Holonomy Groups of Stable Vector Bundles

Let M be a Riemannian manifold and E a vector bundle with a connection ∇. Parallel transport along loops gives a representation of the loop group of M with base point x into the orthogonal group O(E x) of the fiber at x (see, for instance, [KN96, Bry00]). If X is a complex manifold and E a holomorphic vector bundle, then usually there are no holomorphic connections on E. One can, nonetheless, define a close analog of the holonomy representation in the complex setting if E is a stable vector bundle and X is projective algebraic. Assume for simplicity that c 1 (E) = 0, a condition that we remove later. By Mehta–Ramanathan [MR82], if x ∈ C ⊂ X is a sufficiently general complex curve, then E| C is also stable and so by a result of Narasimhan–Seshadri [NS65] it corresponds to a unique unitary representation ρ : π 1 (C) → U (E x). We call it the Narasimhan–Seshadri representation of E| C. The image of the representation, and even the Hermitian form on E x implicit in its definition, depend on the choice of C, but the picture stabilizes if we look at the Zariski closure of the image in GL(E x). The resulting group can also be characterized in different ways. Theorem 1. Let X be a smooth projective variety, H an ample divisor on X, E a stable vector bundle with det E ∼ = O X and x ∈ X a point. Then there is a unique reductive subgroup H x (E) ⊂ SL(E x), called the holonomy group of E, characterized by either of the two properties: (1) H x (E) ⊂ SL(E x) is the smallest algebraic subgroup satisfying the following: For every curve x ∈ C ⊂ X such that E| C is stable, the image of the Narasimhan–Seshadri representation ρ : π 1 (C) → U (E x) is contained in H x (E). (2) If C is sufficiently general, then the image of the Narasimhan–Seshadri representation is Zariski dense in H x (E).