Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.
Scalar curvatures in almost Hermitian geometry and some applications
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abstract
On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we get some inequalities of various total scalar curvatures and some characterization results of the K\"{a}hler metric, balanced metric, locally conformally K\"{a}hler metric and the k-Gauduchon metric. As corollaries, we show some results related to a problem given by Lejmi-Upmeier \cite{LeU} and a conjecture given by Angella-Otal-Ugarte-Villacampa \cite{AOUV}.
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2019 1verdicts
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On Strominger K\"ahler-like manifolds with degenerate torsion
Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.