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

2015/08/05 by Qilin Yang, Yang, Qilin, Dan Zaffran +1
Mathematics · #14J45 #53D20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1508.01089

openalex publication_date 2015/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a Fano threefold and \C ^* × X→ X an algebraic action. Then X has a S¹-invariant Kähler structure and the corresponding S¹-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.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

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