Hecke cycles associated to rank 2 twisted Higgs bundles on a curve

TL;DR

This paper constructs a geometric cycle related to rank 2 twisted Higgs bundles on a curve, using Hecke modifications to connect moduli spaces and rational maps, advancing the understanding of Higgs bundle geometry.

Contribution

It introduces a new cycle in the product space involving Higgs bundle moduli and Picard varieties via Hecke modifications, linking different moduli spaces.

Findings

Constructed a cycle in the product of a stack of rational maps and Picard variety.

Connected moduli spaces of twisted Higgs bundles with Hecke modifications.

Provides a new geometric tool for studying Higgs bundle moduli.

Abstract

Let $X$ be a smooth complex projective curve of genus $g$ and let $L$ be a line bundle on $X$ with $\mathrm{deg}\,L>0$. Let $\mathbf{M}$ be the moduli space of semistable rank 2 $L$-twisted Higgs bundles with trivial determinant on $X$. Let $\mathbf{M}_{X}$ be the moduli space of stable rank 2 $L$-twisted Higgs bundles with determinant $\mathcal{O}(-x)$ for some $x\in X$ on $X$. We construct a cycle in the product of a stack of rational maps from nonsingular curves to $\mathbf{M}_{X}$ and $\mathrm{Pic}^{2\mathrm{deg}\,L}(X)$ by using Hecke modifications of a stable $L$-twisted Higgs bundle in $\mathbf{M}$.