Hitchin fibration under ramified coverings

TL;DR

This paper investigates how the Hitchin fibration behaves under ramified coverings of Riemann surfaces, focusing on the transformation of Higgs bundles and the induced maps between Abelian varieties.

Contribution

It describes the construction of a map between Hitchin fibrations under ramified coverings and analyzes its properties, including the preservation of fibrations and the induced Abelian variety maps.

Findings

The pullback of Higgs bundles under ramified coverings can be birationally transformed to remove apparent singularities.

The Hitchin fibration is preserved under this pullback and birational transformation.

A new map between Abelian varieties associated with the Hitchin fibers is explicitly described.

Abstract

We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $\pi$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we 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. The aim of this paper is to describe this map.