On Fano threefolds with semi-free ${\mathbb C}^*$-actions, I

Abstract

Let $X$ be a Fano threefold and $\C ^* \times X\rightarrow X$ an algebraic action. Then $X$ has a $S^1$-invariant K\"ahler 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 ${\mathbb C}^*$-actions using 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 ${\mathbb C}^*$-action. For the actions whose fixed point sets have only two connected components, and in a few other cases, we give the realizations of the semi-free $\C^*$-actions.