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

Xiaojun Wu

TL;DR

This work establishes Bott–Chern–Aeppli duality on non-compact complex manifolds under pseudoconvexity: for strongly $r$-convex $X$ with finite Betti numbers and $p,q\ge r$, $H^{p,q}_{BC}(X)$ is canonically the topological dual of $H^{n-p,n-q}_{A,c}(X)$, and $H^{p,q}_{A}(X)$ is dual to $H^{n-p,n-q}_{BC,c}(X)$. In the special case $r=1$, duality holds for all degrees, and Stein manifolds enjoy full duality. The results rely on Andreotti–Grauert finiteness and Serre’s criterion for duality in Fréchet spaces, using hypercohomological descriptions of Bott–Chern and Aeppli cohomology. The paper also exhibits counterexamples showing that the duality can fail without pseudoconvexity, notably in dimension two, emphasizing the sharpness of the hypotheses. Overall, it sharpens Serre duality in the non-compact setting by linking refined cohomologies through functional-analytic dualities.

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.

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

TL;DR

This work establishes Bott–Chern–Aeppli duality on non-compact complex manifolds under pseudoconvexity: for strongly -convex with finite Betti numbers and , is canonically the topological dual of , and is dual to . In the special case , duality holds for all degrees, and Stein manifolds enjoy full duality. The results rely on Andreotti–Grauert finiteness and Serre’s criterion for duality in Fréchet spaces, using hypercohomological descriptions of Bott–Chern and Aeppli cohomology. The paper also exhibits counterexamples showing that the duality can fail without pseudoconvexity, notably in dimension two, emphasizing the sharpness of the hypotheses. Overall, it sharpens Serre duality in the non-compact setting by linking refined cohomologies through functional-analytic dualities.

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.
Paper Structure (6 sections, 8 theorems, 69 equations)

This paper contains 6 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Andreotti--Grauert finiteness, Serre’s criterion, and sheaf-theoretic descriptions
  3. Finiteness and duality results
  4. Counterexamples
  5. The complement of a real line
  6. The complement of a real plane

Key Result

Theorem 2.2

Let $\mathcal{F}$ be a coherent analytic sheaf over a strongly $q$-convex complex manifold $X$. Then $H^k(X,\mathcal{F})$ is Hausdorff and finite dimensional for all $k \ge q$.

Theorems & Definitions (17)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Theorem 2.2: agbook
  • Proposition 2.3: Ser55
  • Proposition 2.4: Ser55
  • Proposition 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 7 more