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In this paper, for compact K\"ahler manifolds with nef cotangent bundle, westudy the abundance conjecture and the associated Iitaka fibrations. We showthat, for a minimal compact K\"ahler manifold, the second Chern class vanishesif and only if the cotangent bundle is nef and the canonical bundle has thenumerical dimension $0$ or $1$. Additionally, in this case, we prove that thecanonical bundle is semi-ample. Furthermore, we give a relation between thevariation of the fibers of the Iitaka fibration and a certain semipositivity ofthe cotangent bundle.

Abundance theorem for minimal compact K\"ahler manifolds with vanishing second Chern class