Holonomy group scheme of an integral curve

Abstract

Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that X_k̅ is normal, math formula being the algebraic closure of k. We define a notion of strong semistability for vector bundles on Y. We show that a vector bundle on Y is strongly semistable if and only if its pull back to X is strongly semistable and hence it is a tensor category. In case dim Y = 1, we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme G_Y of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G-bundle E_G, we construct a holonomy group scheme. We show that if Y is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on Y is the Zariski closure of the (topological) fundamental group of Y.