Remarks on Chern-Einstein Hermitian metrics (Q2193065)

The authors study certain special Hermitian metrics \(\omega\) on compact complex (mostly non-Kähler) manifolds, focusing on the geometry of the Chern connection. In particular, the results concern the Chern-Einstein problem(s), that is search for Hermitian metrics whose Chern-Ricci forms are a (non-necessarily constant) multiple of the metric itself. Various interesting theorems are proven, one of which is the following: There are four-dimensional solvable Lie groups endowed with invariant complex structures admitting strong-second-Chern-Einstein metrics with negative Einstein factors.