Methods and Applications of Analysis

Volume 27 (2020)

Number 3

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

Pages: 275 – 310



Qilin Yang (Department of Mathematics, Sun Yat-Sen University, Guangzhou, China)

Dan Zaffran (Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Fl., U.S.A.)


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.


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

2010 Mathematics Subject Classification

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

Received 20 February 2016

Accepted 19 July 2017

Published 13 August 2021