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On $q$-complete and $q$-concave with corners complex manifolds

Youssef Alaoui

TL;DR

This work addresses the nuanced relation between $q$-complete and cohomologically $q$-complete spaces, especially in the presence of corners. It extends finiteness results to $q$-concave with corners and provides an explicit counterexample to the Andreotti–Grauert conjecture, demonstrating that cohomological $q$-completeness does not imply $q$-completeness in general. Furthermore, it shows that for every $(n,q)$ with $2\le q\le n-1$ there exist $q$-complete with corners domains $D\subset \mathbb{P}^n$ that fail to be cohomologically $\hat{q}$-complete, with $\hat{q}=n-\left\lfloor\frac{n-1}{q}\right\rfloor$. Collectively, these results clarify the boundaries of cohomological finiteness for complex spaces with corners and provide constructive methods to control high-degree cohomology on projective-domain complements.

Abstract

It is proved that if there exists a positive and continuous function $f$ on an $n$-dimensional complex manifold $X$, $q$-convex with corners outside a compact set $K\subset X$ and which exhausts $X$ from below, then $dim_{\mathbb{C}}H^{p}(X,{\mathcal{F}})<+\infty$ for any coherent analytic sheaf ${\mathcal{F}}$ on $X$ if $p<n-q$. It is known from the theory of Andreotti and Grauert that if a complex space $X$ is $q$-complete, then $X$ is cohomoloogically $q$-complete. Until now it is not known in general if these two conditions are equivalent. The aim of section $4$ of this article is to provide a counterexample to the conjecture posed by Andreotti and Grauert ~\cite{ref2} to show that a cohomologically $q$-complete space is not necessarily $q$-complete. In section $5$ of this article, we will prove that there exist for each pair of integers $(n,q)$ with $2\leq q\leq n-1$ a $q$-complete with corners open subset $D$ of $\mathbb{P}^{n}$ and $\mathcal{F}\in coh(\mathbb{P}^{n})$ such that $D$ is not cohomologically $\hat{q}$-complete with respect to ${\mathcal{F}}$. Here $\hat{q}=n-[\frac{n-1}{q}]$, where $[x]$ denotes the integral part of $x$.

On $q$-complete and $q$-concave with corners complex manifolds

TL;DR

This work addresses the nuanced relation between -complete and cohomologically -complete spaces, especially in the presence of corners. It extends finiteness results to -concave with corners and provides an explicit counterexample to the Andreotti–Grauert conjecture, demonstrating that cohomological -completeness does not imply -completeness in general. Furthermore, it shows that for every with there exist -complete with corners domains that fail to be cohomologically -complete, with . Collectively, these results clarify the boundaries of cohomological finiteness for complex spaces with corners and provide constructive methods to control high-degree cohomology on projective-domain complements.

Abstract

It is proved that if there exists a positive and continuous function on an -dimensional complex manifold , -convex with corners outside a compact set and which exhausts from below, then for any coherent analytic sheaf on if . It is known from the theory of Andreotti and Grauert that if a complex space is -complete, then is cohomoloogically -complete. Until now it is not known in general if these two conditions are equivalent. The aim of section of this article is to provide a counterexample to the conjecture posed by Andreotti and Grauert ~\cite{ref2} to show that a cohomologically -complete space is not necessarily -complete. In section of this article, we will prove that there exist for each pair of integers with a -complete with corners open subset of and such that is not cohomologically -complete with respect to . Here , where denotes the integral part of .

Paper Structure

This paper contains 5 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-concavity with corners
  4. A counterexample to the Andreotti-Grauert conjecture
  5. $q$-convexity with corners

Key Result

Lemma 1

Let $X$ be a complex manifold of dimension $n$, and let $\phi: X\rightarrow \mathbb{R}$ be a smooth $q$-convex function $\phi$ on $X$. Let $\xi_{0}\in X$ and $X'_{c}=\{x\in X: \phi(x)>c\},$ where $c=\phi(\xi_{0})$. Then for any coherent analytic sheaf ${\mathcal{F}}$ on $X$ the restriction map is bijective if $p\leq n-q-1$, injective if $p=n-q.$

Theorems & Definitions (13)

  • Lemma 1
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 3
  • ...and 3 more