Time-like killing vector fields on compact indefinite Kahler surfaces

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) circle plus O) --> P-1. Furthermore, several remarks related to the existence of indefinite Kahler metrics are given.