On the moduli space of 𝜆-connections

Source abstract

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