Function Theoretic Properties of Symmetric Powers of Complex Manifolds
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Abstract
In the paper we study properties of symmetric powers of complex manifolds. We investigate a number of function theoretic properties [e. g. (quasi) cfinite compactness, existence of peak functions] that are preserved by taking the symmetric power. The case of symmetric products of planar domains is studied in a more detailed way. In particular, a complete description of the Carathéodory and Kobayashi hyperbolicity and Kobayashi completeness in that class of domains is presented.
Keywords
Symmetric power of complex manifolds (quasi) cfinite compactness Peak functions Symmetrized polydisc Kobayashi and Carathéodory hyperbolicity (completeness)Mathematics Subject Classification
32C15 32T40 32F451 Introduction
1.1 Description of Results
In the paper we present a number of properties of \(X^n_\mathrm{{sym}}\). Some of them may be obtained from general properties of the realization of \(\mathbb {D}^n_\mathrm{{sym}}\) as a domain, i. e. the so called the symmetrized polydisc\(\mathbb {G}_n:=S_n(\mathbb {D})\), (\(\mathbb {D}\) denotes the unit disc in \(\mathbb {C}\)). The last domain has been extensively studied in the last two decades (see for instance [1, 5, 7, 9] and references there).
The starting point for considerations in the paper was inspired by recent developments on the function theory in symmetric powers (see e. g. [2, 3] and [4]).
First we present a more general result to that of Theorem 1.4 in [2] where the proof of the Kobayashi completeness of symmetric powers of some of Riemann surfaces relies on the proof of existence of peak functions. Similarly as in [2] the presentation below actually deals with a stronger version of completeness—the cfinite compactness and shows that the notion (more precisely, the weaker notion of quasi cfinite compactness) is preserved under taking the symmetric power, which is done in a general case of complex manifolds (Theorem 1). Following the same line of argument relying upon analoguous results in the symmetrized polydisc, we present a result on the existence of peak functions in symmetric powers (Theorem 6).
In Sect. 3, we concentrate on properties of symmetric products of planar domains in \(\mathbb {C}\). We show the linear convexity of such domains (Proposition 9), we present a Riemanntype mapping theorem for them (Theorem 12) and then we discuss to which extent the Kobayashi hyperbolicity (completeness) is preserved under taking the symmetric power—a complete description of Kobayashi hyperbolicity (completeness) in that class is given in Theorem 16. Finally, we present a result on preserving the Carathéodory hyperbolicity under taking the symmetric powers of planar domains (Proposition 18).
2 General Case
In this Section, we present results for a general class of symmetric powers of manifolds.
2.1 (Quasi) cFinite Compactness
If, on some complex structure X (e. g. a complex manifold), we may welldefine the Carathéodory pseudodistance we call Xquasicfinitely compact if for any sequence \((z_k)_k\subset X\) without the accummulation point we have \(c_X(z_1,z_k)\rightarrow \infty \). Recall that if X is additionally Carathéodory hyperbolic , i. e. \(c_X(w,z)>0\), \(w,z\in X\), \(w\ne z\) then X is called cfinitely compact. As to the basic properties related to the Carathéodory pseudodositance (as well as to other holomorphically invariant functions) we refer the Reader to e. g. [9, 11].
As we shall see below a natural property that is inherited by the symmetric power is the quasi cfinite compactness.
Theorem 1
Let X be a connected complex manifold. Then \(X^n_\mathrm{{sym}}\) is quasi cfinitely compact iff X is quasi cfinitely compact.
Proof
Remark 2
In the case when X is a bounded domain in \(\mathbb {C}\) the above theorem is a reformulation of Theorem 4.1 in [12] (applied to the proper holomorphic mapping \((\pi _n)_{D^n}:D^n\rightarrow S_n(D)\)). We also should be aware of the fact that the idea of the proof of the above theorem is exactly the same as that of Theorem 4.1 in [12].
