arXiv subjects

Aeryeong Seo

Publications and source records attributed to Aeryeong Seo.

At least 19 recordsLinked to original sources

arXiv2026

Intermediate Pseudoconvexity of Fiber Bundles

In this paper, we investigate the pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces of genus $\geq 2$, as well as the intermediate pseudoconvexity of their complements in the associated projective space bundles. Inspired by Brunella's work, we prove that any such ball bundle is $1$-convex, while its complement is $n$-convex, where $n$ denotes the dimension of the ball fiber, provided that the bundle admits a harmonic section with a regular point.

math.CV↗

arXiv2026

Schwarz-Pick Lemma for Invariant Harmonic Functions on the Complex Unit Ball

This paper establishes a sharp Schwarz-Pick type inequality for real-valued invariant harmonic functions defined on the complex unit ball $\mathbb B^n$. The proof of this main result simultaneously provides a solution to a natural extension of the Khavinson conjecture for invariant harmonic functions, demonstrating that the sharp constants for the gradient and the radial derivative coincide. As further consequences of the main theorem, we derive two corollaries.

math.CV↗

arXiv2025

Symmetric differentials and jets extension of $L^2$ holomorphic functions II: Explicit form

For a symmetric differential on the compact quotient $Σ= \mathbb{B}^n / Γ$ of the complex unit ball $\mathbb{B}^n \subset \mathbb{C}^n$ by a discrete subgroup $Γ\subset \mathrm{Aut}(\mathbb{B}^n)$, there exists a corresponding weighted $L^2$-holomorphic function on $(\mathbb{B}^n \times \mathbb{B}^n)/Γ$, where $Γ$ acts diagonally on $\mathbb{B}^n \times \mathbb{B}^n$. In this paper, we give an explicit description of this correspondence and derive several applications based on its explicit form.

math.CV↗

arXiv2025

On the Kähler-hyperbolicity of bounded symmetric domains

In this paper, we characterize the Kähler-hyperbolicity length of a bounded symmetric domain, defined by its rank and genus, as a unique constant determined by a constant gradient length of a special Bergman potential. Additionally, we establish a characterization of the lower bound of $L^\infty$ norm of the gradient length of any Bergman potential.

math.CV↗

arXiv2025

Proper holomorphic maps between bounded symmetric domains with small rank differences

In this paper we study the rigidity of proper holomorphic maps $f\colon Ω\toΩ'$ between irreducible bounded symmetric domains $Ω$ and $Ω'$ with small rank differences: $2\leq \text{rank}(Ω')< 2\,\text{rank}(Ω)-1$. More precisely, if either $Ω$ and $Ω'$ have the same type or $Ω$ is of type~III and $Ω'$ is of type~I, then up to automorphisms, $f$ is of the form $f=\imath\circ F$, where $F = F_1\times F_2\colon Ω\to Ω_1'\times Ω_2'$. Here $Ω_1'$, $Ω_2'$ are bounded symmetric domains, the map $F_1\colon Ω\to Ω_1'$ is a standard embedding, $F_2: Ω\to Ω_2'$, and $\imath\colon Ω'_1\times Ω'_2 \to Ω'$ is a totally geodesic holomorphic isometric embedding. Moreover we show that, under the rank condition above, there exists no proper holomorphic map $f: Ω\to Ω'$ if $Ω$ is of type~I and $Ω'$ is of type~III, or $Ω$ is of type~II and $Ω'$ is either of type~I or III. By considering boundary values of proper holomorphic maps on maximal boundary components of $Ω$, we construct rational maps between moduli spaces of subgrassmannians of compact duals of $Ω$ and $Ω'$, and induced CR-maps between CR-hypersurfaces of mixed signature, thereby forcing the moduli map to satisfy strong local differential-geometric constraints (or that such moduli maps do not exist), and complete the proofs from rigidity results on geometric substructures modeled on certain admissible pairs of rational homogeneous spaces of Picard number 1.

math.CV↗

arXiv2023

Cohomology Isomorphism of Symmetric Power of Cotangent Bundle of Ball quotient and Its Toroidal Compactification

In this paper, we investigate the $L^2$-Dolbeault cohomology of the symmetric power of cotangent bundles of ball quotients with finite volume, as well as their toroidal compactification. Through the application of Hodge theory for complete hermitian manifolds, we establish the existence of Hodge decomposition and Green's operator. Moreover, we extend the results by Adachi [A21] and Lee-Seo [LS23-2] from compact complex hyperbolic spaces to complex hyperbolic spaces with finite volume.

