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# Methods and Applications of Analysis

## Volume 27 (2020)

### Number 3

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

Pages: 275 – 310

DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n3.a3

#### Authors

#### Abstract

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

Received 20 February 2016

Accepted 19 July 2017

Published 13 August 2021