:\sup _{j}\langle q_j(Z)\rangle <\infty \}. \end{aligned}$$
It follows that \(E={\text {Conv}}(g)\), where \(g(x)=\sum _{j=1}^\infty q_j(x)\).

Proposition 4.3

If \(E\in {\text {Conv}}(\mathbb {P}^{n})\) and \(F\in {\text {Conv}}(\mathbb {P} ^{n})\) then \(E\cap F\in {\text {Conv}}(\mathbb {P}^{n})\).

Proof

Suppose that E and F are convergence sets. By Lemma 4.2, there are countable families \(\mathscr {E}\) and \(\mathscr {F}\) of homogeneous polynomials such that
$$\begin{aligned} E=\{Z\in \mathbb {P}^n: \sup _{p\in \mathscr {E}}\langle p(Z)\rangle <\infty \},\;\; F=\{Z\in \mathbb {P}^n: \sup _{p\in \mathscr {F}}\langle p(Z)\rangle <\infty \}. \end{aligned}$$
It follows that
$$\begin{aligned} E\cap F=\{Z\in \mathbb {P}^n: \sup _{p\in \mathscr {E}\cup \mathscr {F}}\langle p(Z)\rangle <\infty \}. \end{aligned}$$
Therefore, \(E\cap F\in {\text {Conv}}(\mathbb {P}^{n})\). \(\square \)

We do not know whether the union of two convergence sets is necessarily a convergence set.

Proposition 4.4

Suppose that K is a compact pluripolar subset of \(\mathbb {P}^{n}\) . Then \(\hat{K}\in {\text {Conv}}(\mathbb {P}^{n})\) . In particular, each projectively convex compact pluripolar set in \(\mathbb {P}^n\) is a convergence set.

Proof

The proposition is proved by following the approach of [10, Theorem 5.6]. Let
$$\begin{aligned} \mathscr {F}=\{p: p\in \mathcal H_j(\mathbb {C}^{n+1}) \;\text { with }\; j\ge 1, \;\langle p\rangle _K\le 1\}. \end{aligned}$$
Then
$$\begin{aligned} \hat{K}=\{Z\in \mathbb {P}^n: \sup _{p\in \mathscr {F}} \langle p(Z)\rangle <\infty \}. \end{aligned}$$
Since K is a compact pluripolar set, \(\hat{K}\ne \mathbb {P}^n\) (see [6, Corollary 4.4]). Let \(\mathscr {E}\) be the set of polynomials in \(\mathscr {F}\) whose coefficients belong to \(\mathbb {Q}+i\mathbb {Q}\), the set of complex numbers whose real and imaginary parts are rational numbers. Since \(\mathbb {Q}+i\mathbb {Q}\) is dense in \(\mathbb {C}\), we see that
$$\begin{aligned} \hat{K}=\{Z\in \mathbb {P}^n: \sup _{p\in \mathscr {E}} \langle p(Z)\rangle <\infty \}. \end{aligned}$$
Since \(\mathscr {E}\) is countable, the set \(\hat{K}\) is a convergence set in \(\mathbb {P}^n\) by Lemma 4.2. \(\square \)

Remark

Proposition 4.4 is motivated by Theorem 5.6 in [10], which states that if U is an affine open set in \(\mathbb {P}^n\) and if K is a compact complete pluripolar set in U then K is the intersection of U and a convergence set in \(\mathbb {P}^n\). We observe that the same reasoning which proves that theorem can be applied to prove the more general statement Proposition 4.4. Proposition 4.4 is more general than Theorem 5.6 in [10] because (a) \(\hat{K}\) may be non-compact, (b) \(\hat{K}\) is not necessarily a complete pluripolar set, and (c) \(\hat{K}\) does not have to lie in an affine open set.

Corollary 4.5

Each algebraic variety in \(\mathbb {P}^n\) is a convergence set. In particular, each finite set is a convergence set.

Theorem 4.6

Let E be a convergence set in \(\mathbb {P}^{n}\) . Then there exists an ascending sequence \(\{K_{j}\}\) of compact pluripolar sets such that \(E=\cup K_j=\cup \hat{K}_{j}\) . In particular, E is a pluripolar \(F_\sigma \) set.

