Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$. Is $c_k$ necessarily vertical, i.e. $$ c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots i_{k}} \, \omega^{i_1} \wedge \dots \wedge \omega^{i_k} $$ for $\omega^i$ a basis of $H^{2}(M, \mathbb{Z})$ and some suitable coefficients $\alpha_{i_1 \dots i_{k}}$?

If this is not true in general, does it hold for Kähler (or Calabi-Yau) manifolds?

compactKähler manifold $H^2$ is nonzero, because its symplectic structure gives a nontrivial class, but for noncompact Kähler manifolds it can be zero: for example, $\mathbb{C}^n$. $\endgroup$