Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds

TL;DR

This paper investigates conditions under which almost Hermitian and almost K"ahler 4-manifolds are actually K"ahler, providing specific curvature and divergence criteria that guarantee integrability of the complex structure.

Contribution

It establishes new curvature and divergence conditions that ensure almost Hermitian and almost K"ahler 4-manifolds are K"ahler, extending known results and providing explicit criteria involving the Weyl tensor.

Findings

Almost K"ahler 4-manifolds with nonnegative scalar curvature are K"ahler if they satisfy a specific Weyl tensor relation.

Compact almost K"ahler 4-manifolds are K"ahler under certain curvature and divergence conditions.

Certain almost Hermitian 4-manifolds are K"ahler if they satisfy relations involving the Weyl tensor, scalar curvature, and star scalar curvature.

Abstract

If $W_+$ denotes the self dual part of the Weyl tensor of any K\"ahler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost K\"ahler 4-manifold with $S \ge 0$, this condition forces the K\"ahler property. A compact almost K\"ahler 4-manifold is already K\"ahler if it satisfies the conditions $| W_+ |^2 = S^2/6$ and $\delta W_+=0$ 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 K\"ahler if it satisfies the condition $| W_+ |^2 = 3 (S_{\star} - S/3)^2 /8$ together with $|\nabla W_+ | = | \nabla |W_+||$ or with $\delta W_+ + \nabla \log | W_+ | \lrcorner W_+ =0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\star}$ only.