Proof

Put \(E={\text {Conv}}(f)\) and \(f(z)=\sum _{m=1}^{\infty }f_{m}(z)\), where \(f_{m}\in \mathcal {H}_m(\mathbb {C}^{n+1})\). Then \(E=\cup _{j}K_{j}\) where
$$\begin{aligned} K_{j}=\{Z\in \mathbb {P}^n: \langle f_m(Z)\rangle \le j\;,\forall m\}. \end{aligned}$$
Fix a j and suppose that \(X\in \hat{K}_{j}\). There is a positive integer \(\ell \) such that
$$\begin{aligned} \langle p(X)\rangle \le \ell \langle p\rangle _{K_j} \end{aligned}$$
for each \(p\in \mathcal {H}(\mathbb {C}^{n+1})\), and in particular for \(p=f_m\). Thus \(X\in K_{\ell j}\subset E\). It follows that \(\hat{K}_j\subset E\) for each j and \(E=\cup _{j}\hat{K}_{j}\). For a compact set \(K\subset \mathbb {P}^n\), \(\hat{K}\ne \mathbb {P}^n\) if and only if K and \(\hat{K}\) are pluripolar (see, e.g., [6, Corollary 4.4]). Since \(E\ne \mathbb {P}^n\), we see that \(K_j\) and \(\hat{K}_j\) are pluripolar for each j. \(\square \)

Lemma 4.7

Let \(\{K_j\}\) be a sequence of compact pluripolar sets in \(\mathbb {P}^n\) such that \(\cup _{j=1}^k K_j\) is projectively convex for each k , and let \(E=\cup K_j\) . Let \(\{U_j\}\) be a sequence of open sets such that \(U_k\supset \cup _{j=1}^k K_j\) and \(\cap _{j=k}^\infty U_j\subset E\) for each k . Then \(E\in {\text {Conv}}(\mathbb {P}^n)\).

Proof

Let \(E_j=\cup _{i=1}^j K_i\) and \(\Gamma _j=\mathbb {P}^n{\setminus } U_j\) for \(j=1,2,\dots \). Fix a j and let \(X\in \Gamma _j\). Since X does not belong to \(E_j\), a projectively convex set, there exists a homogeneous polynomial \(p_{X }\) such that
$$\begin{aligned} \langle p_X(X)\rangle >j \quad \text { and } \quad \langle p_X\rangle _{E_j}\le 1. \end{aligned}$$
Since the sets \(\{Z\in \mathbb {P}^n: \langle p_X(Z)\rangle >j\}\), where \(X\in \Gamma _j\), form an open cover of \(\Gamma _j\), there exist \(X_1,\dots , X_{\ell _j}\in \Gamma _j\) such that
$$\begin{aligned} \cup _{i=1}^{\ell _j}\{Z: \langle p_{X_i}(Z)\rangle >j\}\supset \Gamma _j. \end{aligned}$$
Put \(p_{ji}=p_{X_i}\). Then
$$\begin{aligned} \langle p_{ji}\rangle _{E_j}\le 1, \end{aligned}$$
and
$$\begin{aligned} \max _{1\le i\le \ell _j}\langle p_{ji}(X)\rangle >j,\quad \text {for}\, X\in \Gamma _j. \end{aligned}$$
(13)
Suppose that \(X\in E\). Then there is a positive integer k such that \(X\in E_k\). Thus \(\langle p_{ji}(X)\rangle \le 1\) for \(j\ge k\). Since the set \(\{p_{ji}: j<k, 1\le i\le \ell _j\}\) is finite, we see that
$$\begin{aligned} \sup _{j,i}\langle p_{ji}(X)\rangle <\infty .\end{aligned}$$
(14)
Conversely, suppose that \(X\in \mathbb {P}^n\) and (13) holds. Then \( \sup _{j,i}\langle p_{ji}(X)\rangle \le k\) for some positive integer k. It follows from (12) that \(X\not \in \Gamma _j\) for \(j\ge k\). Thus \(X\in \cap _{j=k}^\infty U_j\subset E\).

To summarize, \(X\in E\) if and only if (13) holds. By Lemma 4.2, \(E\in {\text {Conv}}(\mathbb {P}^n)\). \(\square \)

Proposition 4.8

Let \(\{K_{j}\}\) be an ascending sequence of projectively convex, compact, pluripolar sets such that \(E:=\cup _{j}K_{j}\) is \(G_{\delta }\) . Then \(E\in {\text {Conv}}(\mathbb {P}^{n})\).

