Let $\pi:Z\rightarrow Y$ be a smooth projective morphism of complex manifolds with connected rationally connected fibers. We prove holomorphic approximation on arbitrary compact sets and finite-jet interpolation on arbitrary closed discrete sets for continuous liftings defined on open Riemann surfaces and holomorphic n...
We introduce the local approximation propert (LAP) and prove that it characterizes Oka manifolds. We prove that LAP yields convex approximation and, in an $n$-dimensional target, propagates across sets of zero $(2n-1)$-dimensional Hausdorff measure. Consequently, strong dominability characterizes Oka manifolds, convex...
Si-Cheng An, Yun Du, Bin Guo et al.· 4 citations· ⚡1
We prove that a holomorphic family of connected complex Lie groups has Oka total space if and only if its base is Oka. Extension presentations and generating sections give Oka results for stable and parabolic bundle moduli and for holomorphic mapping and embedding spaces. Every nonempty degree component of maps from a...
We give the first known examples of Oka Calabi--Yau manifolds in any dimension: every smooth hypersurface of multidegree $(2,\ldots,2)$ in a product of projective lines is Oka. Building on this example, we prove that in every local Kuranishi family of K3 surfaces the points with Oka fibers form a dense $G_\delta$ subse...
Starting from Alp\"oge's torus fibration with exactly three special fibres on a complex six-sphere, we construct a locally miniversal family of compact complex threefolds. Every member is Oka, and the projection over each relatively compact parameter disc is an Oka map. Deleting or simultaneously blowing up finitely ma...
Let $D\subset\mathbb{P}^n$, $n\geqslant2$, be an arbitrary cubic hypersurface, and let $D_{\mathrm{red}}$ denote its reduced support. We prove that $\mathbb{P}^n\setminus D$ is holomorphically elliptic, and hence Oka, unless $D_{\mathrm{red}}$ is the union of three distinct hyperplanes containing a common codimension-t...
For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty....
Yun Du, Song-Yan Xie· 0 citations
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