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 Kähler 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 first Ricci vanishes or equivalently that the restricted holonomy is inside \(\textsf{SU}(m)\) $\mathsf{SU}(m)$ ). The second result concerns when a compact Kähler manifold with a generic restricted holonomy group is projective, under some nonnegativity assumptions in terms of the Ricci and other curvatures.