Remark 3
In Theorem 1.4 in [2] a result on Kobayashi completeness of symmetric powers of some Riemann surfaces is formulated. The proof relies on the existence of some peak functions together with the application of Result 3.6 from [2], in which the fact of cfinite compactness is claimed under assumption of the existence of some peak functions. It is however not explained us how the necessary fact of the Carathéodory hyperbolicity is obtained only with the help of the existence of peak functions (in the case studied in the reasoning from [11], to which the paper [2] appeals, the hyperbolicity is trivially satisfied).
Remark 4
X is dhyperbolic \(\implies X^n_\mathrm{{sym}}\) is dhyperbolic
the observation in the next remark shows that it is not true in general.
Remark 5
It would be tempting to formulate a similar equivalence as in Theorem 1 for the notion of the Kobayashi quasi completeness. However, the example \(\mathbb {C}\setminus \{0,1\}\) shows that the Kobayashi completeness of X does not guarantee any reasonable property of the Kobayashi pseudodistance of \(X^n_\mathrm{{sym}}\).
2.2 Peak Functions
In the paper [2], the proof of the Kobayashi completeness (Theorem 1.4) was conducted with the help of the existence of peak functions (that was done for some Riemann surfaces). We generalize the result and simplify the proof below. We also see that we may reduce the proof of the existence of peak functions in symmetric powers to the existence of some of peak functions in the original complex manifold.
For a domain Y in a complex manifold X and \(K\subset \overline{Y}\) we define \(A(Y,K):=\mathcal {O}(Y)\cap \mathcal {C}(Y\cup K)\). We call a point \(z\in K\) an A(Y, K) peak point if there is an \(f\in A(Y,K)\) such that \(f<1\) on Y and \(f(z)=1\). We call ftheA(Y, K) peak function atz.
Theorem 6
Let Y be a domain in a complex manifold X. Assume that \(z_1\in \partial Y\) is the \(A(Y,\{z_1,\ldots ,z_n\})\) peak point, where \(z_2,\ldots ,z_n\in \overline{Y}\). Then the point \(\langle z_1,\ldots ,z_n\rangle \) is an \(A(Y^n_\mathrm{{sym}},\{\langle z_1,\ldots ,z_n\rangle \})\) peak point.
Proof
3 Remarks on Symmetric Products of Planar Domains
In this section we present properties of symmetric products of planar domains.
3.1 General Properties
Recall that if D is a domain in \(\mathbb {C}\) then we have a nice representation of \(D^n_\mathrm{{sym}}\) as a domain in \(\mathbb {C}\). We work therefore on this representation, i. e. the domain \(S_n(D)\). First we collect some facts concerning \(S_n(D)\). Below we list some known or straightforward properties of \(S_n(D)\).
Remark 8

\(S_n(D)\) is a domain in \(\mathbb {C}^n\) and \((\pi _n)_{D^n}:D^n\rightarrow S_n(D)\) is proper onto the image,

if \(\Sigma _n:=\{z\in D^n: z_j= z_k \text { for some}\ j\ne k\}\) then \(\pi _n{:}D^n\setminus \Sigma _n\rightarrow S_n(D)\setminus \pi _n(\Sigma _n)\) is a holomorphic covering,

\(S_n(D)\) is bounded iff D is bounded,

\(\overline{S_n(D)}=\pi _n(\overline{D}^n)\), \(\partial S_n(D)=\pi _n(\partial D\times \overline{D}^{n1})\),

if D is additionally bounded then the mapping \((\pi _n)_{D^n}\) maps \(A(D^n)\) peak points (\(A(\Omega ) :=\mathcal {O}(\Omega )\cap \mathcal {C}(\overline{\Omega })\)) onto \(A(S_n(D))\) peak points. In particular, the Shilov boundary \(\partial _S(S_n(D))\) equals \(S_n(\partial _S(D))\) (see Theorem 3.1 in [12]).
Recall that a domain \(\Omega \subset \mathbb {C}^n\) is called linearly convex if for any \(w\in \mathbb {C}^n\setminus \Omega \) we may find an affine hyperplane H passing through w and disjoint from \(\Omega \). Following step by step the idea from [15] we get the linear convexity of \(S_n(D)\).
