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A characterization of Oeljeklaus-Toma manifolds in locally conformally Kähler geometry

Shuho Kanda

TL;DR

The paper establishes a deep link between LCK solvmanifolds and OT manifolds by showing that, for a class of unimodular meta-abelian Lie groups, the existence of a lattice together with a left-invariant non-Vaisman LCK metric forces the group to originate from the OT construction via a number field $K$ and an admissible unit group $U$. It develops a two-pronged approach: (i) reconstructing OT data from lattice information using Mostow-type splitting to obtain a matrix $C$ and a field $K$, and (ii) classifying LCK structures on meta-abelian Lie algebras, yielding OT-like models when non-Vaisman. The results explain why number-theoretic data is indispensable in OT lattices and provide a partial converse linking Lie-theoretic realizations to OT manifolds, including a complete description in the meta-abelian non-Vaisman setting for small $m$. The work has implications for the classification of LCK solvmanifolds and for understanding the rigidity of OT-type geometries in relation to arithmetic data.

Abstract

We show that for a certain class of solvable Lie groups, if they admit a left-invariant non-Vaisman locally conformally Kähler metric and a lattice, they must arise from the construction of Oeljeklaus-Toma manifolds. This result provides a natural explanation for why number-theoretic considerations play a role in the construction of Oeljeklaus-Toma manifolds.

A characterization of Oeljeklaus-Toma manifolds in locally conformally Kähler geometry

TL;DR

The paper establishes a deep link between LCK solvmanifolds and OT manifolds by showing that, for a class of unimodular meta-abelian Lie groups, the existence of a lattice together with a left-invariant non-Vaisman LCK metric forces the group to originate from the OT construction via a number field and an admissible unit group . It develops a two-pronged approach: (i) reconstructing OT data from lattice information using Mostow-type splitting to obtain a matrix and a field , and (ii) classifying LCK structures on meta-abelian Lie algebras, yielding OT-like models when non-Vaisman. The results explain why number-theoretic data is indispensable in OT lattices and provide a partial converse linking Lie-theoretic realizations to OT manifolds, including a complete description in the meta-abelian non-Vaisman setting for small . The work has implications for the classification of LCK solvmanifolds and for understanding the rigidity of OT-type geometries in relation to arithmetic data.

Abstract

We show that for a certain class of solvable Lie groups, if they admit a left-invariant non-Vaisman locally conformally Kähler metric and a lattice, they must arise from the construction of Oeljeklaus-Toma manifolds. This result provides a natural explanation for why number-theoretic considerations play a role in the construction of Oeljeklaus-Toma manifolds.

Paper Structure

This paper contains 23 sections, 51 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Background
  4. Locally conformally Kähler geometry
  5. Solvmanifolds
  6. Oeljeklaus-Toma manifolds
  7. Results
  8. Strategy of the proof
  9. Organization of this paper
  10. Preliminaries
  11. Solvmanifolds and lattices
  12. Locally conformally Kähler structures on manifolds
  13. Locally conformally Kähler structures on Lie algebras
  14. Oeljeklaus-Toma manifolds
  15. Construction
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $X(K,U)$ be an OT manifold of type $(s,t)$. Take the basis $(a_i)_{i=1}^s$ of $U$ and a matrix $C=(c_{ij})_{ij} \in \mathop{\mathrm{Mat}}\nolimits_{t \times s}({\mathbb C})$ so that for all $1 \le i \le t$. Then, there is a lattice $\Gamma$ in $G_C$ such that $X(K,U) \simeq \Gamma \backslash G_C$.

Figures (1)

  • Figure 1:

Theorems & Definitions (97)

  • Definition 1.1
  • Theorem 1.2: Kas13b
  • Theorem 1.3: = Theorem \ref{['Thm:OT-like Lie group with simple lattice is constructed by OT']}
  • Theorem 1.4: = Corollary \ref{['Cor:LCK OT-like with lattice is OT']}
  • Theorem 1.5: = Theorem \ref{['Thm:classification of LCK meta-abelian']}
  • Theorem 1.6: = Theorem \ref{['Thm:classification when m=1,2']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 87 more