Several results for time-like Killing vector fields on a compact indefinite Kahler surface are introduced. For example, under certain assumptions, the existence of an indefinite Kahler metric and a time-like Killing vector field implies that the underlying complex surface is biholomorphic to a Hirzebruch surface F_d. Moreover, for any indefinite Kahler metric g on a Hirzebruch surface F_d, every time-like Killing vector field associated with an S~1-symmetry on (F_(d,g)) is induced from the standard C~*-action along the fibers of F_d = P(O(d) ? O) → P~1. Furthermore, several remarks related to the existence of indefinite Kahler metrics are given.
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