math.CV↗

arXiv2023

Ampleness of normal bundles of base cycles in flag domains

Flag domains are open orbits of noncompact real forms of complex semisimple Lie groups acting on flag manifolds. To each flag domain one can associate a compact complex manifold called the base cycle. The ampleness of the normal bundle of the base cycle in a flag domain measures the concavity near the base cycle. In this paper we compute the ampleness of normal bundles of base cycles in flag domains in various cases, including flag domains in the full flag manifolds $G/B$ when $G$ is classical, and period domains parameterizing polarized Hodge structures with fixed Hodge numbers.

math.CV↗

arXiv2023

Weighted $L^2$ Holomorphic functions on ball-fiber bundles over compact Kähler manifolds

Let $\widetilde{M}$ be a complex manifold and $Γ$ be a torsion-free cocompact lattice of $\text{Aut}(\widetilde{M})$. Let $ρ\colonΓ\to SU(N,1)$ be a representation and $M:=\widetilde M/Γ$ be an $n$-dimensional compact complex manifold which admits a holomorphic embedding $\imath$ into $Σ:=\mathbb B^N/ρ(Γ)$. In this paper, we investigate a relation between weighted $L^2$ holomorphic functions on the fiber bundle $Ω:=M\times_ρ\mathbb B^N$ and the holomorphic sections of the pull-back bundle $\imath^{-1}(S^mT^*_Σ)$ over $M$. In particular, $A^2_α(Ω)$ has infinite dimension for any $α>-1$ and if $n -1$, $A^2_α(\mathbb B^n\times_ρ \mathbb B^N)$ has infinite dimension. If $n<N$, then $A_{-1}^2(\mathbb B^n\times_ρ \mathbb B^N)$ also has the same property.

math.CV↗

arXiv2023

A Kähler potential on the unit ball with constant differential norm

Let $\mathbb B^n$ be the unit ball in $\mathbb C^n$ and $\mathbb H^n$ be the homogeneous Siegel domain of the second kind which is biholomorphic to $\mathbb B^n$. We show that the Kähler potential of $\mathbb H^n$ is unique up to the automorphisms among Kähler potentials whose differentials have constant norms. As an application, we consider a domain $Ω$ in $\mathbb C^n$, which is biholomorphic to $\mathbb B^n$. We show that if $Ω$ is affine homogeneous, then it is affine equivalent to $\mathbb H^n$. Assume next that its canonical potential with respect to the Kähler--Einstein metric has a differential with a constant norm. If the biholomorphism between $Ω$ and $\mathbb B^n$ is a restriction of a Möbius transformation, then the map is affine equivalent to a Cayley transform.

math.CV↗

arXiv2022

A characterization of the unit ball by a Kähler-Einstein potential

We will show that a universal covering of a compact Kähler manifold with ample canonical bundle is the unit ball if it admits a global potential function of the Kähler-Einstein metric whose gradient length is a minimal constant. As an application, we will extend the Wong-Rosay theorem to a complex manifold without boundary.

math.CV↗

arXiv2022

Weakly 1-completeness of holomorphic fiber bundles over compact Kähler manifolds

In 1985 Diederich and Ohsawa proved that every disc bundle over a compact Kähler manifold is weakly 1-complete. In this paper, under certain conditions we generalize this result to the case of fiber bundles over compact Kähler manifolds whose fibers are bounded symmetric domains. Moreover if the bundle is obtained by the diagonal action on the product of irreducible bounded symmetric domains, we show that it is hyperconvex.

math.CV↗

arXiv2022

Totally geodesic discs in bounded symmetric domains

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon Ω\toΩ'$ between bounded symmetric domains $Ω$ and $Ω'$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $Ω$ is irreducible, there exist totally geodesic bounded symmetric subdomains $Ω_1$ and $Ω_2$ of $Ω'$ such that $f = (f_1, f_2)$ maps into $Ω_1\times Ω_2\subset Ω$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text{rank}(Ω')<2\text{rank}(Ω)$, then either $f$ or $\bar f$ is a standard holomorphic embedding.

math.CV↗

arXiv2022

Holomorphicity of totally geodesic Kobayashi isometry between bounded symmetric domains

