On the integral homology and counterexamples to the Andreotti-Grauert conjecture

TL;DR

This paper provides a counterexample in complex geometry showing that certain cohomologically p-complete open sets in complex n-space are not p-complete, challenging the Andreotti-Grauert conjecture.

Contribution

It constructs explicit counterexamples demonstrating the failure of p-completeness despite cohomological p-completeness, refuting a longstanding conjecture.

Findings

Counterexamples for (n,p) with n≥3 and 2≤p≤n-1

Existence of open sets cohomologically p-complete but not p-complete

Non-vanishing of specific homology groups in these sets

Abstract

In this paper, we prove by means of a counterexample that there exist pair of integers (n,p) with $n\geq 3$, $2\leq p\leq n-1$, and open sets $D$ in $C^{n}$ which are cohomologically $p$-complete with respect to the structure sheaf of $D$ such that the cohomology group $H_{n+p}(D,Z)$ does not vanish. In particular $D$ is not $p$-complete.