Proof

Since E is \(G_\delta \), there is a descending sequence \(\{U_{j}\}\) of open sets such that \(E=\cup K_{j}=\cap U_{j}\). It follows from Lemma 4.7 that \(E\in {\text {Conv}}(\mathbb {P}^{n})\). \(\square \)

Proposition 4.9

Let \(\{K_j\}\) be a sequence of pairwise disjoint compact pluripolar sets in \(\mathbb {P}^n\) such that for each positive integer k the set \(\cup _{j=1}^k K_j\) is projectively convex. Then \(K:=\cup K_j\) is a convergence set in \(\mathbb {P}^n\).

Proof

Let d denote a distance function on \(\mathbb {P}^n\) that induces the standard topology. Let
$$\begin{aligned} r_k=(1/2)\min (1/k, \min \{d(K_i,K_j): 1\le i<j\le k+1\}). \end{aligned}$$
Let \(U_k\) be the \(r_k\)-neighborhood of \(\cup _{j=1}^k K_j\).
Suppose that m is a positive integer and that \(X\in \cap _{k=m}^\infty U_k\). We claim that
$$\begin{aligned} d(X, \cup _{j=1}^{m} K_j)<r_{\ell },\;\;\ell =m,m+1,\dots . \end{aligned}$$
(15)
We will prove (14 ) by induction on \(\ell \). If \(\ell =m\), then (14) holds because \(X\in U_m\). Suppose that \(\ell >m\) and (14) holds with \(\ell \) replaced by \(\ell -1\). Since \(X\in U_\ell \), we have
$$\begin{aligned} d(X, \cup _{j=1}^\ell K_j)<r_\ell .\end{aligned}$$
(16)
On the other hand,
$$\begin{aligned} d(X, \cup _{j=m+1}^\ell K_j)\ge d(\cup _{j=1}^m K_j, \cup _{j=m+1}^\ell K_j)-d(X,\cup _{j=1}^m K_j)\ge 2 r_{\ell -1}-r_{\ell -1}=r_{\ell -1}\ge r_\ell , \end{aligned}$$
hence
$$\begin{aligned} d(X, \cup _{j=m+1}^\ell K_j)\ge r_\ell .\end{aligned}$$
(17)
The inequalities (15) and (16) imply that \(d(X, \cup _{j=1}^{m} K_j)<r_{\ell }\). Therefore, (14) holds for all \(\ell \ge m\). This implies that \(X\in \cup _{j=1}^{m} K_j\) for each \(X\in \cap _{k=m}^\infty U_k\). It follows that \(\cap _{k=m}^\infty U_k=\cup _{j=1}^m K_j\). By Lemma 4.7, \(K:=\cup K_j\) is a convergence set in \(\mathbb {P}^n\). \(\square \)

Corollary 4.10

Each countable set in \(\mathbb {P}^n\) is a convergence set.

Let \(\varphi (z)\) be a non-polynomial entire function defined on the complex plane. For \(S\subset \mathbb {C}\), let \({\tilde{S}}\) be the subset of \(\mathbb {P}^2\) defined by
$$\begin{aligned} {\tilde{S}}=\{[1: z: \varphi (z)]\in \mathbb {P}^2: z\in S\}. \end{aligned}$$
By Theorem 9.2 in [6], which depends on a deep theorem in [14], for a compact set \(K\subset \mathbb {C}\), the set \({\tilde{K}}\) is projectively convex if and only if \(\mathbb {C}\backslash K\) is connected.

Example 4.11

The set \({\tilde{\mathbb {C}}}\) is a convergence set. To see this, write \({\tilde{\mathbb {C}}}=\cup {\tilde{\Delta }}_j\), where \(\Delta _j=\{z\in \mathbb {C}: |z|\le j\}\). Since \({\tilde{\mathbb {C}}}\) is a \(G_\delta \) set in \(\mathbb {P}^2\) and since each \({\tilde{\Delta }}_j\) is projectively convex, we see that \({\tilde{\mathbb {C}}}\in {\text {Conv}}(\mathbb {P}^2)\) by Proposition 4.8.

Example 4.12

Let \(\Gamma \) be the unit circle in \(\mathbb {C}\). Then \({\tilde{\Gamma }}\in {\text {Conv}}(\mathbb {P}^2)\). To see this, write \({\tilde{\Gamma }}=\cup {\tilde{S}}_j\), where
$$\begin{aligned} S_j=\{e^{it}: 0\le t\le 2\pi -1/j\}. \end{aligned}$$
Since \({\tilde{\Gamma }}\) is \(G_\delta \) and since each \({\tilde{S}}_j\) is projectively convex, we see that \({\tilde{\Gamma }}\) is a convergence set by Proposition 4.8.

