Stable Higgs Bundles on ruled surfaces

Let $\pi : X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $\pi$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,\theta)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= \pi^*(\bf d \rm)$ for some divisor $\bf d \rm$ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,\psi)$ of degree $d$ on $C$ whose pullback under $\pi$ is isomorphic to $(V,\theta)$. As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on $X$ whenever $g(C)\geq 2$ and the base field is $\mathbb{C}$.