Let $M$ be an $m$-dimensional compact complex manifold and $\Omega$ a K\"ahler class of $M$.
Assume that $M$ admits an $\Omega$-preserving $0$-pseudofree $S^1$-action and that $\Omega$ contains a K\"ahler metric of constant scalar curvature.
Then using the fixed point formula for the Bando-Calabi-Futaki character obtained in [5], we can obtain information on the fixed point data of the $S^1$-action.
Our main result is Theorem 2.
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