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On a class of almost Hermitian 4-manifolds

Ethan Addison, Tedi Draghici, Mehdi Lejmi

TL;DR

This paper investigates 4-dimensional almost Hermitian manifolds under Gray's first curvature condition $(G_1)$, defining the class $\\mathcal{AH}_1$ and using Sekigawa's integral identity to obtain a global, globalizable characterization from weaker hypotheses. It develops a detailed $U(2)$-curvature decomposition, analyzes the canonical Chern form through $\,\gamma=\rho^*+\Phi$, and derives componentwise differential Bianchi identities tailored to 4D AH-manifolds to enable precise local classifications. The authors prove that, for compact manifolds, $J$-invariant Ricci tensors together with vanishing $\int ρ^*\wedgeΦ$ place the manifold in $\\mathcal{AH}_1$, with equality cases linking to Kähler geometry; they then pursue a classification program, establishing strong local constraints and a complete Lie-algebra picture. Their main result in the Lie-algebra setting is the uniqueness of a non-Kähler, left-invariant $\,\mathcal{AH}_1$-structure on the flat algebra $\\mathcal{A}_{3,6}\\oplus\\mathcal{A}_1$, while all other 4D Hermitian (or unimodular) realizations are forced to be Kähler. Together, these findings illuminate the geometry near the Kähler boundary in dimension four and identify a canonical non-Kähler model among 4D Lie algebras.

Abstract

Using an integral identity proved by Sekigawa \cite{Sek87} on compact almost Hermitian 4-manifolds, we naturally obtain a global characterization of the class $\mathcal{AH}_1$ of almost Hermitian 4-manifolds satisfying the first Gray curvature condition from apparently weaker conditions. Then we take steps towards a classification of almost Hermitian 4-manifolds of class $\mathcal{AH}_1$, including proving a uniqueness result on 4-dimensional Lie algebras.

On a class of almost Hermitian 4-manifolds

TL;DR

This paper investigates 4-dimensional almost Hermitian manifolds under Gray's first curvature condition , defining the class and using Sekigawa's integral identity to obtain a global, globalizable characterization from weaker hypotheses. It develops a detailed -curvature decomposition, analyzes the canonical Chern form through , and derives componentwise differential Bianchi identities tailored to 4D AH-manifolds to enable precise local classifications. The authors prove that, for compact manifolds, -invariant Ricci tensors together with vanishing place the manifold in , with equality cases linking to Kähler geometry; they then pursue a classification program, establishing strong local constraints and a complete Lie-algebra picture. Their main result in the Lie-algebra setting is the uniqueness of a non-Kähler, left-invariant -structure on the flat algebra , while all other 4D Hermitian (or unimodular) realizations are forced to be Kähler. Together, these findings illuminate the geometry near the Kähler boundary in dimension four and identify a canonical non-Kähler model among 4D Lie algebras.

Abstract

Using an integral identity proved by Sekigawa \cite{Sek87} on compact almost Hermitian 4-manifolds, we naturally obtain a global characterization of the class of almost Hermitian 4-manifolds satisfying the first Gray curvature condition from apparently weaker conditions. Then we take steps towards a classification of almost Hermitian 4-manifolds of class , including proving a uniqueness result on 4-dimensional Lie algebras.

Paper Structure

This paper contains 9 sections, 19 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $U(2)$-decomposition of curvature and the condition $(G_1)$.
  4. The canonical Chern form and the form $\Phi$
  5. A global characterization of compact $\mathcal{AH}_1$ 4-manifolds
  6. The differential Bianchi identity for almost Hermitian 4-manifolds
  7. Towards a classification of $\mathcal{AH}_1$ 4-manifolds
  8. $\mathcal{H}_1$-surfaces
  9. A unique non-Kähler 4-dimensional $\mathcal{AH}_1$ Lie algebra

Key Result

Theorem 1.1

Let $(M^4, g, J, \omega)$ be a compact Einstein almost Hermitian 4-manifold. Then the following inequality is satisfied with equality if and only if the manifold is Kähler-Einstein, or if the metric is Ricci flat and anti-self-dual ($W^+ = 0$). In the second case, $(M^4, g)$ is the 4-torus (or a quotient of the torus) with a flat metric, or a K3 surface (or a quotient of a K3 surface) with a Ricc

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • ...and 31 more