Example 4.13

Let S be the subset of \(\mathbb {C}\) obtained by removing an open triangle from a closed triangle. Then \({\tilde{S}}\) is a convergence set. This follows from an argument very similar to the previous example.

Example 4.14

Let S be the subset of \(\mathbb {C}\) obtained by removing a finite number of open triangles from a closed triangle. Then \({\tilde{S}}\) is a convergence set. This follows from the previous example and Proposition 4.3.

Let \(\Lambda \) be a closed triangle in \(\mathbb {C}\). The “open middle triangle” of \(\Lambda \) is the open triangle whose vertices are the midpoints of the sides of \(\Lambda \). Let \(V_1\) be the open middle triangle of \(\Lambda \). The set \(\Lambda \backslash V_1\) is the union of three congruent closed triangles. Let \(V_2, V_3, V_4\) denote the open middle triangles of the three closed triangles whose union is \(\Lambda \backslash V_1\). Similarly, let \(V_5,\dots , V_{13}\) denote the open middle triangles of the nine closed triangles whose union is \(\Lambda \backslash \cup _{j=1}^4 V_j\). Continuing in this way, we obtain a sequence \(\{V_j\}\) of open triangles. The set \(E:=\Lambda \backslash \cup _{j=1}^\infty V_j\) is called Sierpiński’s triangle. Let \(E_k=\Lambda \backslash \cup _{j=1}^k V_j\) for \(k=1, 2,\dots \). By Example 4.14, each \({\tilde{E}}_k\) is a convergence set in \(\mathbb {P}^2\). Note that \({\tilde{E}}=\cap _{k=1}^\infty {\tilde{E}}_k\).

Proposition 4.15

The set \({\tilde{E}}\) is not a convergence set.

Proof

Seeking for a contradiction, suppose that \({\tilde{E}}={\text {Conv}}(f)\) and \(f=\sum p_m\), where \(p_m\) is a homogeneous polynomial of degree m in three variables \(z_0, z_1, z_2\). Let
$$\begin{aligned} T_k=\{z\in \mathbb {C}: |p_m(1, z, \varphi (z))|^{1/m}\le k\quad \forall m\}. \end{aligned}$$
Then each \(T_k\) is a closed subset of \(\mathbb {C}\) and \(E=\cup T_k\). By the maximum modulus principle, \(\mathbb {C}{\setminus } T_k\) is connected for each k. We observe that the set E has the following property: if S is a closed set in the topological space E with nonempty interior, then S contains the sides of a small triangle and hence \(\mathbb {C}{\setminus } S\) is disconnected. It follows that each \(T_k\) has empty interior, which contradicts the Baire category theorem. \(\square \)

Let \(\{{\tilde{E}}_k\}\) be the sequence defined above. Note that each \({\tilde{E}}_k\in {\text {Conv}}(\mathbb {P}^2)\), but \(\cap {\tilde{E}}_k\not \in {\text {Conv}}(\mathbb {P}^2)\).

Fix a positive number M. Let
$$\begin{aligned} S_0&=\{[1:0:\dots :0]\},\;\; \text {and for }k=1,\dots ,n,\\ S_k&=\{X\in \mathbb {P}^n: |X_0|^2+\cdots +|X_{k-1}|^2\le M^2|X_k|^2, X_{k+1}=\cdots =X_n=0\}. \end{aligned}$$
Put
$$\begin{aligned} K_M=\cup _{k=0}^n S_k. \end{aligned}$$
(18)
Then \(\{K_m\}\) is an ascending sequence of closed sets with \(\mathbb {P}^{n}=\cup _{m=1}^\infty K_m\).

Recall that the set \(\Pi \) of all hyperplanes in \(\mathbb {P}^n\) is naturally isomorphic to \(\mathbb {P}^n\). The set of all hyperplanes in \(\mathbb {P}^{n}\) passing through a fixed point is a hyperplane in \(\Pi \).

Lemma 4.16

If K is a closed subset of \(\mathbb {P}^{n}\) contained in an affine open set, and if \(\xi \in \mathbb {P}^{n}\) , then \(K\cup \{\xi \}\) is contained in an affine open set.

