Holonomy and the Ricci curvature of complex Hermitian manifolds

We prove two results on geometric consequences of the representation of restricted holonomy group of a Hermitian connection. The first result concerns when such a Hermitian manifold is K\"ahler in terms of the torsion and the irreducibility of the holonomy action. As a consequence we obtain a criterion of when a Hermitian manifold (and connection) is a generalized Calabi-Yau (in the sense that the Chern Ricci vanishes or equivalently that the restricted holonomy is inside $\mathsf{SU}(m)$). The second result concerns when a compact K\"ahler manifold with a generic restricted holonomy group is projective.