Exploring Holomorphic Retracts
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909775124496384 |
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| author | Balakumar, G. P. Mammen, Jiju |
| author_facet | Balakumar, G. P. Mammen, Jiju |
| contents | The purpose of this article is towards systematically characterizing (holomorphic) retracts of domains of holomorphy; to begin with, bounded balanced pseudoconvex domains $B \subset \mathbb{C}^N$. Specifically, we show that every retract of $B$ passing through its center (origin), is the graph of a holomorphic map over a linear subspace of $B$. To deal with a case when may fail to have sufficiently many extreme points, we consider products of bounded balanced domains of holomorphy with holomorphically extreme boundaries and obtain a complete description of retracts passing through its center. This can be applied to solve a special case of the union problem with a degeneracy, namely: to characterize those Kobayashi corank one complex manifolds which can be expressed as an increasing union of submanifolds which are biholomorphic to a prescribed homogeneous bounded balanced domain. In this course, we prove a generalization of Mazet's Schwarz lemma. Results about non-existence of retracts of each possible dimension is established for the simplest non-convex but pseudoconvex domain: the $\ell^q$-`ball' for all $0<q <1$ as well as `anisotropic' analogues. The same is also done for balanced analytic polyhedra. This enables an illustration of applying retracts to establishing biholomorphic inequivalences. To go beyond balanced domains, we then first obtain a complete characterization of retracts of the Hartogs triangle and `analytic complements' thereof. Thereafter, similar characterization results for domains which are neither bounded nor topologically trivial. We conclude with some expositions about retracts of $\mathbb{C}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exploring Holomorphic Retracts Balakumar, G. P. Mammen, Jiju Complex Variables The purpose of this article is towards systematically characterizing (holomorphic) retracts of domains of holomorphy; to begin with, bounded balanced pseudoconvex domains $B \subset \mathbb{C}^N$. Specifically, we show that every retract of $B$ passing through its center (origin), is the graph of a holomorphic map over a linear subspace of $B$. To deal with a case when may fail to have sufficiently many extreme points, we consider products of bounded balanced domains of holomorphy with holomorphically extreme boundaries and obtain a complete description of retracts passing through its center. This can be applied to solve a special case of the union problem with a degeneracy, namely: to characterize those Kobayashi corank one complex manifolds which can be expressed as an increasing union of submanifolds which are biholomorphic to a prescribed homogeneous bounded balanced domain. In this course, we prove a generalization of Mazet's Schwarz lemma. Results about non-existence of retracts of each possible dimension is established for the simplest non-convex but pseudoconvex domain: the $\ell^q$-`ball' for all $0<q <1$ as well as `anisotropic' analogues. The same is also done for balanced analytic polyhedra. This enables an illustration of applying retracts to establishing biholomorphic inequivalences. To go beyond balanced domains, we then first obtain a complete characterization of retracts of the Hartogs triangle and `analytic complements' thereof. Thereafter, similar characterization results for domains which are neither bounded nor topologically trivial. We conclude with some expositions about retracts of $\mathbb{C}^2$. |
| title | Exploring Holomorphic Retracts |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2401.14700 |