Proof

Let \(R=\{H\in \Pi : H\cap K=\emptyset \}\) and \(S=\{H\in \Pi : \xi \in H\}\). Then R is a non-empty open set in \(\Pi \) and S is a hyperplane in \(\Pi \). Thus \(R{\setminus } S\) is non-empty. This means that there is a hyperplane H with \(H\cap (K\cup \{\xi \})=\emptyset \). Therefore, \(K\cup \{\xi \}\) is contained in the affine open set \(\mathbb {P}^n{\setminus } H\). \(\square \)

If \(1\le \nu \le n\) and if \(u_0,\dots , u_{\nu }\) are linearly independent vectors in \(\mathbb {C}^{n+1}\), let \({\text {span}}(u_0,\dots , u_{\nu })\) be the \(\nu \)-dimensional linear space in \(\mathbb {P}^n\) defined by
$$\begin{aligned} {\text {span}}(u_0,\dots , u_{\nu }):=\{\pi (c_0 u_0+\cdots +c_\nu u_\nu ): (c_0,\dots ,c_\nu )\in \mathbb {C}^{\nu +1}{\setminus }\{0\}\}, \end{aligned}$$
where \(\pi : \mathbb {C}^{n+1}{\setminus } \{0\}\rightarrow \mathbb {P}^n\) is the standard projection.

Lemma 4.17

For each \(M>0\) , the set \(K_M\) is contained in an affine open set.

Proof

Let \(e_0,\dots , e_n\) be the standard basis of \(\mathbb {C}^{n+1}\), and let \(\varepsilon \) be a sufficiently small positive number. Let \(v_j=e_j+\varepsilon e_{j+1}\) for \(j=0,\dots ,n-1\).

Put \(V_j=\mathrm {span}(v_0,\dots ,v_j)\) for \(j=0,\dots , n-1\), and \(V=V_{n-1}\). Also, let \(W_j={\text {span}}(e_0,\dots ,e_j)\). Note that
$$\begin{aligned} V\cap W_j\subset V_{j-1},\quad \text {for }j=1,\dots ,n. \end{aligned}$$
Since \(S_j\subset W_j\), it follows that \(V\cap S_j\subset V_{j-1}\) for \(j\ge 1\). Since the vectors \(e_0, v_0, v_1, \dots , v_{n-1}\) are linearly independent we see that \(V\cap S_0=\emptyset \). For \(j\ge 1\), since \(W_{j-1}\cap S_j=\emptyset \), and since \(V_{j-1}\) is close to \(W_{j-1}\) for sufficiently small \(\varepsilon \), we see that \(V_{j-1}\cap S_j=\emptyset \). It follows that
$$\begin{aligned} V\cap K_M=\cup _{j=0}^n (V\cap S_j)\subset \cup _{j=1}^n (V_{j-1}\cap S_j)=\emptyset . \end{aligned}$$
Therefore \(K_M\) is contained in the affine open set \(\mathbb {P}^n{\setminus } V\). \(\square \)

Theorem 4.18

The union of a countable collection of closed complete pluripolar sets in \(\mathbb {P}^n\) is a convergence set in \(\mathbb {P}^n\).

Proof

Let \(\{E_m\}\) be a sequence of closed complete pluripolar sets in \(\mathbb {P}^n\) and let \(E=\cup E_m\). Without loss of generality, we assume that the sequence \(\{E_m\}\) is ascending, since the union of a finite number of closed complete pluripolar sets in \(\mathbb {P}^n\) is a closed complete pluripolar set. Recall that \(\mathbb {P}^n=\cup K_m\), where \(\{K_m\}\) is ascending and each \(K_m\) is a compact set contained in an affine open set. We have \(E=\cup (E_m\cap K_m)\).

For each positive integer m, we shall construct a sequence \(\{h_{mk}\}_{k=1}^{\infty }\) of homogeneous polynomials such that for all k,
  1. (i)

    \(\langle h_{mk}\rangle _{K_m\cap E_m}\le 1\),

     
  2. (ii)

    \(\langle h_{mk}\rangle \le m\),

    and

     
  3. (iii)

    \(\cup _{k=1}^\infty \{X\in \mathbb {P}^{n}:\langle h_{mk}(X)\rangle >m/2\}\supset \mathbb {P}^{n}{\setminus } E_m\).

