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

Let $X$ be a Fano threefold and $\mathbb{C}^{\ast} \times X \to X$ an algebraic action. Fix a maximal compact subgroup $S^1$ of $\mathbb{C}^{\ast}$. Then $X$ has a $S^1$-invariant Kähler 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 $\mathbb{C}^{\ast}$-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 $\mathbb{C}^{\ast}$-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 $\mathbb{C}^{\ast}$-actions.

Keywords

Hamiltonian action, moment map, Morse theory, holomorphic Lefschetz formula, equivariant localization

2010 Mathematics Subject Classification

14J45, 32M05, 53C55, 53D20, 57R20

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Received 20 February 2016

Accepted 19 July 2017

Published 13 August 2021