On the moduli space of $\lambda$-connections

Abstract:

Let $X$ be a compact Riemann surface of genus $g \geq 3$. Let $\mathcal {M}_{Hod}$ denote the moduli space of stable $\lambda$-connections over $X$ and let $\mathcal {M}’_{Hod} \subset \mathcal {M}_{Hod}$ denote the subvariety whose underlying vector bundle is stable. Fix a line bundle $L$ of degree zero. Let $\mathcal {M}_{Hod}(L)$ denote the moduli space of stable $\lambda$-connections with fixed determinant $L$ and let $\mathcal {M}’_{Hod}(L) \subset \mathcal {M}_{Hod}(L)$ be the subvariety whose underlying vector bundle is stable. We show that there is a natural compactification of $\mathcal {M}’_{Hod}$ and $\mathcal {M}’_{Hod} (L)$ and study their Picard groups. Let $\mathbb {M}_{Hod}(L)$ denote the moduli space of polystable $\lambda$-connections. We investigate the nature of algebraic functions on $\mathcal {M}_{Hod}(L)$ and $\mathbb {M}_{Hod}(L)$. We also study the automorphism group of $\mathcal {M}’_{Hod}(L)$.