No 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.

No abstract

Let X be a compact Riemann surface, fix integers n and d with n positive, and let ∆ be a parabolic datum of rank n on X (see Section 3 below). Denote by U X (n, d, ∆) the moduli space of parabolic vector bundles of rank n and degree d, which are parabolic semistable with respect to ∆. Fix a holomorphic line bundle L of degree d on X, and let SU X (n, L, ∆) be the subvariety of U X (n, d, ∆) consisting of vector bundles with determinant isomorphic to L. We make the following two hypotheses on the parameters n, d and ∆.Assumption 1.1 Every parabolic vector bundle of rank n and degree d on X which is parabolic semistable with respect to the parabolic datum ∆ is in fact parabolic stable.Assumption 1.2 There exists a universal parabolic bundle (or briefly, a universal bundle) on U X (n, d, ∆) × X.Recall that a universal bundle on U X (n, d, ∆) × X is a vector bundle U on U X (n, d, ∆) × X together with a flag of subbundles j *x U for each x ∈ J (J being the set of parabolic points), where j x : U X (n, d, ∆)→U X (n, d, ∆)×X is the map E → (E, x), such that for each E ∈ U X (n, d, ∆), the restriction of U to {E}×X is parabolically isomorphic to E. We use the same symbols to denote the restrictions of U and U x,i etc. to SU X (n, L, ∆). Notation 1.3 • If S and T are topological spaces, and if α ∈ H * (S × T, Q), then σ(α) : H * (T, Q) → H * (S, Q) is the map σ(α)z = α/z, where / denotes the slant product.

No abstract