On q-Runge domains

TL;DR

This paper constructs specific complex domains demonstrating nuanced properties related to q-Runge conditions and cohomology density, revealing intricate relationships between domain geometry and sheaf cohomology.

Contribution

It generalizes previous examples by constructing domains in complex spaces that are not (q-1)-Runge but still exhibit dense cohomology restriction maps, for various dimensions and q.

Findings

Existence of domains not (q-1)-Runge with dense cohomology restriction maps.

Extension of Coltoiu's example to higher dimensions and different q-values.

Detailed conditions linking domain properties to cohomology behavior.

Abstract

In $[2]$, Coltoiu gave an example of a domain $D\subset\complexes^{6}$ which is 4-complete such that for every ${\mathcal{F}}\in Coh(\complexes^{6})$ the restriction map $H^{3}(\complexes^{6},{\mathcal{F}})\to H^{3}(D,{\mathcal{F}})$ has a dense image but $D$ is not 4-Runge in $\complexes^{6}$. Here, we prove that for every integers $n\geq 4$ and $1\leq q\leq n$ there exists a domain $D\subset \complexes^{n}$ which is not ($\tilde{q}-1$)-Runge in $\complexes^{n}$ but such that for any coherent analytic sheaf ${\mathcal{F}}$ on $\complexes^{n}$ the restriction map $H^{p}(\complexes^{n},{\mathcal{F}})\to H^{3}(D,{\mathcal{F}})$ has a dense image for all $p\geq \tilde{q}-2$ if $q$ does not divide $n$, where $\tilde{q}=n-[\frac{n}{q}]+1$ and $[\frac{n}{q}]$ denotes the integral part of $\frac{n}{q}$.