Proposition 9
Proof
Remark 10
We show how some properties of D induce the same ones of \(S_n(D)\) (compare Theorem 1). The first notion that we discuss is the hyperconvexity.
Proposition 11
Let D be a domain in \(\mathbb {C}\), \(n\ge 2\). Then \(S_n(D)\) is hyperconvex iff D is hyperconvex.
Proof
To prove the opposite implication fix some \(\lambda _1,\ldots ,\lambda _{n1}\in D\) and let v be the negative plurisubharmonic exhaustion function on \(S_n(D)\). Let \(u(\cdot ):=v(\pi _n(\lambda _1,\ldots ,\lambda _{n1},\cdot ))\) be defined on D. Then u is a negative exhaustion subharmonic function on D. \(\square \)
3.2 RiemannType Mapping Theorem
In the next result we show a Riemanntype mapping theorem for symmetric powers of planar domains.
Theorem 12
Let D be a bounded, hyperconvex domain in \(\mathbb {C}\), \(n\ge 2\). Assume that \(c_{S_n(D)}\equiv l_{S_n(D)}\). Then D is biholomorphic to \(\mathbb {D}\) and \(n=2\).
Proof
Define \(\tilde{g}\) (respectively, \(\tilde{g}_k\)) to be the n components of the multivalued function \(\pi _n^{1}\circ f\) (respectively, \(\pi _n^{1}\circ f_k\)). We know that all the components of \(\tilde{g}\) (respectively, \(\tilde{g}_k\)) have values in \(\overline{D}\) (respectively, D). Additionally at the points 0 and \(\sigma \) all but one components of \(\tilde{g}\) are from \(\partial D\). Note that the values of f at these two points are regular values for the proper holomorphic mapping \(\pi _n\) and thus the functions \(\pi _n^{1}\circ f\) near these two points (0 and \(\sigma \)) may be chosen to be a holomorphic mapping.
The openness of holomorphic functions and the description of the closure of \(\pi _n(D)\) together with the fact that \(\lambda _j^0\in \partial (\hbox {int}\,\overline{D})\), \(j=1,\ldots ,n1\), imply that near these two points all but one components of \(\tilde{g}\) are constant (and equal to \(\lambda _1^0,\ldots ,\lambda _{n1}^0\)) whereas the last one is from D. The fact that some nonempty open part of \(f(\mathbb {D})\) is lying in the complex line \(L:=\{\pi _n(\lambda _1^0,\ldots , \lambda _{n1}^0,\lambda ):\lambda \in \mathbb {C}\}\) implies easily that \(f(\mathbb {D})\subset L\) and consequently all but one components of \(\tilde{g}\) are constant (equal to \(\lambda _j^0\), \(j=1,\ldots ,n1\)) and the last component is a nonconstant holomorpic function \(g:\mathbb {D}\rightarrow \hbox {int}\,(\overline{D})\) with \(g(0)=\lambda _n^0\) and \(g(\sigma )=\mu _n^0\). We show below that we have even more; namely, \(g(\mathbb {D})\subset D\). In fact, following the same line of argument we get more; the multivalued functions \(\tilde{g}_k\) have the following property: there exists a sequence \(r_k\rightarrow 1\), \(0<r_k<1\), such that the multivalued functions \(\tilde{g}_k\) are actually holomorphic functions when restricted to \(r_k\mathbb {D}\) with values in D. Moreover, taking the last component of the multifunction \(\tilde{g}_k\) (which we denote by \(g_k\)) we get that \(g_k\rightarrow g\) locally uniformly on \(\mathbb {D}\). Recall that \(g_k:r_k\mathbb {D}\rightarrow D\) so the Hurwitz theorem implies that \(g(\mathbb {D})\subset D\).