     
Fix a positive integer m. Let \(Y\in \mathbb {P}^n{\setminus } E_m\). By Lemmas 4.16 and 4.17, we see that \((K_m\cap E_m)\cup \{Y\}\) is contained in an affine open set V. Since \(V\cap E_m\) is a (relatively) closed complete pluripolar set in \(V\approx \mathbb {C}^n\), the set \(V\cap E_m\) has Property J in V by Theorem 2.7. Hence there is a number r with \(0<r<1\) such that \(\tau (Y, K_m\cap E_m, r)=0\), which means, in terms of homogeneous coordinates, that there is a sequence \(\{p_j\}\) in \(\mathcal H(\mathbb {C}^{n+1})\) such that
$$\begin{aligned} \lim _{j\rightarrow \infty } \langle p_j\rangle _{K_m\cap E_m}=0, \;\text {and for all}\, j, \;\langle p_j(Y)\rangle \ge r,\;\;\langle p_j\rangle \le 1. \end{aligned}$$
Choose a positive rational number \(\beta =a/b<1\), where ab are positive integers, such that \((r/m)^\beta > 1/2\). There is a homogeneous polynomial \(p\in \mathcal H(\mathbb {C}^{n+1})\) such that
$$\begin{aligned} \langle p\rangle _{K_m\cap E_m}<m^{-1/\beta }, \;\langle p(Y)\rangle \ge r,\quad \text { and }\quad \langle p\rangle _{\mathbb {P}^n}\le 1. \end{aligned}$$
Let \(v={{\mathrm{deg}}}p\) and \(y\in \pi ^{-1}(Y)\). Define \(q\in \mathcal H(\mathbb {C}^{n+1})\) by
$$\begin{aligned} q(x)=(mx\cdot \overline{y}/|y|)^v=[m|y|^{-1}(x_0\overline{y}_0+\cdots +x_n\overline{y}_n)]^v. \end{aligned}$$
By the Cauchy–Schwarz inequality, we have
$$\begin{aligned} \langle q\rangle _{\mathbb {P}^n}=m=\langle q(Y)\rangle . \end{aligned}$$
Let \(h=p^aq^{b-a}\). Then \(h\in \mathcal H(\mathbb {C}^{n+1})\) and \({{\mathrm{deg}}}\, h=bv\). For each \(X\in \mathbb {P}^n\), \(\langle h(X)\rangle =\langle p(X)\rangle ^\beta \langle q(X)\rangle ^{1-\beta }\). Hence,
$$\begin{aligned} \langle h\rangle _{\mathbb {P}^n}&\le m^{1-\beta }\le m,\\ \langle h\rangle _{E_m\cap K_m}&< m^{-1}m^{1-\beta }=m^{-\beta }\le 1,\\ \langle h(Y)\rangle&\ge r^\beta m^{1-\beta }=(r/m)^\beta m>m/2. \end{aligned}$$
To summarize, there is an \(h_Y\in \mathcal H(\mathbb {C}^{n+1})\) such that
$$\begin{aligned} \langle h_Y\rangle _{\mathbb {P}^n}&\le m,\;\ \langle h_Y\rangle _{E_m\cap K_m}<1,\quad \text { and }\quad \langle h_Y(Y)\rangle >m/2. \end{aligned}$$
Let \(U_Y:=\{X: \langle h_Y(X)\rangle >m/2\}\). Then \(U_Y\) is a neighborhood of Y. The open cover \(\{U_Y: Y\in \mathbb {P}^{n}{\setminus } E_m\}\) of \(\mathbb {P}^{n}{\setminus } E_m\) contains a countable subcover \(\{U_{Y_k}: k=1,2,\dots \}\). Put \(h_{mk}=h_{Y_k}\). Then the sequence \(\{h_{mk}\}\) satisfies (i), (ii) and (iii).

Suppose that \(X\in E\). Then there is an \(m_0\ge 2\) such that \(X\in (E_{m_0}\cap K_{m_0})\). We have \(\langle h_{mk}(X)\rangle \le 1\) for \(m\ge m_0\), and \(\langle h_{mk}(X)\rangle \le m_0-1\) for \(m< m_0\). Hence, \(\sup _{m,k} \langle h_{mk}(X)\rangle \le m_0-1<\infty \).

Suppose that \(X\in \mathbb {P}^n{\setminus } E\). For each m, \(X\in \mathbb {P}^n{\setminus } E_m\), hence \(\sup _k \langle h_{mk}(X)\rangle \ge m/2\). Thus \(\sup _{m,k} \langle h_{mk}(X)\rangle =\infty \).

Therefore,
$$\begin{aligned} E=\{X\in \mathbb {P}^n: \sup _{m,k} \langle h_{mk}(X)\rangle <\infty \}. \end{aligned}$$
By Lemma 4.2, \(E\in {\text {Conv}}(\mathbb {P}^n)\). \(\square \)

Corollary 4.19

The union of a countable collection of algebraic varieties in \(\mathbb {P}^n\) is a convergence set in \(\mathbb {P}^n\).

