Holomorphic Convexity Research Articles

Sufficiency criteria for a class of convex functions connected with tangent function

<abstract><p>The research here was motivated by a number of recent studies on Hankel inequalities and sharp bounds. In this article, we define a new subclass of holomorphic convex functions that are related to tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic.</p></abstract>

A Remark on Oka’s Lemma and a Geometric Property of Pseudoconvex Domains

A direct proof of Oka’s lemma on the relation of holomorphic convexity and the properties of the distance to the boundary function is provided. Some related problems for subharmonicity properties of this function are also studied. A new geometric property of pseudoconvex domains is described.

On the holomorphic convexity of reductive Galois coverings over compact Kähler surfaces

This article generalizes the result of Katzarkov and Ramachandran from algebraic surfaces to Kähler surfaces. We follow their argument to prove the holomorphic convexity of a reductive Galois covering over a compact Kähler surface which does not have two ends, except that we replace the p-adic factorization theorem by an analysis of the singularities of the continuous subanalytic plurisubharmonic exhaustion function.

Holomorphic convexity of pseudoconvex surfaces

Holomorphic convexity of pseudoconvex surfaces

Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf

Abstract We prove that a pseudoconvex complex space X of pure dimension n + 1 {n+1} is holomorphically convex provided that its singular set has only compact connected components (e.g., X has isolated singularities) and rank x ⁡ 𝒪 X = n {\operatorname{rank}_{x}\mathscr{O}_{X}=n} for all smooth points x outside a compact set of X. This extends a known result due to Ohsawa asserting that a weakly 1-complete, smooth complex surface is holomorphically convex if it admits a globally defined non-constant holomorphic function. We also revise an example of Markoe concerning non-invariance of holomorphic convexity under normalization.

Stein neighbourhoods of bordered complex curves attached to holomorphically convex sets

In this paper we construct open Stein neighbourhoods of compact sets of the form $A\cup K$ in a complex space, where $K$ is a compact holomorphically convex set, $A$ is a compact complex curve with boundary $bA$ of class $\mathscr C^2$ which may intersect $K$, and $A\cap K$ is $\mathscr O(A)$-convex.

Euclidean domains in complex manifolds

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets K∪M in a complex manifold X, where K is a compact holomorphically convex set in an open Stein neighbourhood, M is an embedded Stein submanifold, and K∩M is compact O(M)-convex. We prove a Docquier–Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of K∪M, biholomorphic to domains in Cn with n=dim⁡X, such that M is mapped onto a closed complex submanifold of Cn.

Convexity with respect to weakly $q$-concave line bundles and reduction

We reconsider intermediate holomorphic convexity introduced by Barlet and Silva in the more general case of convexity with respect to hermitian holomorphic line bundles whose weight functions are (weakly) $q$-convex. Also we give completeness and Kahler properties of the canonical reduction.

On the fundamental group of a smooth projective surface with a finite group of automorphisms

In this article we prove new results on fundamental groups for some classes of fibered smooth projective algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of the surfaces under study upto finite index. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori’s well-known question on fundamental groups and free abelianness of second homotopy groups for these surfaces. We also prove a theorem that bounds the multiplicity of the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms $G$ in terms of the multiplicities of the induced fibration on $X/G$. If $X/G$ is a $\mathbb{P}^1$-fibration, we show that the multiplicity actually divides $|G|$. This theorem on multiplicity, which is of independent interest, plays an important role in our theorems.

