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

Let X be a compact Riemann surface of genus g ≥ 3 {g≥ 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 ) {MHod(X,S,ξ)} of logarithmic λ-connections singular over S ⊂ X {S⊂ X} of rank n and degree d with fixed residues in the center of 𝔤 ⁢ 𝔩 ⁢ ( n , ℂ ) {gl(n,C)} , where n and d are mutually coprime. We determine the Picard group and investigate the automorphism group of ℳ Hod ⁢ ( X , S , ξ ) {MHod(X,S,ξ)} . Let ℳ Hod ′ ⁢ ( X , S , ξ ) ⊂ ℳ Hod ⁢ ( X , S , ξ ) {M^{}Hod(X,S,ξ)Hod(X,S,ξ)} 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 ) {M^{}Hod(X,S,ξ)}