The converse of Theorem 4.18 is not true. The set \(\Lambda :=\{[1:z:\varphi (z)]: z\in \mathbb {C}, |z|=1\}\) in Example 4.12 is a convergence set, but it is not a countable union of complete pluripolar sets. Recall that \(\varphi \) is a non-polynomial entire function defined on \(\mathbb {C}\). Seeking for a contradiction, suppose that \(\Lambda =\cup F_j\), where \(F_j\) are complete pluripolar sets in \(\mathbb {P}^2\). Let \(Q:=\{[1:z:\varphi (z)]: z\in \mathbb {C}\}\). For each j, since \(F_j\subsetneqq Q\), and since \(F_j\) is a complete pluripolar set in \(\mathbb {P}^2\), it follows that \(F_j\) is polar in Q. Therefore, \(\Lambda =\cup F_j\) is polar in Q, which is false.

5 Affine convergence sets

Let \(\Lambda _n\) be the set of series \(f(t,x)=\sum _{j=0}^\infty P_j(x) t^j\in \mathbb {C}[x_1,\dots ,x_n][[t]]\) such that \(P_j\in \mathcal P_k(\mathbb {C}^n)\). (Here we use the convention that the degree of the zero polynomial is \(-1\); hence \(0\in \mathcal P_k(\mathbb {C}^n)\).) For \(f\in \Lambda _n\), let \({\text {Conv}}_a(f)\) be the set of \(x\in \mathbb {C}^n\) for which f(tx) converges as a series of a single indeterminate t:
$$\begin{aligned} {\text {Conv}}_a(f):=\{x\in \mathbb {C}^n: f(t,x)\in \mathbb {C}\{t\}\}. \end{aligned}$$
By Hartogs’ theorem, \({\text {Conv}}_a(f)=\mathbb {C}^n\) if and only if \(f\in \mathbb {C}\{t, x_1,\dots , x_n\}\), the set of convergent series in \(t, x_1,\dots , x_n\).

Definition 5.1

A subset E of \(\mathbb {C}^n\) is said to be an affine convergence set (in \(\mathbb {C}^n\)) if \(E={\text {Conv}}_a(f)\) for some divergent power series \(f\in \Lambda _n\). Let \({\text {Conv}}_a(\mathbb {C}^n)\) denote the collection of all affine convergence sets in \(\mathbb {C}^n\):
$$\begin{aligned} {\text {Conv}}_a(\mathbb {C}^n):=\{{\text {Conv}}_a(f): f\in \Lambda _n,\; f \text {diverges}\}. \end{aligned}$$

Affine convergence sets were studied in [12, 13].

Note that if \(x\in {\text {Conv}}_a(f)\) for a divergent series f it does not follow that \(\lambda x\in {\text {Conv}}_a(f)\quad \forall \lambda \in \mathbb {C}\), and therefore \(\pi ({\text {Conv}}_a(f))\) in general is not a convergence set in \(\mathbb {P}^{n-1}\).

In analogy with Lemma 4.2, we have the following lemma.

Lemma 5.2

Suppose that \(E\subsetneqq \mathbb {C}^{n}\) . Then \(E\in {\text {Conv}}_a(\mathbb {C}^n)\) if and only if there exists a sequence \(\{k_j\}\subset \mathbb {N}\) and a sequence \(\{P_j\}\) of polynomials with \(P_j\in \mathcal P_{k_j}(\mathbb {C}^n)\) for all j such that
$$\begin{aligned} E=\{x\in \mathbb {C}^n: \sup _{j}\langle P_j(x)\rangle _{k_j} <\infty \}. \end{aligned}$$
(19)
Let \(\iota :\mathbb {C}^n\rightarrow \mathbb {P}^n\) be defined by \(\iota (x_1,\dots , x_n)=[1:x_1:\dots : x_n]\). Then \(\iota \) embeds \(\mathbb {C}^n\) into \(\mathbb {P}^n\) and identifies \(\mathbb {C}^n\) with the affine open set \(\iota (\mathbb {C}^n)=U_0:=\{Z\in \mathbb {P}^n: Z_0\ne 0\}\). Define \(\tau :\Lambda _n\rightarrow \mathbb {C}[[x_0,\dots ,x_n]]\) by
$$\begin{aligned} {\tau \left( \sum _{j=0}^\infty P_j(x) t^j \right) =\sum _{j=0}^\infty x_0^jP_j(x/x_0).} \end{aligned}$$
Note that \(p_j(x_0,x):=x_0^jP_j(x/x_0)\) belongs to \(\mathcal H_j(\mathbb {C}^{n+1})\) for \(j\ge 0\). We also have \(\langle P_j(x)\rangle _j=\langle p_j(\iota (x))\rangle \) for \(j\ge 1\). It follows from Lemmas 4.2 and 5.2 that
$$\begin{aligned} \iota ({\text {Conv}}_a(f))=U_0\cap {\text {Conv}}(\tau (f)). \end{aligned}$$
Consequently, we have the following proposition.

