On the Definition and Smooth Examples of Weakly Kähler--Finsler Metrics

Abstract: The generalization of the Kähler metric in the complex Finsler setting admits four natural versions. Among these, the strongly Kähler--Finsler condition is known to coincide with one of the intermediate conditions. In this paper, we establish the equivalence between the weakly Kähler--Finsler condition and the remaining intermediate condition. Moreover, we obtain new necessary and sufficient conditions for a weakly Kähler--Finsler metric to be strongly Kähler--Finsler. Using these criteria, we construct, for the first time, smooth examples of weakly Kähler--Finsler metrics which are not Kähler--Finsler, via the Kobayashi metrics on smooth bounded strongly convex domains.