Semiprojectivity and very stability in moduli of symplectic and orthogonal parabolic Higgs bundles

TL;DR

This paper proves the semiprojectivity of moduli spaces of semistable symplectic or orthogonal parabolic Higgs bundles on a Riemann surface and characterizes very stability via the properness of the Hitchin morphism.

Contribution

It establishes the semiprojectivity of these moduli spaces and provides a criterion for very stability based on the properness of the Hitchin morphism.

Findings

Semiprojectivity of moduli spaces proven.

Characterization of very stability via Hitchin morphism properness.

Criterion applies to both symplectic and orthogonal cases.

Abstract

Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. We prove the semiprojectivity of the moduli space of semistable symplectic or orthogonal parabolic Higgs bundles over $X$. We show that a stable symplectic parabolic bundle $E$ on $X$ is strongly very stable, meaning $E$ does not have any nonzero strongly parabolic nilpotent Higgs field, if and only if the symplectic parabolic Hitchin morphism induced on the affine space $$H^0(X,\mathrm{SPEnd}_\mathrm{Sp}(E) \otimes K(D))$$ is a proper morphism, where $\mathrm{SPEnd}_\mathrm{Sp}(E)$ denotes the set of symplectic strongly parabolic endomorphisms of $E$. We remark that the same criterion for very stability applies to the orthogonal case.