Related papers: Oka tubes in holomorphic line bundles

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact K\"ahler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang…

Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano…

Let $\,G/K\,$ be a non-compact irreducible Hermitian symmetric space of rank $\,r\,$ and let $\,NAK\,$ be an Iwasawa decomposition of $\,G$. By the polydisc theorem, $\,AK/K\,$ can be regarded as the base of an $\,r$-dimensional tube domain…

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete K\"ahler metric, or are hyperconvex but have no nonconstant holomorphic functions. For…

Let $(X,g)$ be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure $J$ compatible with $g$, then the canonical bundle $K_X$ is not pseudo-effective and the Kodaira dimension…

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a…

Noncommutative K\"ahler structures were recently introduced by the second author as a framework for studying noncommutative K\"ahler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector…

We prove that if $E$ is a compact subset of the unit disk ${\mathbb D}$ in the complex plane, if $E$ contains a sequence of distinct points $a_n\not= 0$ for $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and for all $n$ we have $ |a_{n+1}|…

Hyperholomorphic bundle is a bundle with connection defined over a hyperkaehler manifold such that this connection is holomorphic with respect to all complex structures induced by a hyperkaehler structure. A hyperholomorphic connection is…

Using $L^2$-methods for the $\bar\partial$-equation we prove that the Ohsawa-Takegoshi extension theorem also holds for holomorphic sections of a vector bundle, over compact K\"ahler manifolds. We then proceed to show that the conditions…

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational…

In an earlier article the second author introduced three families of tube domains in ${\mathbf C}^2$ with holomorphic automorphism group isomorphic to ${\mathbf R}\ltimes{\mathbf R}^2$ and envelope of holomorphy equal to ${\mathbf C}^2$. In…

Consider a domain $\varOmega$ in $\mathbb{C}^n$ with $n\geqslant 2$ and a compact subset $K\subset\varOmega$ such that $\varOmega\backslash K$ is connected. We address the problem whether a holomorphic line bundle defined on…

We study holomorphic geometric structures on non-K\"ahler compact complex manifolds with trivial canonical line bundle. For Vaisman Calabi-Yau manifolds we prove that all holomorphic geometric structures of affine type on them are locally…

In this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used in the sequel to show that the algebraic dimension of OT-manifolds is zero. In the…

We prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the…

A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle $K_X$ of a Kobayashi hyperbolic compact complex manifold $X$ is ample. In this note we prove that $K_X$ is ample if $X$ is projective and satisfies a stronger…

Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat K\"ahler metric on $D$. Then $h$ is…

We study the homological mirror symmetry statement where A-side is the conic bundle Hori--Vafa mirror $\mathcal{Y} = \{uv = f(z)\} \subset \mathbb{C}^2 \times (\mathbb{C}^\ast)^n$ for a Laurent polynomial $f$ in $(\mathbb{C}^\ast)^n$, and…

Let $A$ be a commutative Banach algebra. Let $M$ be a complex manifold on $A$ (an $A$-manifold). Then, we define an $A$-holomorphic vector bundle $(\wedge^kT^*)(M)$ on $M$. For an open set $U$ of $M$, $\omega$ is said to be an…