ON FANO THREEFOLDS WITH SEMI-FREE C*-ACTIONS, I

Let X be a Fano threefold and C* x X -> X an algebraic action. Fix a maximal compact subgroup S-1 of C*. Then X has a S-1-invariant Kahler structure and the corresponding S-1-action admits an equivariant moment map which is at the same time a perfect Bott-Morse function. We will initiate a program to classify the Fano threefolds with semi-free C*-actions using the Morse theory and the holomorphic Lefschetz fixed point formula as the main tools. In this paper we give a complete list of all possible Fano threefolds without "interior isolated fixed points" for any semi-free C*-action. Moreover when the actions whose fixed point sets have only two connected components and a few of the rest cases, we give the realizations of the semi-free C*-actions.