Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry

A. Latorre, L. Ugarte

TL;DR

This work advances the understanding of invariant complex structures on nilmanifolds by completing the classification of 8-dimensional nilpotent Lie algebras that admit non-nilpotent, weakly non-nilpotent ($\text{WnN}$) complex structures, building on the known strongly non-nilpotent ($\text{SnN}$) cases. It provides an explicit constructive procedure to derive $8$-dimensional WnN structures from SnN data via central extensions, followed by a rigorous reduction and equivalence analysis of the complex structure equations, yielding a finite, complete list (Theorem) and a corresponding real-algebra classification with eight $(f_1,\dots,f_8)$ types. The authors then classify which of these algebras admit invariant pseudo-Kähler metrics, discovering an infinite family of counterexamples to a prior conjecture, and construct new neutral Calabi–Yau metrics that are Ricci-flat but not hypersymplectic. They also prove a topological restriction $b_1(X)\ge 3$ for pseudo-Kähler nilmanifolds with invariant complex structure in complex dimension $\le 4$, highlighting structural obstructions in this setting. Overall, the paper provides a detailed algebraic and geometric map from $\text{WnN}$ $J$ on 8D NLAs to pseudo-Kähler geometry and neutral Calabi–Yau metrics on the associated nilmanifolds, with implications for deformation stability and topology.

Abstract

We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi-Yau structures. Moreover, we arrive at the topological restriction $b_1(X)\geq 3$ for every pseudo-Kähler nilmanifold $X$ with invariant complex structure, up to complex dimension four.

Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry

TL;DR

This work advances the understanding of invariant complex structures on nilmanifolds by completing the classification of 8-dimensional nilpotent Lie algebras that admit non-nilpotent, weakly non-nilpotent () complex structures, building on the known strongly non-nilpotent () cases. It provides an explicit constructive procedure to derive -dimensional WnN structures from SnN data via central extensions, followed by a rigorous reduction and equivalence analysis of the complex structure equations, yielding a finite, complete list (Theorem) and a corresponding real-algebra classification with eight types. The authors then classify which of these algebras admit invariant pseudo-Kähler metrics, discovering an infinite family of counterexamples to a prior conjecture, and construct new neutral Calabi–Yau metrics that are Ricci-flat but not hypersymplectic. They also prove a topological restriction for pseudo-Kähler nilmanifolds with invariant complex structure in complex dimension , highlighting structural obstructions in this setting. Overall, the paper provides a detailed algebraic and geometric map from on 8D NLAs to pseudo-Kähler geometry and neutral Calabi–Yau metrics on the associated nilmanifolds, with implications for deformation stability and topology.

Abstract

We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi-Yau structures. Moreover, we arrive at the topological restriction for every pseudo-Kähler nilmanifold with invariant complex structure, up to complex dimension four.
Paper Structure (10 sections, 22 theorems, 97 equations, 2 tables)

This paper contains 10 sections, 22 theorems, 97 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Complex structures on nilpotent Lie algebras
  3. Different types of complex structures on NLAs
  4. Equivalence of quasi-nilpotent complex structures
  5. Classification of weakly non-nilpotent complex structures in real dimension 8
  6. Construction of WnN from SnN
  7. Reduction of the complex structure equations
  8. Classification of complex structures up to equivalence
  9. Classification of 8-dimensional nilpotent Lie algebras with WnN complex structures
  10. Pseudo-Kähler and neutral Calabi-Yau structures

Key Result

Lemma 2.3

CFGU-proceeding Let $J$ be a quasi-nilpotent complex structure on ${\mathfrak{g}}$. Consider a non-trivial term ${\mathfrak{a}}_q(J)$ of the ascending $J$-compatible series of ${\mathfrak{g}}$. Then, the ascending $\tilde{J}_q$-compatible series $\{{\mathfrak{a}}_l(\tilde{J}_q)\}_l$ of the quotient

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 37 more