Abstract
Let \({f : Y \longrightarrow M}\) be a surjective holomorphic map between compact connected Kähler manifolds such that each fiber of f is a finite subset of Y. Let ω be a Kähler 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 Kähler 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.
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Biswas, I., Subramanian, S. Semistability and finite maps. Arch. Math. 93, 437–443 (2009). https://doi.org/10.1007/s00013-009-0059-7
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DOI: https://doi.org/10.1007/s00013-009-0059-7