Somebody tell me that:
For a bundle(maybe polystable) over algebraic manifold, take a symmetric power of the bundle and tensor with its determinant line bundle to some power. Assume that the resulting bundle has zero first chern class.Then if it has sections, the structure group reduces to smaller group.
Who can tell me however to prove it or any reference? Thank you!
The correct version of this statement is the following. Let $B$ be a stable vector bundle with structure group $G$, and $TB$ a tensor component in the bundle of all tensors over $B$ (that is, a sub-bundle determined by a symmetric group representation) with trivial determinant. Then holomorphic sections of $TB$ are $G$-invariant, and (unless structure group of $B$ can be reduced to a smaller subgroup) there are no other holomorphic sections.
For slope stability, this result follows from Donaldson-Uhlenbeck-Yau theorem, because the structure group is holonomy group of the Yang-Mills connection. Originally this notion was conceived by Bogomolov, and called "Bogomolov stability"; see e.g. Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective varieties. Math. USSR Izvestija 13/3 (1979), 499-555.
