I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3.

Warning: the holonomy is computed with respect to the Chern connection of the hermitian metric, which is, since $(X,\omega)$ is not necessarily Kähler, not equal in general to the Levi-Civita connection of the underlying riemannian metric!

Have a nice reading!

I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3.

Warning (in view of the comment below by S.S.): the holonomy is computed with respect to the Chern connection of the hermitian metric, which is, since $(X,\omega)$ is not necessarily Kähler, not equal in general to the Levi-Civita connection of the underlying riemannian metric!

Have a nice reading!

I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4).

Have a nice reading!

I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3.

Warning: the holonomy is computed with respect to the Chern connection of the hermitian metric, which is, since $(X,\omega)$ is not necessarily Kähler, not equal in general to the Levi-Civita connection of the underlying riemannian metric!

Have a nice reading!