Function theoretic properties of symmetric powers of complex manifolds. (English) Zbl 1436.32030
Summary: In the paper we study properties of symmetric powers of complex manifolds. We investigate a number of function theoretic properties [e. g. (quasi) \(c\)-finite 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.
MSC:
| 32C15 | Complex spaces |
| 32T40 | Peak functions |
| 32F45 | Invariant metrics and pseudodistances in several complex variables |
Keywords:
symmetric power of complex manifolds; (quasi) \(c\)-finite compactness; peak functions; symmetrized polydisc; Kobayashi and Carathéodory hyperbolicity (completeness)References:
| [1] | Agler, J.; Young, NJ, The hyperbolic geometry of the symmetrised bidisc, J. Geom. Anal., 14, 375-403 (2004) · Zbl 1055.32010 |
| [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) · Zbl 1402.32017 |
| [3] | Chakrabrati, D.; Gorai, S., Function theory and holomorphic maps on symmetric products of planar domains, J. Geom. Anal., 25, 4, 2196-2225 (2015) · Zbl 1334.32001 |
| [4] | Chakrabarti, D.; Grow, C., Proper holomorphic self-maps of symmetric powers of balls, Arch. Math., 110, 45-52 (2018) · Zbl 1388.32011 |
| [5] | Costara, C., Lempert’s theorem and the symmetrized bidisc, Bull. Lond. Math. Soc., 36, 5, 656-662 (2004) · Zbl 1065.32006 |
| [6] | Edigarian, A.: Carathéodory completeness on the plane. (2018). arXiv:1803.08714 · Zbl 1508.30096 |
| [7] | Edigarian, A.; Zwonek, W., Geometry of the symmetrized polydisc, Arch. Math. (Basel), 84, 364-374 (2005) · Zbl 1068.32012 |
| [8] | Green, ML, 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) · Zbl 0366.32013 |
| [9] | Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, de Gruyter, 2nd extended edition (2013) · Zbl 1273.32002 |
| [10] | Kiernan, P., Hyperbolic submanifolds of complex projective space, Proc. Am. Math. Soc., 22, 603-606 (1969) · Zbl 0182.11101 |
| [11] | Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehre d. mathematischen Wissenschaften, vol. 318 (1998) · Zbl 0917.32019 |
| [12] | Kosiński, Ł.; Zwonek, W., Proper holomorphic mappings vs. peak points and Shilov boundary, Ann. Polon. Math., 107, 97-108 (2013) · Zbl 1271.32036 |
| [13] | Lempert, L., La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. Fr., 109, 427-474 (1981) · Zbl 0492.32025 |
| [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) · Zbl 1123.32007 |
| [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) · Zbl 1155.32009 |
| [16] | Nikolov, N.; Trybuła, M., Gromov hyperbolicity of the Kobayashi metric on “convex” sets, J. Math. Anal. Appl., 468, 2, 1164-1178 (2018) · Zbl 1400.53033 |
| [17] | Snurnitsyn, VE, The complement of \(2n\) hyperplanes in \(CP^n\) is not hyperbolic, Mat. Zametki, 40, 455-459 (1986) · Zbl 0611.51019 |
| [18] | Whitney, H., Complex Analytic Varieties (1972), Reading, London: Addison-Wesley Publishing Co., Reading, London · Zbl 0265.32008 |
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