Since \(G\circ g(0)=0\), \(G\circ g(\sigma )=\sigma \), the Schwarz Lemma implies that \(G\circ g\) is the identity. Consequently, D is biholomorphic to \(\mathbb {D}\) (with biholomorphisms given by g or G). The results on the symmetrized polydisc (see [1, 5, 14]) imply that \(n=2\). \(\square \)
Remark 13
3.3 Kobayashi Hyperbolicity and Completeness of Symmetric Products of Planar Domains
Recall that the Kobayashi (pseudo)distance\(k_{\Omega }\) of a domain \(\Omega \subset \mathbb {C}^n\) may be defined as the largest pseudodistance smaller than or equal to \(l_{\Omega }\). The domain \(\Omega \) is called Kobayashi hyperbolic if \(k_{\Omega }\) is a distance. If additionally, \((\Omega ,k_{\Omega })\) is a complete metric space then \(\Omega \) is called Kobayashi complete. Recall that the Kobayashi completeness of a Kobayashi hyperbolic domain is equivalent to thekfinite compactness, i. e. the fact that \(k_{\Omega }(z,z^k)\rightarrow \infty \) for some (any) \(z\in \Omega \) and any sequence \((z^k)_k\subset \Omega \) having no accummulation point (see e. g. [9, 11]).
We already know that representations of symmetric products of planar domains are linearly convex. It is worth mentioning that a bounded linearly convex domain \(\Omega \subset \mathbb {C}^n\) is automatically Kobayashi complete—as mentioned by N. Nikolov to the author it follows directly from Lemma 3.3 in [16].
Proposition 14
(see Lemma 3.3 in [16]) Let \(\Omega \subset \mathbb {C}^n\) be a bounded linearly convex domain. Then \(\Omega \) is Kobayashi complete.
The above proposition allows us to conclude that a domain \(S_n(D)\) is Kobayashi complete if \(D\subset \mathbb {C}\) is bounded. In the unbounded case we should be more careful. Below we present a complete description of Kobayashi hyperbolicity and completeness of symmetric products of planar domains. We start with the special case.
Proposition 15
Fix \(n,N\ge 2\). Let \(\mu _1,\ldots ,\mu _N\in \mathbb {C}\) be pairwise different.
If \(N\ge 2n\) then the domain \(S_n(\mathbb {C}\setminus \{\mu _1,\ldots ,\mu _N\})\) is Kobayashi complete.
If \(N<2n\) then the domain \(S_n(\mathbb {C}\setminus \{\mu _1,\ldots ,\mu _N\}\) contains a nonconstant holomorphic image of \(\mathbb {C}\) and thus it is not Kobayashi hyperbolic.
Proof
Then the theorem on Kobayashi completeness of the complement of the unions of \((2n+1\)) hyperplanes in general position in the projective space (see [8, 11]) and results on nonhyperbolicity of complements of 2n hyperplanes (see [10] and [17] or [11]) finish the proof. \(\square \)
Theorem 16
Let \(D\subset \mathbb {C}\) be a domain and let \(n\ge 2\) be fixed. If \(\#(\mathbb {C}\setminus D)\ge 2n\) then \(S_n(D)\) is Kobayashi complete. If \(\#(\mathbb {C}\setminus D)<2n\) then \(S_n(D)\) contains a nonconstant holomorphic image of \(\mathbb {C}\) and thus \(S_n(D)\) is not Kobayashi hyperbolic.
Proof
Remark 17
As we saw in the proof of Theorem 16 the description of Kobayashi complete symmetric products of planar domains relied not only on the linear convexity of \(S_n(D)\) but also on the special geometry of \(S_n(D)\). It could be interesting to see whether the following could be true: a linearly convex domain, which admits a certain number (at least 2n) of hyperplanes in a general position disjoint from the domain, is Kobayashi complete.
3.4 Carathéodory Hyperbolicity
It turns out that in the class of symmetric products of planar domains the Carathéodory hyperbolicity is preserved under taking symmetric powers.
Proposition 18
Let D be a domain in \(\mathbb {C}\). Then D is Carathéodory hyperbolic if and only if \(S_n(D)\) is Carathéodory hyperbolic. Consequently, D is cfinitely compact if and only if \(S_n(D)\) is cfinitely compact.
