Classification results for expanding and shrinking gradient Kähler–Ricci solitons

Conlon, Ronan J¹ ; Deruelle, Alix² ; Sun, Song³

¹ Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, United States
² Université Paris-Saclay, Laboratoire de mathématiques d’Orsay, Orsay, France
³ Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China, Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States

Geometry & topology, Tome 28 (2024) no. 1, pp. 267-351.

Résumé

We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler–Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler–Ricci solitons with quadratic curvature decay with derivatives. We then show that any two-dimensional complete shrinking gradient Kähler–Ricci soliton whose scalar curvature tends to zero at infinity is, up to pullback by an element of $\mathrm{GL}(2,ℂ)$, either the flat Gaussian shrinking soliton on ${ℂ}^2$ or the $U(2)$–invariant shrinking gradient Kähler–Ricci soliton of Feldman, Ilmanen and Knopf on the blowup of ${ℂ}^2$ at one point. Finally, we show that up to pullback by an element of $\mathrm{GL}(n,ℂ)$, the only complete shrinking gradient Kähler–Ricci soliton with bounded Ricci curvature on ${ℂ}^n$ is the flat Gaussian shrinking soliton, and on the total space of $\mathcal{𝒪}(-k)\to {\mathbb{P}}^{n-1}$ for $0<k<n$ is the $U(n)$–invariant example of Feldman, Ilmanen and Knopf. In the course of the proof, we establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler–Ricci soliton with bounded Ricci curvature in the Lie algebra of a torus. A key tool used to achieve this result is the Duistermaat–Heckman theorem from symplectic geometry. This provides the first step towards understanding the relationship between complete shrinking gradient Kähler–Ricci solitons and algebraic geometry.

Détail

DOI : 10.2140/gt.2024.28.267

Keywords: Kähler–Ricci solitons, Kähler–Ricci flow, self-similar solutions, Duistermaat–Heckman formula

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