Classification results for expanding and shrinking gradient Kähler-Ricci solitons

Abstract

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 $GL(2,\mathbb{C})$, either the flat Gaussian shrinking soliton on ${\mathbb{C}}^2$ or the $U(2)$-invariant shrinking gradient Kähler-Ricci soliton of Feldman-Ilmanen-Knopf on the blowup of ${\mathbb{C}}^2$ at one point. Finally, we show that up to pullback by an element of $GL(n,\mathbb{C})$, the only complete shrinking gradient Kähler-Ricci soliton with bounded Ricci curvature on ${\mathbb{C}}^n$ 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-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.

66 pages; Appendix B incorporated into statement of Theorem B, statement and proof of Theorems B and C refined, topology on Lie groups in proof of Theorem D clarified, curvature assumption in items (1) and (2) of Theorem E weakened to bounded Ricci curvature, two additional open problems added to Section 7.2, Section A.6 added