Suppose we have a cyclic cover of smooth complex surfaces. I'm wondering if there is a nice description, in terms of known bundles on the base, for the pushforward of the bundle of symmetric differentials on the cover.

I mainly care about the simplest possible circumstance, a double cover. So let $Y$ be a smooth complex surface with a smooth divisor $D$ and line bundle $\mathcal{L}$ such that $D \in |2\mathcal{L}|$. Let $X$ be the double cover of $Y$ branched over $D$, with finite morphism $\pi: X \rightarrow Y$.

Esnault and Viehweg show that $\pi_* \Omega_X = \Omega_Y \oplus (\Omega_Y(log D)\otimes \mathcal{L}^{-1})$. I'm wondering if we can find a nice description of $\pi_* S^m \Omega_X$?

Let $R \subset X$ be the ramification curve. We know that \begin{equation*} \pi_*S^m \Omega_X \hookrightarrow \pi_* S^m \Omega_X(log R) = S^m \Omega_Y(log D) \otimes (\mathcal{O}_Y \oplus \mathcal{L}^{-1}). \end{equation*} We need to find which local sections are actually contained in $\pi_* S^m\Omega_X$.

Even for $m = 2$ it seems that difficulties start to arise because $\pi_* S^2 \Omega_X \cap S^2 \Omega_Y(log D) \not= S^2 \Omega_Y$ because it contains elements like $b (db/b)^2$, where ${b = 0}$ is a local equation for the branch locus. But perhaps there is a nice filtration for this bundle?