This is an announcement of our result showing that the Bando-Calabi-Futaki character for an integral Kähler class, if any, of a compact complex manifold is regarded as an obstruction to semistability of the manifold. This develops a pioneering work of Fujiki .
Toshiki Mabuchi. Yasuhiro Nakagawa. "An obstruction to semistability of manifolds." Proc. Japan Acad. Ser. A Math. Sci. 77 (4) 47 - 49, April 2001. https://doi.org/10.3792/pjaa.77.47