Holonomy Groups of Stable Vector Bundles

Abstract

We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan-Seshadri unitary representation of its restriction to curves.Next we relate the holonomy group to the minimal structure group and to the decomposition of tensor powers of F . Finally we illustrate the principle that either the holonomy is large or there is a clear geometric reason why it should be small. 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.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 NarasimhanSeshadri [NS65], it corresponds to a unique unitary representation ρ :If c 1 (E| C ) = 0, one gets a special type of unitary representation ρ : π 1 (C \ x) → U (E x ), see (12). We call these the NarasimhanSeshadri representation of E| C .

Monodromy for principal bundles

“…When X is a curve and G is semisimple, the monodromy group-scheme and the monodromy bundle coincide with those constructed in [4]. For a polystable vector bundle over a complex projective manifold, the monodromy group coincides with the holonomy group constructed in [2].…”

“…In [2], Balaji and Kollár constructed a group for any polystable vector bundle over a complex projective manifold, which in [2] is called the holonomy group. The construction of the holonomy group crucially uses the fact that a polystable vector bundle of degree zero over a compact Riemann surface admits a unique unitary flat connection.…”

On the Algebraic Holonomy of Stable Principal Bundles

“…The algebraic holonomy H x 0 (E G ) of E G constructed in [BK] is an algebraic subgroup of Ad(E G ) x 0 ; recall that Ad(E G ) x 0 is the group of automorphisms of (E G ) x 0 that commute with the action of G. Hence H x 0 (E G ) gives a conjugacy class of algebraic subgroups of G. Take any subgroup H x 0 ⊂ G in this conjugacy class. The principal G-bundle admits a minimal reductive reduction of structure group to this subgroup H x 0 [BK,p. 193,Theorem 20(3)].…”

“…We note that from a theorem of Aubin and Yau it follows that the holomorphic tangent bundle T 1,0 M of M is polystable provided K M is ample. If the anticanonical line bundle K −1 M is ample, Aut(X) < ∞, and the holomorphic tangent bundle T 1,0 M is polystable, the Question 48 of [BK,p. 209] asks whether the algebraic holonomy of T 1,0 M is GL(n, C).…”

“…In [BK,p. 187,Question 9] it is asked whether the algebraic holonomy of the holomorphic tangent bundle T 1,0 M coincides with the complexification of the differential geometric holonomy of M.…”