Weakly complex Einstein-Finsler vector bundle

Abstract

Let (E, F) be a complex Finsler vector bundle over a compact Kahler manifold (M, g) with Kahler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kahler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E1, F1) ⨁ · · · ⨁ (Ek; Fk); where F j := F |E j , and each (E j , F j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).