Semistability and finite maps

Abstract

Let \({f : Y \longrightarrow M}\) be a surjective holomorphic map between compact connected Kahler manifolds such that each fiber of f is a finite subset of Y. Let ω be a Kahler form on M. Using a criterion of Demailly and Paun (Ann. Math. 159 (2004), 1247–1274) it follows that the form f*ω represents a Kahler class. Using this we prove that for any semistable sheaf \({E\, \longrightarrow\,M}\) , the pullback f*E is also semistable. Furthermore, f*E is shown to be polystable provided E is reflexive and polystable. These results remain valid for principal bundles on M and also for Higgs G-sheaves.