K\"ahlerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures

We prove several K\"ahlerness criteria for compact Hermitian surfaces under semi-definiteness assumptions on natural Ricci curvatures of the Strominger-Bismut connection. The key tools for proving these results are explicit identities relating these Ricci curvatures to the torsion of the Strominger-Bismut connection, together with corresponding Chern number identities on compact Hermitian surfaces. The results may be viewed as Strominger-Bismut analogues and reformulations of Yang's K\"ahlerness criteria for compact complex surfaces.