Hitchin fibration under ramified coverings (Q6161093)

The main goal of this paper is to study the behavior of the Hitchin fibration under a degree two ramified covering \(p:Y\longrightarrow X\) between compact Riemann surfaces \(X\) and \(Y\). The first objective is to establish a dictionary that allows the authors to express elementary transformations in terms of the Beauville-Narasimhan-Ramanan (BNR) correspondence. As a consequence, they present a study of the variation of the Hitchin fibration with respect to a degree two ramified covering. Their main motivation is the application of these techniques to explicit examples. Given the degree two map \(p\), the authors derive a Higgs bundle from \(X\) to \(Y\), the lifted Higgs bundle tends to have many apparent singularities, then they perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between abelian varieties. Among the objectives of this paper is to describe this map. This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with basic tools. Section 3 is devoted to parabolic Higgs bundles and Hitchin fibration. Section 4 is devoted to spectral curves under degree two ramified coverings. Section 5 consider the case of a rational map between moduli spaces of parabolic Higgs bundles. The author finish the paper by applying the result to a degree two morphism \(p:Y\longrightarrow \mathbb{P}^1\) where the genus of \(Y\) belongs to \(\{1, 2\}\).