Function Theoretic Properties of Symmetric Powers of Complex Manifolds

Zwonek, Włodzimierz

doi:10.1007/s12220-019-00233-z

Cited by 4 publications

(3 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is easy to see that Σ n (Ω) is a domain in C n . The following result of Zwonek [23,Theorem 16] shows that for most planar domains Ω, Σ n (Ω) is Kobayashi complete (see Section 2 for the definition) which plays a crucial role in this article.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On a spectral version of Cartan's theorem

Bera¹,

Chandel²,

Londhe³

2021

Preprint

View full text Add to dashboard Cite

For a domain Ω in the complex plane, we consider the domain Sn(Ω) consisting of those n × n complex matrices whose spectrum is contained in Ω. Given a holomorphic self-map Ψ of Sn(Ω) such that Ψ(A) = A and the derivative of Ψ at A is identity for some A ∈ Sn(Ω), we investigate when the map Ψ would be spectrum-preserving. We prove that if the matrix A is either diagonalizable or non-derogatory then for most domains Ω, Ψ is spectrum-preserving on Sn(Ω). Further, when A is arbitrary, we prove that Ψ is spectrumpreserving on a certain analytic subset of Sn(Ω).

show abstract

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On a spectral version of Cartan's theorem

Bera¹,

Chandel²,

Londhe³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Under the condition on Ω as in the statement of the lemma, a result of Zwonek [20,Theorem 16] implies that Σ n (Ω) is Kobayashi hyperbolic. Then Result 2.1 implies that Ψ is a constant function.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

Chandel

2021

Int. J. Math.

View full text Add to dashboard Cite

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider [Formula: see text]-point and [Formula: see text]-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

show abstract

“…The symmetrized bidisk and polydisk have been the subject of intense research for the past two decades; see, for instance, [AY01, EZ05,Nik06,ALY13]. More recently, the symmetric product of more general objects has also been studied by several researchers [CG15,BBDJ18,CG18,Zwo18]. The symmetric product of a Riemann surface can be given a natural complex structure that makes it into a complex manifold.…”

Section: Introductionmentioning

confidence: 99%

Schwarz Lemmas via the Pluricomplex Green’s Function

Janardhanan

2019

J Geom Anal

View full text Add to dashboard Cite

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.In this section, we de ne and prove basic facts about an extremal function de ned using plurisubharmonic functions. Our treatment is from [Kob98, p. 184] where the de nition is attributed to Klimek [Kli85]. The paper by Demailly [Dem87] contains further properties of this function.De nition 4. Let X be a complex manifold. Fix z 0 ∈ X and de ne the extremal functionwhere P X (z 0 ) is the collection of functions ϕ on X that satisfy:1. ϕ is upper semi-continuous,

show abstract

Function Theoretic Properties of Symmetric Powers of Complex Manifolds

Cited by 4 publications

References 18 publications

On a spectral version of Cartan's theorem

On a spectral version of Cartan's theorem

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

Schwarz Lemmas via the Pluricomplex Green’s Function

Contact Info

Product

Resources

About