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Almost abelian complex nilmanifolds

Adrián Andrada, Romina M. Arroyo, María L. Barberis, Sönke Rollenske, Konstantin Wehler

TL;DR

This work identifies and exploits a rigidity phenomenon for complex structures on nilpotent almost abelian Lie algebras: whenever such a structure exists, it is unique up to isomorphism. Building on this, the authors establish that these manifolds admit a stable torus bundle filtration, allowing them to realize every almost abelian complex nilmanifold as an iterated holomorphic torus bundle and to compute cohomology via left-invariant forms. They prove the Dolbeault cohomology, de Rham cohomology, and Frölicher spectral sequence are governed by the Lie algebra data, with the spectral sequence degenerating at $E_1$, and show that small deformations are unobstructed and remain complex nilmanifolds. The paper also provides explicit, representation-theoretic methods to compute Betti and Hodge numbers across the almost abelian family, including detailed results for the 2-step nilpotent case, highlighting the practical impact for understanding a broad class of complex nilmanifolds.

Abstract

We show that a complex structure on a nilpotent almost abelian real Lie algebra is unique if it exists. As a consequence, we get full control over the cohomology and deformations of almost abelian complex nilmanifolds.

Almost abelian complex nilmanifolds

TL;DR

This work identifies and exploits a rigidity phenomenon for complex structures on nilpotent almost abelian Lie algebras: whenever such a structure exists, it is unique up to isomorphism. Building on this, the authors establish that these manifolds admit a stable torus bundle filtration, allowing them to realize every almost abelian complex nilmanifold as an iterated holomorphic torus bundle and to compute cohomology via left-invariant forms. They prove the Dolbeault cohomology, de Rham cohomology, and Frölicher spectral sequence are governed by the Lie algebra data, with the spectral sequence degenerating at , and show that small deformations are unobstructed and remain complex nilmanifolds. The paper also provides explicit, representation-theoretic methods to compute Betti and Hodge numbers across the almost abelian family, including detailed results for the 2-step nilpotent case, highlighting the practical impact for understanding a broad class of complex nilmanifolds.

Abstract

We show that a complex structure on a nilpotent almost abelian real Lie algebra is unique if it exists. As a consequence, we get full control over the cohomology and deformations of almost abelian complex nilmanifolds.

Paper Structure

This paper contains 15 sections, 17 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Complex nilmanifolds
  4. Lie algebra Dolbeault cohomology
  5. Fibrations and torus bundle series
  6. Remarks on the Jordan normal form
  7. Almost abelian complex nilmanifolds
  8. Almost abelian Lie algebras
  9. Nilpotent almost abelian Lie algebras with complex structure
  10. Proof of Theorem \ref{['thm: stbs']} $(i)$ and $(ii)$
  11. Cohomology, Frölicher spectral sequence, and deformations
  12. Consequences of the Hochschild-Serre spectral sequence
  13. Proof of Theorem \ref{['thm: deform']}
  14. Explicit computations
  15. The 2-step nilpotent case

Key Result

Theorem 1.1

Let $X = (\Gamma\backslash G, J)$ be an almost abelian complex nilmanifold with associated Lie algebra $\mathfrak{g}$. Then the following hold:

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.6
  • Remark 3.1
  • Theorem 3.2: ABDGH24
  • Remark 3.5
  • Remark 3.6
  • ...and 31 more