[Paper Review] Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds

Klaus-Dieter Kirchberg|ArXiv.org|May 23, 2006

Geometry and complex manifolds10 references3 citations

TL;DR

This paper establishes curvature conditions that force almost Hermitian and almost Kähler 4-manifolds to be Kähler. It proves that if the self-dual Weyl tensor satisfies |W₊|² = S²/6 and either δW₊ = 0 or ∇|W₊| = |∇W₊|, then the almost complex structure is integrable. The key result is that compact almost Kähler 4-manifolds satisfying |W₊|² = S²/6 and δW₊ = 0 are Kähler, providing a partial solution to the Goldberg conjecture for non-negative scalar curvature.

ABSTRACT

If $W_+$ denotes the self dual part of the Weyl tensor of any Kähler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost Kähler 4-manifold with $S \ge 0$, this condition forces the Kähler property. A compact almost Kähler 4-manifold is already Kähler if it satisfies the conditions $| W_+ |^2 = S^2/6$ and $δW_+=0$ and also if it is Einstein and $| W_+|$ is constant. Some further results of this type are proved. An almost Hermitian 4-manifold $(M,g,J)$ with $\mathrm{supp} (W_+)=M$ is already Kähler if it satisfies the condition $| W_+ |^2 = 3 (S_{\star} - S/3)^2 /8$ together with $| abla W_+ | = | abla |W_+||$ or with $δW_+ + abla \log | W_+ | \lrcorner W_+ =0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\star}$ only.

Motivation & Objective

• To identify curvature conditions on almost Hermitian 4-manifolds that imply integrability of the almost complex structure.
• To determine when almost Kähler 4-manifolds are necessarily Kähler, particularly under curvature constraints.
• To provide sufficient conditions for integrability using the self-dual Weyl tensor and scalar curvature.
• To contribute to the Goldberg conjecture by proving that compact almost Kähler Einstein 4-manifolds with constant |W₊| are Kähler.
• To generalize known Kähler curvature identities to sufficient conditions for the Kähler property in non-Kähler settings.

Proposed method

• Analyzes the self-dual part of the Weyl tensor W₊ and its norm |W₊|² in relation to scalar curvature S and star scalar curvature S⋆.
• Uses the identity |W₊|² = S²/6 as a key curvature condition known to hold on Kähler 4-manifolds.
• Applies differential identities involving δW₊ and ∇|W₊|, including δW₊ + ∇log|W₊|⌟W₊ = 0.
• Employs the obstruction Q(J) to the Kähler property in compact almost Hermitian manifolds.
• Utilizes conformal equivalence arguments and curvature identities on M₊ (where W₊ ≠ 0).
• Applies integration and Bochner-type identities to show ∇J = 0 under curvature and topological constraints.

Experimental results

Research questions

• RQ1 Under what curvature conditions is an almost Hermitian 4-manifold with supp(W₊) = M necessarily Kähler?
• RQ2 Can the condition |W₊|² = S²/6 together with δW₊ = 0 or ∇|W₊| = |∇W₊| imply integrability of J?
• RQ3 Does a compact almost Kähler 4-manifold satisfying |W₊|² = S²/6 and δW₊ = 0 have to be Kähler?
• RQ4 Is the Goldberg conjecture true for compact almost Kähler Einstein 4-manifolds with constant |W₊|?
• RQ5 What curvature conditions force a compact almost Kähler 4-manifold with constant negative scalar curvature to be Kähler?

Key findings

• An almost Hermitian 4-manifold with supp(W₊) = M and |W₊|² = 3(S⋆ − S/3)²/8 is Kähler if |∇W₊| = |∇|W₊||.
• If an almost Hermitian 4-manifold satisfies |W₊|² = 3(S⋆ − S/3)²/8 and δW₊ + ∇log|W₊|⌟W₊ = 0, then it is Kähler.
• A compact almost Kähler 4-manifold is Kähler if |W₊|² = S²/6 and δW₊ = 0, as shown in Theorem 4.2.
• A compact almost Kähler Einstein 4-manifold is Kähler if and only if |W₊| is constant, as established in Theorem 4.4.
• For compact almost Kähler 4-manifolds with constant negative scalar curvature S, the conditions det(W₊) = S³/108 and δW₊ = 0 imply the Kähler property.
• The condition det(W₊) = S|W₊|²/18 characterizes Kähler 4-manifolds of constant scalar curvature, and under Einstein or harmonic curvature assumptions, it implies the Kähler property.