The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern–Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

中文翻译：

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几乎Hermitian几何中的可积定理和共形的Chern标量曲率度量

研究了几乎埃尔米特流形上的各种标量曲率，特别是关于保形变化。我们展示了几个可积性定理，这些定理指出其中两个只能在Kähler情况下一致。我们的主要问题是存在几乎具有一致的Chern标量曲率的Kähler度量。对于规则流形和在补充情况下完全解决了该问题，在这种情况下，来自Chern-Yamabe问题的方法适用于不可积情况。还给出了该问题的矩图解释，从而得出了Futaki不变式和几何不变式理论的常用图像。