A proof of the Muir–Suffridge conjecture for convex maps of the unit ball in $${\mathbb {C}}^n$$ C n

We prove (and improve) the Muir–Suffridge conjecture for holomorphic convex maps. Namely, let \(F:{\mathbb {B}}^n\rightarrow {\mathbb {C}}^n\) be a univalent map from the unit ball whose image D is convex. Let \({\mathcal {S}}\subset \partial {\mathbb {B}}^n\) be the set of points \(\xi \) such that \(\lim _{z\rightarrow \xi }\Vert F(z)\Vert =\infty \). Then we prove that \({\mathcal {S}}\) is either empty, or contains one or two points and F extends as a homeomorphism \(\tilde{F}:\overline{{\mathbb {B}}^n}{\setminus } {\mathcal {S}}\rightarrow \overline{D}\). Moreover, \({\mathcal {S}}=\emptyset \) if D is bounded, \({\mathcal {S}}\) has one point if D has one connected component at \(\infty \) and \({\mathcal {S}}\) has two points if D has two connected components at \(\infty \) and, up to composition with an automorphism of the ball and renormalization, F is an extension of the strip map in the plane to higher dimension.

Szegő kernels and Poincaré series

Let \(M = {{\widetilde M} \mathord{\left/ {\vphantom {{\widetilde M} \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma }\) be a Kahler manifold, where Γ ~ π1(M) and \(\widetilde M\) is the universal Kahler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L2 Szegő projector \({\widetilde \Pi _N}\) for L2-holomorphic sections on the lifted bundle \({\widetilde L^N}\) is related to the Szegő projector for H0(M, LN) by \({\widehat \Pi _N}\left( {x,y} \right) = \sum\nolimits_{\gamma \in \Gamma } {{{\widetilde {\widehat \Pi }}_N}} \left( {\gamma \cdot x,y} \right)\). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of \(\widetilde M\) with respect to \({\widetilde L^N}\) and to surjectivity of Poincare series.

Grauert’s line bundle convexity, reduction and Riemann domains

We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H 0(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space P n , where n is the complex dimension of X and L is the pull-back of O(1).

Exponential estimate for the asymptotics of Bergman kernels

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. Further, we give Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.

Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

AbstractWe extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

FUNDAMENTAL GROUP OF SOME GENUS-2 FIBRATIONS AND APPLICATIONS

We will prove that given a genus-2 fibration f : X → C on a smooth projective surface X such that b1(X) = b1(C) + 2, the fundamental group of X is almost isomorphic to π1(C) × π1(E), where E is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces X with genus-2 fibration X → C such that b1(X) > b1(C).

On the fundamental group of hyperelliptic fibrations and some applications

We prove that for a hyperelliptic fibration on a surface of general type with irreducible fibers over a (possibly) non-complete curve, the image of the fundamental group of a general fiber in the fundamental group of the surface is finite. Examples show that the result is optimal. As a corollary of this result we prove two conjectures; the Shafarevich conjecture on holomorphic convexity for the universal cover of these surfaces, and a conjecture of Nori on the finiteness of the fundamental groups of some surfaces. We also prove a striking general result about the multiplicities of multiple fibers of a hyperelliptic fibration on a smooth, projective surface.

Holomorphic convexity and Carleman approximation by entire functions on Stein manifolds

We give necessary and sufficient conditions for totally real sets in Stein manifolds to admit Carleman approximation of class $${\mathcal C^k}$$ , k ≥ 1, by entire functions.

Stochastic Calculus of Variations on complex line bundle and construction of unitarizing measures for the Poincaré disk

Holomorphic representation of Lie algebra can be realized through Kählerian symplectic formalism; underlying holomorphic convexity requires then the introduction of elliptic operators with complex coefficients. We construct the Stochastic Calculus of Variations for those elliptic operators; remote past vanishing of projections of the underlying process implies convergence in law; then limit laws lead to the unitarizing measure of the given representation; this general approach is developed in full details on Poincaré disk.

Stein compacts in Levi-flat hypersurfaces

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface $M$ and holomorphic convexity of compact sets in $M$, or bounded in part by $M$. Applications include extendability of Cauchy-Riemann functions, solvability of the $\overline {\partial }_b$-equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the $\overline {\partial }$-Neumann operator.

Holomorphic Generation of Continuous Inverse Algebras

AbstractWe study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixedn-tuple of elements are dense. In particular, we characterize the compact subsets of ℂnwhich appear as joint spectra of suchn-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.