In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon Ω\toΩ'$ where $Ω$, $Ω'$ are bounded symmetric domains, if $Ω$ is irreducible and $\text{rank}(Ω) \geq \text{rank}(Ω')$ or more generally, $\text{rank}(Ω) \geq \text{rank}(f_*v)$ for any tangent vector $v$ of $Ω$, then $f$ is either holomorphic or anti-holomorphic. Secondly we characterize $C^1$ Kobayashi isometries from a reducible bounded symmetric domain to itself.

math.CV↗

arXiv2021

Symmetric differentials and jets extension of $L^2$ holomorphic functions

Let $Σ= \mathbb B^n/Γ$ be a complex hyperbolic space with discrete subgroup $Γ$ of the automorphism group of the unit ball $\mathbb B^n$ and $Ω$ be a quotient of $\mathbb B^n \times\mathbb B^n$ under the diagonal action of $Γ$ which is a holomorphic $\mathbb B^n$-fiber bundle over $Σ$. The goal of this article is to investigate the relation between symmetric differentials of $Σ$ and the weighted $L^2$ holomorphic functions of $Ω$. If there exists a holomorphic function on $Ω$ and it vanishes up to $k$-th order on the maximal compact complex variety in $Ω$, then there exists a symmetric differential of degree $k+1$ on $Σ$. Using this property, we show that $Σ$ always has a symmetric differential of degree $N$ for any $N \geq n+2$. Moreover if $Σ$ is compact, for each symmetric differential over $Σ$ we construct a weighted $L^2$ holomorphic function on $Ω$. We also show that any bounded holomorphic function on $Ω$ is constant when $H^0 (Σ, S^{m} T_Σ^* )=0$ for every $0 < m \leq n+1$.

math.CV↗

arXiv2019

Generalizations of linear fractional maps for classical symmetric domains and related fixed point theorems for generalized balls

We extended the study of the linear fractional self maps (e.g. by Cowen-MacCluer and Bisi-Bracci on the unit balls) to a much more general class of domains, called generalized type-I domains, which includes in particular the classical bounded symmetric domains of type-I and the generalized balls. Since the linear fractional maps on the unit balls are simply the restrictions of the linear maps of the ambient projective space (in which the unit ball is embedded) on a Euclidean chart with inhomogeneous coordinates, and in this article we always worked with homogeneous coordinates, here the term linear map was used in this more general context. After establishing the fundamental result which essentially says that almost every linear self map of a generalized type-I domain can be represented by a matrix satisfying the "expansion property" with respect to some indefinite Hermitian form, we gave a variety of results for the linear self maps on the generalized balls, such as the holomorphic extension across the boundary, the normal form and partial double transitivity on the boundary for automorphisms, the existence and the behavior of the fixed points, etc. Our results generalize a number of known statements for the unit balls, including, for example, a theorem of Bisi-Bracci saying that any linear fractional map of the unit ball with more than two boundary fixed points must have an interior fixed point.

math.CV↗

arXiv2018

Normal bundles of cycles in flag domains

A real semisimple Lie group G_0 embedded in its complexification G has only finitely many orbits in any G-fag manifold Z = G/Q. The complex geometry of its open orbits D (flag domains) is studied from the point of view of compact complex submanifolds C (cycles) which arise as orbits of certain distinguished subgroups. Normal bundles E of the cycles are analyzed in some detail. It is shown that E is trivial if and only if D is holomorphically convex, in fact a product of C and a Hermitian symmetric space, and otherwise D is pseudoconcave.

math.AG↗

arXiv2018

A Kobayashi pseudo-distance for holomorphic bracket generating distributions

In this paper, we generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow's theorem in sub-Riemannian geometry. Let G be a linear semisimple Lie group. For a complex $G$-homogeneous manifold M with a G-invariant holomorphic bracket generating distribution D, we prove that (M,D) is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain and the induced distribution is the superhorizontal distribution.

math.CV↗

arXiv2018

Biholomorphisms between Hartogs domains over homogeneous Siegel domains

In this paper, we characterize the Hartogs domains over homogeneous Siegel domains of type II and explicitly describe their automorphism groups. Moreover we prove that any proper holomorphic map between Hartogs domains over homogeneous Siegel domains over type II is a biholomorphism.

math.CV↗