I'm new on the forum.
I've a strange question but it's puzzling me. Consider a compact projective Calabi-Yau $M$ of complex dimension $n$, and two finite ramified covers : $$\gamma : M \longrightarrow X \,\, and \,\, f: M \longrightarrow Y$$ respectively ramified over $\Delta$ and $D$. Suppose $X$ and $Y$ have ample tangent bundles. Can we deduce relations between the Chern classes of $\gamma^* TX$ and $f^* TY$ ?
In the particular case where $X = \mathbb{P}^n$ and $Y$ is a flag manifold, can $\gamma^* T\mathbb{P}^n$ and $f^* TY$ have no relation ?
Can express the Chern classes from those of $TM$ and $\gamma^* \mathcal{K}_{\mathbb{P}^n}$ (resp. $f^* \mathcal{K}_Y$)?
Thanks for reading,