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A note on almost abelian groups with constant holomorphic sectional curvature

Yulu Li, Fangyang Zheng

TL;DR

This work investigates the constant holomorphic sectional curvature problem within Lie-Hermitian manifolds, focusing on compact quotients $M=G/\Gamma$ with left-invariant complex structure $J$ and metric $g$. It establishes the conjecture for two structural classes of the underlying Lie algebra: almost abelian and those containing a $J$-invariant abelian ideal of codimension $2$, showing that constant curvature $c$ forces flatness ($c=0$) with Chern-flat or Levi-Civita-flat outcomes (the Levi-Civita case requires unimodularity). The Chern-flat results hold without unimodularity, while the Levi-Civita results rely on unimodularity; a non-unimodular counterexample with negative curvature illustrates limitations in the non-Kähler setting. Together, these results advance the understanding of the conjecture by validating it in broad, nontrivial Lie-homogeneous contexts and clarifying the roles of torsion and unimodularity in determining flatness.

Abstract

A long-standing conjecture in non-Kähler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant $c$, then the metric must be Kähler when $c\neq 0$ and must be Chern (or Levi-Civita) flat when $c=0$. The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon, Sato-Sekigawa, and Apostolov-Davidov-Muskarov in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov, for locally conformally Kähler manifolds (when $c\leq 0$) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients $G/Γ$ where $G$ is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and $Γ$ is a discrete subgroup. We confirm the conjecture when the Lie algebra ${\mathfrak g}$ of $G$ either is almost abelian, or contains a $J$-invariant abelian ideal of codimension 2.

A note on almost abelian groups with constant holomorphic sectional curvature

TL;DR

This work investigates the constant holomorphic sectional curvature problem within Lie-Hermitian manifolds, focusing on compact quotients with left-invariant complex structure and metric . It establishes the conjecture for two structural classes of the underlying Lie algebra: almost abelian and those containing a -invariant abelian ideal of codimension , showing that constant curvature forces flatness () with Chern-flat or Levi-Civita-flat outcomes (the Levi-Civita case requires unimodularity). The Chern-flat results hold without unimodularity, while the Levi-Civita results rely on unimodularity; a non-unimodular counterexample with negative curvature illustrates limitations in the non-Kähler setting. Together, these results advance the understanding of the conjecture by validating it in broad, nontrivial Lie-homogeneous contexts and clarifying the roles of torsion and unimodularity in determining flatness.

Abstract

A long-standing conjecture in non-Kähler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant , then the metric must be Kähler when and must be Chern (or Levi-Civita) flat when . The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon, Sato-Sekigawa, and Apostolov-Davidov-Muskarov in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov, for locally conformally Kähler manifolds (when ) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients where is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and is a discrete subgroup. We confirm the conjecture when the Lie algebra of either is almost abelian, or contains a -invariant abelian ideal of codimension 2.

Paper Structure

This paper contains 4 sections, 12 theorems, 52 equations.

Table of Contents

  1. Introduction and statement of result
  2. Preliminaries
  3. The almost abelian case
  4. Lie algebra with $J$-invariant abelian ideal of codimension 2

Key Result

Theorem 1

Let $(M^n,g)$ be a compact Hermitian manifold with universal cover $(G,J,g)$, where $G$ is a Lie group and both $J$ and $g$ are left-invariant. Assume that the Chern (or Levi-Civita) holomorphic sectional curvature of $g$ is a constant $c$, and ${\mathfrak a}$ is an abelian ideal in the Lie algebra

Theorems & Definitions (19)

  • Conjecture 1: Constant Holomorphic Sectional Conjecture
  • Conjecture 2
  • Theorem 1
  • Lemma 1: YZ
  • Lemma 2
  • Proposition 2
  • proof
  • Lemma 3: LZ
  • Lemma 4: LZ
  • Lemma 5: GZ
  • ...and 9 more