On the Moduli space of $λ$-connections

Abstract

Let X be a compact Riemann surface of genus g ≥ 3. Let M Hod denote the moduli space of stable λ-connections over X and M ′ Hod ⊂ M Hod denote the subvariety whose underlying vector bundle is stable. Fix a line bundle L of degree zero. Let M Hod (L) denote the moduli space of stable λ-connections with fixed determinant L and M ′ Hod (L) ⊂ M Hod (L) be the subvariety whose underlying vector bundle is stable. We show that there is a natural compactification of M ′ Hod and M ′ Hod (L), and study their Picard groups. Let M Hod (L) denote the moduli space of polystable λ-connections. We investigate the nature of algebraic functions on M Hod (L) and M Hod (L). We also study the automorphism group of M ′ Hod (L).

Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties

“…, it has been shown that there are no nonconstant algebraic functions on M λ (X, r, O X ) [35,43]. Hence, σλ is a constant map, in other words, σ maps the fiber π −1 (λ) to another fiber π −1 (λ ′ ) for some λ ′ ∈ C. By Riemann-Hilbert correspondence, M dR (X, r, O X ) is biholomorphic to the affine variety M B (X, SL(2, C)), hence it does not contain any compact submanifold of positive dimension, however, by properness of Hitchin map [20,21,39], M Dol (X, r, O X ) contains many compact submanifolds of positive dimensions, namely, π −1 (λ) is not biholomorphic to π −1 (0) for any λ ∈ C * .…”