Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Abstract

Let S be a finite subset of a compact connected Riemann surface X of genus g ≥ 2. Let M lc (n, d) denote the moduli space of pairs (E, D), where E is a holomorphic vector bundle over X and D is a logarithmic connection on E singular over S, with fixed residues in the centre of gl(n, C), where n and d are mutually corpime. Let L denote a fixed line bundle with a logarithmic connection D L singular over S. Let M ′ lc (n, d) and M lc (n, L) be the moduli spaces parametrising all pairs (E, D) such that underlying vector bundle E is stable andbe the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of M ′ lc (n, d) and M ′ lc (n, L) and compute their Picard groups. We also show that M ′ lc (n, L) and hence M lc (n, L) do not have any nonconstant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over S, with arbitrary residues.

On the Moduli space of $λ$-connections

“…Nevertheless, we give a different proof for the compactification of M ′ Hod , which is constructive in nature. The statement and the steps involved in the following Proposition (3.1) is exactly similar to the statement and proof of the Theorem 4.3 in [12]. The main difference is that the fibre of p defined in (3.1) is a finite dimensional vector space, while in [12], the fibre of p [see [12] equation(4.1)] is an affine space modelled over H 0 (X, Ω 1 X ⊗ End(E)).…”

“…The statement and the steps involved in the following Proposition (3.1) is exactly similar to the statement and proof of the Theorem 4.3 in [12]. The main difference is that the fibre of p defined in (3.1) is a finite dimensional vector space, while in [12], the fibre of p [see [12] equation(4.1)] is an affine space modelled over H 0 (X, Ω 1 X ⊗ End(E)). Nonetheless, we shall sketch the construction of algebraic vector bundle Q over U, because the morphisms (3.4) and (3.5) in proof of the following Proposition 3.1 are very crucial in proving Theorem (3.3).…”

“…The Picard group of moduli space of vector bundles have been studied extensively by several algebraic geometers, for instance [7], [11], [2] to name a few. Also, the Picard group and algebraic functions for the moduli space of logarithmic connections have been studied in [5] and [12]. So, the purpose of this article is to study the Picard group, algebraic function and automorphism group of the moduli spaces of λ-connections.…”

“…Define a map ι : of Picard groups which restricts any line bundle ξ over M Hod to M ′ Hod . Now, we state the following theorem whose first part (that is, p * : Pic(U) → Pic(M ′ Hod ) is an isomorphism) is similar to Theorem 4.4 in [12], but in different context, that is for moduli space of logarithmic connections. Proof.…”

Moduli space of rank one logarithmic connections over a compact Riemann surface

“…The moduli space of logarithmic connections over a complex projective variety singular over a smooth normal crossing divisor has been constructed in [33]. Several algebro-geometric invariants like Picard group, algebraic functions of the moduli space of holomorphic and logarithmic connections have been studied, see [7], [9], [10], [36] [27], and [26].…”