Line bundles on the moduli space of parabolic connections over a compact Riemann surface

TL;DR

This paper studies the geometry of moduli spaces of parabolic connections over compact Riemann surfaces, including their compactifications, divisor properties, Picard groups, and the algebraic functions on spaces of holomorphic connections.

Contribution

It introduces natural compactifications of moduli spaces, describes their divisor classes, computes their Picard groups, and proves the non-existence of non-constant algebraic functions on certain connection spaces.

Findings

Existence of natural compactifications by smooth divisors

Determination of the Picard group of the moduli spaces

Proof that the space of holomorphic connections admits no non-constant algebraic functions

Abstract

Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $\xi$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $\alpha$, (respectively, with the fixed determinant $\xi$) singular over the parabolic points $S \subset X$. Let $\mathcal{M}'_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}'_{pc}(r, \alpha, \xi)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ )parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces $\mathcal{M}'_{pc}(r, d, \alpha)$, and $\mathcal{M}'_{pc}(r, \alpha, \xi)$ by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces $\mathcal{M}_{pc}(r, d, \alpha)$, and $\mathcal{M}_{pc}(r, \alpha, \xi)$. Let $\mathcal{C}(L)$ denote the space of holomorphic connections on an ample line bundle $L$ over the moduli space $\mathcal{M}(r, d, \alpha)$ of parabolic bundles. We show that $\mathcal{C}(L)$ does not admit any non-constant algebraic function.