Stable Higgs Bundles on ruled surfaces

Let $\pi :X={\mathbb{P}}_C(E)⟶C$ 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 $c_1(V)= \pi^*(\bf d \rm)$forsomedivisor$\bf d \rm $ofdegree$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 $g(C)\ge 2$ and the base field is $\mathbb{C}$.