Cohomologies of unipotent harmonic bundles over noncompact curves

Cohomologies of unipotent harmonic bundles over quasi-projective varieties I: The case of noncompact curves

Let S be a compact Riemann surface (holomorphic curve) of genus g. Let p1, p2, · · · , ps be s > 0 points on it; these points define a divisor, and we denote the open Riemann surface S \ {p1, . . . , ps} by S. When 3g− 3+ s > 0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points p1, p2, · · · , ps then become cusps at infinity. Even in the remain...

Cohomologies of Harmonic Bundles on Quasi-Compact Kähler manifolds

If M is compact Kählerian and ρ is semi-simple, that is that the Zariski closure of the image of ρ in GL(n,C) is semi-simple 1, by means of a result of Donaldson (in the case of Riemann surfaces, [5]) and Corlette (in the higher dimensional case, [4]), there exists a unique harmonic metric u on V, equivalently, a ρ-equivariant harmonic map from the universal covering M̃ of M into Pn := GL(n,C)/U...

Vector bundles over algebraic curves

Orthogonal Bundles over Curves in Characteristic Two

Let X be a smooth projective curve of genus g ≥ 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend’s conjecture on the canonical reduction of principal G-bundles for G = SO(n) with n ≥ 7. Let X be a smooth projective curve of genus g ≥ 2 and let G be a connected reduc...

Holomorphic Connections on Filtered Bundles over Curves

Let X be a compact connected Riemann surface and EP a holomorphic principal P–bundle over X , where P is a parabolic subgroup of a complex reductive affine algebraic group G. If the Levi bundle associated to EP admits a holomorphic connection, and the reduction EP ⊂ EP × P G is rigid, we prove that EP admits a holomorphic connection. As an immediate consequence, we obtain a sufficient condition...