Weakly complex Einstein-Finsler vector bundle
In [Manuscr. Math. 42, 245--257 (1983; Zbl 0558.53037)], \textit{M. Lübke} proved the following result: ``Lübke's theorem. Let \((E,h)\) be an Einstein-Hermitian vector bundle over a compact Kähler manifold \((M,g)\) with Kähler form \(\Phi\). Then \(E\) is \(\Phi\)-semistable and \((E,h)\) is a direct sum \((E,h) = (E_1,h_1)\oplus\cdots\oplus (E_k,h_k)\) of \(\Phi\)-stable Einstein-Hermitian vector bundles \((E_1,h_1),\cdots,(E_k,h_k)\) with the same factor \(c\) as \((E,h)\)''. In [Publ. Math. 51, No. 3--4, 363--384 (1997; Zbl 0905.53015)], \textit{T. Aikou} generalized Lübke's theorem to complex Einstein-Finsler vector bundles \((E,F)\) under the assumption that \((E,F)\) is modeled on a complex Minkowski space. In this paper, the authors prove the following main result: ``If \((E,F)\) is a weakly complex Einstein-Finsler vector bundle over a Hermitian manifold \((M,g)\) in the sense of Aikou, then \((E,F)\) is modeled on a complex Minkowski space''. Using the above result, the authors are able to generalize Lübke's theorem to the complex Einstein-Finsler case without the assumption of Aikou: ``Let \((E,F)\) be a complex Einstein-Finsler vector bundle over a compact Kähler manifold \((M,g)\) with Kähler form \(\Phi\). Then \(E\) is \(\Phi\)-semistable and \((E,F)\) is a direct sum \((E, F) = (E_1,F_1)\oplus\cdots\oplus(E_k,F_k)\), 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 \(\Phi\)-stable Einstein-Hermitian vector bundle with the same factor \(c\) as \((E,F)\)''. The paper is concluded with an example of a complex Einstein-Finsler vector bundle.
scientific article; zbMATH DE number 6898602
scientific article