Line bundles over a moduli space of logarithmic connections on a Riemann surface

Abstract

We consider logarithmic connections, on rank n and degree d vector bundles over a compact Riemann surface X, singular over a fixed point x 0 ∈ X with residue in the center of gl(n, C); the integers n and d are assumed to be mutually coprime. A necessary and sufficient condition is given for a vector bundle to admit such a logarithmic connection. We also compute the Picard group of the moduli space of all such logarithmic connections. Let N D (L) denote the moduli space of all such logarithmic connections, with the underlying vector bundle being of fixed determinant L, and inducing a fixed logarithmic connection on the determinant line L. Let N D (L) ⊂ N D (L) be the Zariski open dense subset parametrizing all connections such that the underlying vector bundle is stable. The space of all global sections of certain line bundles on N D (L) are computed. In particular, there are no nonconstant algebraic functions on N D (L). Therefore, there are no nonconstant algebraic functions on N D (L), although N D (L)is biholomorphic to a representation space which admits nonconstant algebraic functions. The moduli space N D (L) admits a natural compactification by a smooth divisor. We investigate numerically effectiveness of this divisor at infinity. It turns out that the divisor is not numerically effective in general.

A Torelli type theorem for the moduli space of rank two connections on a curve

“…Let ω denote the first Chern class of the ample generator of Pic(N X ) ∼ = Z. Since the homomorphism 2 …”

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

“…Now, observe that C(Θ) = P(At(Θ)) \ P(T U(n, L 0 )). Now we are in the same situation as in [9,p.797,Theorem 4.3], and the same techniques will work in our case too.…”

“…Similarly, for any line bundle L over M ′ lc (n, L), we have L = p * Θl , for some l ∈ Z, where p is the morphism defined in (3.3) andΘ is the generalised theta line bundle over U(n, L). Then we have a natural generalisation of [9,p.797,Theorem 4.3], and the same ideas can be used to prove the following.…”

On the moduli spaces of framed logarithmic connections on a Riemann surface

“…It is known that there is no non-constant global algebraic function on the moduli space of logarithmic connections with central residues on a curve of genus at least 3 [8]. On the other hand, the character variety, which is a moduli space of representations of a fundamental group, is affine.…”