Proposition 5.3

Suppose that \(E\subsetneqq \mathbb {C}^{n}\) . Then \(E\in {\text {Conv}}_a(\mathbb {C}^n)\) if and only if \(\iota (E)=U_0\cap F\) for some \(F\in {\text {Conv}}(\mathbb {P}^n)\).

For an affine convergence set E in \(\mathbb {C}^n\), the set \(\iota (E)\) may or may not be a convergence set in \(\mathbb {P}^n\).

Example 5.4

Let \(E=\{(x, e^x): x\in \mathbb {C}\}\subset \mathbb {C}^2\). Then the set \(\iota (E)\) equals the set \({\tilde{\mathbb {C}}}\) in Example 4.11 when \(\varphi (x)=e^x\). It follows that \(\iota (E)\) is a convergence set in \(\mathbb {P}^2\), and hence E is an affine convergence set in \(\mathbb {C}^2\).

Example 5.5

Let \(E=\{(x, 0): x\in \mathbb {C}\}\subset \mathbb {C}^2\). Then \(\iota (E)=U_0\cap H\), where \(H=\{[Z_0:Z_1:Z_2]: Z_2=0\}\). Since H is a convergence set in \(\mathbb {P}^2\) by Corollary 4.5, we see that E is an affine convergence set in \(\mathbb {C}^2\). Since H is a copy of \(\mathbb {P}^1\), it follows from Theorem 4.6 that for each convergence set Q in \(\mathbb {P}^2\), either \(H\subset Q\), or \(H\cap Q\) is polar in H. Therefore, \(\iota (E)=U_0\cap H\) is not a convergence set in \(\mathbb {P}^2\).

For a compact subset K of \(\mathbb {C}^n\), let
$$\begin{aligned} \hat{K}_a=\iota ^{-1}(U_0\cap \widehat{\iota (K)}). \end{aligned}$$
Intuitively, \(\hat{K}_a\) is the projective hull of K minus the part at \(\infty \). Equivalently, \(\hat{K}_a\) is the set of all \(x\in \mathbb {C}^n\) for which there is a constant \(C=C_x>0\) such that \(\langle P(x)\rangle _j\le C\langle P\rangle _{j,K}\) for all \(j\in \mathbb {N}\) and all \(P\in \mathcal P_j(\mathbb {C}^n)\).

Theorem 5.6

Let E be an affine convergence set in \(\mathbb {C}^{n}\) . Then there exists an ascending sequence \(\{K_{j}\}\) of compact pluripolar sets such that \(E=\cup K_j=\cup \hat{K}_{j,a}\) . In particular, E is a pluripolar \(F_\sigma \) set.

Theorem 5.7

The union of a countable collection of closed complete pluripolar sets in \(\mathbb {C}^n\) is an affine convergence set in \(\mathbb {C}^n\).

Corollary 5.8

The union of a countable collection of analytic varieties in \(\mathbb {C}^n\) is an affine convergence set in \(\mathbb {C}^n\).

Corollary 5.9

The finite sets and the countable sets in \(\mathbb {C}^n\) are affine convergence sets.

The proofs of Theorems 5.6 and 5.7 are routine modifications of the respective proofs of Theorems 4.6 and 4.18. Note that Theorem 5.7 is not a consequence of Theorem 4.18, since a complete pluripolar set in an affine open set in \(\mathbb {P}^n\) in general does not extend to a complete pluripolar set in \(\mathbb {P}^n\).

Footnotes
1

All sets considered in this paper are assumed to be Borel.

 

Declarations

Authors’ contributions

DM and TSN discussed the topic and obtained the results and proofs. DM wrote the first and the third versions, while TSN wrote the second version. Both authors read and approved the final manuscript.

Acknowledgements

We thank the referee for his comments and suggestions that greatly improved the paper. We thank B. Fridman, N. Levenberg, J. Ribón and A. Sadullaev for helpful discussions. We are very grateful to E. Poletsky for patiently answering our many questions.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Wichita State University
(2)
Department of Mathematics, California State University San Marcos

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© Ma and Neelon. 2015