Example of stable bundle whose pullback is polystable

Question

Kempf (1992): "Pulling back bundles" has the following theorem:

Let $f: Y \rightarrow X$ be a finite morphism. If $\mathscr{W}$ is a bundle on $X$ that is stable with respect to an ample divisor $D$, then $f^* \mathscr{W}$ is poly-stable on $Y$ with respect to $f^{-1}D$.

Huybrechts, Lehn (2010): "The Geometry of Moduli Spaces of Sheaves", Lemma 3.2.3, says that the pullback of a polystable bundle is polystable.

My questions are:

1. What is an example of a finite morphism $f: Y \rightarrow X$ and a stable bundle $\mathscr{W}$ on $X$, such that $f^*\mathscr{W}$ is not stable?

2. In Donaldson, Kronheimer (1990): "The Geometry of Four-Manifolds", Lemma 9.1.9, it is proven that the pullback of the tangent bundle from $\mathbb{P}^2$ to a double cover branched over a sextic is stable. I am not sure, but that proof seems to suggest that the pullback of any stable $SL(2,\mathbb{C})$ bundle to a branched double cover is stable, not just poly-stable. In which situations is it true that the pullback of a stable bundle is stable, not just poly-stable?

Answer 1

An example with $\deg(f)=2$: let $f :\tilde{C}\rightarrow C $ be an étale double covering of curves, and let $\sigma $ be the corresponding involution. Let $L$ be a line bundle on $\tilde{C} $ which is not the pull back of a line bundle on $C$. Then the rank 2 bundle $E:=f_*L$ on $C$ is stable, but $f ^*E\cong L\oplus \sigma ^*L$.