  Assume that $M$ is a compact connected unitary 2n-dimensional manifold and
admits a non-trivial circle action preserving the given complex structure. If
the first Chern class of $M$ equals to $k_0x$ for a certain 2nd integral
cohomology class $x$ with $|k_0|\geq n + 2$, and its first integral cohomology
group is zero, this short paper shows that the Todd genus and $A_k$-genus of
$M$ vanish.
