Suppose a compact K"{a}hler manifold $(M,\omega)$ admits a smooth circle action $g_t,~t\in S^1$. So the pull back of the K"{a}hler form $g_t^{\ast}(\omega)$ is a nondegenerate two-form. Since the circle action is only smooth and may not preserve the complex structure, $g_t^{\ast}(\omega)$ may not be $(1,1)$-from. My question is if it is possible for some special manifolds or under some special cases $g_t^{\ast}(\omega)$ is $(1,1)$ for any $t$.