  In this paper we establish partial structure results on the geometry of
compact Hermitian manifolds of semipositive Griffiths curvature. We show that
after appropriate arbitrary small deformation of the initial metric, the null
spaces of the Chern-Ricci two-form generate a holomorphic, integrable
distribution. This distribution induces an isometric, holomorphic, almost free
action of a complex Lie group on the universal cover of the manifold. Our proof
combines the strong maximum principle for the Hermitian Curvature Flow (HCF),
new results on the interplay of the HCF and the torsion-twisted connection, and
observations on the geometry of the torsion-twisted connection on a general
Hermitian manifold.
