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

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.