Proof
Remark 19
Recall that in the class of planar domains by a recent result (Theorem 1 in [6]) two closely related notions of Carathéodory completeness and cfinite compactness are equivalent. Moreover, they are both equivalent to the fact that any boundary point z of D is an \(A(D,\{z\})\) peak point. Note that although the cfinite compactness is equivalent to cfinite compactness of \(S_n(D)\) (Proposition 18) we did not prove the equivalence of cfinite compactness of \(S_n(D)\) with the fact that any boundary point z of \(S_n(D)\) is a weak \(A(D,\{z\})\) point.
Notes
Acknowledgements
The author was partially supported by the OPUS Grant No. 2015/17/B/ST1/00996 financed by the National Science Centre, Poland.
References
 1.Agler, J., Young, N.J.: The hyperbolic geometry of the symmetrised bidisc. J. Geom. Anal. 14, 375–403 (2004)MathSciNetCrossRefGoogle Scholar
 2.Bharali, G., Biswas, I., Divakaran, D., Janardhanan, J.: Proper holomorphic mappings onto symmetric products of a Riemann surface. Doc. Math. 23, 1291–1311 (2018)MathSciNetzbMATHGoogle Scholar
 3.Chakrabrati, D., Gorai, S.: Function theory and holomorphic maps on symmetric products of planar domains. J. Geom. Anal. 25(4), 2196–2225 (2015)MathSciNetCrossRefGoogle Scholar
 4.Chakrabarti, D., Grow, C.: Proper holomorphic selfmaps of symmetric powers of balls. Arch. Math. 110, 45–52 (2018)MathSciNetCrossRefGoogle Scholar
 5.Costara, C.: Lempert’s theorem and the symmetrized bidisc. Bull. Lond. Math. Soc. 36(5), 656–662 (2004)CrossRefGoogle Scholar
 6.Edigarian, A.: Carathéodory completeness on the plane. (2018). arXiv:1803.08714
 7.Edigarian, A., Zwonek, W.: Geometry of the symmetrized polydisc. Arch. Math. (Basel) 84, 364–374 (2005)MathSciNetCrossRefGoogle Scholar
 8.Green, M.L.: The hyperbolicity of the complement of \(2n + 1\) hyperplanes in general position in \(P_n\), and related results. Proc. Am. Math. Soc. 66(1), 109–113 (1977)MathSciNetzbMATHGoogle Scholar
 9.Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, de Gruyter, 2nd extended edition (2013)Google Scholar
 10.Kiernan, P.: Hyperbolic submanifolds of complex projective space. Proc. Am. Math. Soc. 22, 603–606 (1969)MathSciNetCrossRefGoogle Scholar
 11.Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehre d. mathematischen Wissenschaften, vol. 318 (1998)CrossRefGoogle Scholar
 12.Kosiński, Ł., Zwonek, W.: Proper holomorphic mappings vs. peak points and Shilov boundary. Ann. Polon. Math. 107, 97–108 (2013)MathSciNetCrossRefGoogle Scholar
 13.Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. Fr. 109, 427–474 (1981)CrossRefGoogle Scholar
 14.Nikolov, N., Pflug, P., Zwonek, W.: The Lempert function of the symmetrized polydisc in higher dimensions is not a distance. Proc. Am. Math. Soc. 135(9), 2921–2928 (2007)MathSciNetCrossRefGoogle Scholar
 15.Nikolov, N., Pflug, P., Zwonek, W.: An example of a bounded \({\mathbb{C}}\)convex domain which is not biholomorphic to a convex domain. Math. Scand. 102(1), 149–155 (2008)MathSciNetCrossRefGoogle Scholar
 16.Nikolov, N., Trybuła, M.: Gromov hyperbolicity of the Kobayashi metric on “convex” sets. J. Math. Anal. Appl. 468(2), 1164–1178 (2018)MathSciNetCrossRefGoogle Scholar
 17.Snurnitsyn, V.E.: The complement of \(2n\) hyperplanes in \(CP^{n}\) is not hyperbolic. Mat. Zametki 40, 455–459 (1986)MathSciNetzbMATHGoogle Scholar
 18.Whitney, H.: Complex Analytic Varieties. AddisonWesley Series in Mathematics. AddisonWesley Publishing Co., Reading, London (1972)zbMATHGoogle Scholar
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