Stable Higgs Bundles on ruled surfaces

Let $π: X = \mathbb{P}_C(E) \longrightarrow C$ 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 $π$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,θ)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= π^*(\bf d \rm)$ for some divisor $\bf d \rm$ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,ψ)$ of degree $d$ on $C$ whose pullback under $π$ is isomorphic to $(V,θ)$. 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 $\mathbb{C}$.

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