Duality between Bott-Chern and Aeppli Cohomology on Non-Compact Complex Manifolds

TL;DR

This paper establishes duality theorems connecting Bott-Chern and Aeppli cohomology on non-compact complex manifolds, extending Serre duality under pseudoconvexity and providing counterexamples when assumptions are not met.

Contribution

It proves duality theorems for Bott-Chern and Aeppli cohomology on non-compact manifolds with pseudoconvexity, extending classical dualities and highlighting limitations without these conditions.

Findings

Duality theorems hold under pseudoconvexity assumptions.

Full Bott-Chern-Aeppli duality extends Serre duality on Stein manifolds.

Counterexamples show duality fails without pseudoconvexity.

Abstract

In this paper we establish duality theorems relating Bott-Chern and Aeppli cohomology, both with and without compact support, on non-compact complex manifolds under suitable pseudoconvexity assumptions. In particular, on Stein manifolds we obtain a full Bott-Chern-Aeppli duality extending Serre duality for Dolbeault cohomology. We also show that these results fail in general without pseudoconvexity assumptions by constructing explicit counterexamples on non-compact complex surfaces.