A remark on the automorphisms of the moduli space of logarithmic λ-connections over a curve

Abstract

Abstract Let X be a compact Riemann surface of genus g ≥ 3 {g\geq 3} and let S be a finite subset of X. Let ξ be a fixed line bundle over X of degree d. We consider the moduli space ℳ Hod ⁢ ( X , S , ξ ) {\mathcal{M}_{\rm Hod}(X,S,\xi)} of logarithmic λ-connections singular over S ⊂ X {S\subset X} of rank n and degree d with fixed residues in the center of 𝔤 ⁢ 𝔩 ⁢ ( n , ℂ ) {\mathfrak{gl}(n,\mathbb{C})} , where n and d are mutually coprime. We determine the Picard group and investigate the automorphism group of ℳ Hod ⁢ ( X , S , ξ ) {\mathcal{M}_{\rm Hod}(X,S,\xi)} . Let ℳ Hod ′ ⁢ ( X , S , ξ ) ⊂ ℳ Hod ⁢ ( X , S , ξ ) {\mathcal{M}^{\prime}_{\rm Hod}(X,S,\xi)\subset\mathcal{M}_{\rm Hod}(X,S,\xi)} be the open subvariety consisting of those logarithmic λ-connections whose underlying vector bundle is stable. We show that there is a natural compactification of the moduli space ℳ Hod ′ ⁢ ( X , S , ξ ) {\mathcal{M}^{\prime}_{\rm Hod}(X,S,\xi)} . We also recall the logarithmic version of Riemann–Hilbert correspondence which arises from Deligne’s extension theorem, and using this we prove the Torelli-type theorem for ℳ Hod ⁢ ( X , S , ξ ) {\mathcal{M}_{\rm Hod}(X,S,\xi)} .