Uniqueness of shrinking gradient Kähler–Ricci solitons on non‐compact toric manifolds

Abstract

We show that, up to biholomorphism, there is at most one complete 𝑇 𝑛 -invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold M.We also establish uniqueness without assuming 𝑇 𝑛invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra 𝔱 of 𝑇 𝑛 . As an application, we show that, up to isometry, the unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on ℂℙ 1 × ℂ is the standard product metric associated to the Fubini-Study metric on ℂℙ 1 and the Euclidean metric on ℂ.

Weighted K-stability for a class of non-compact toric fibrations

“…As such, given such an M we will sometimes denote the polyhedron (2.21) by P −K M . With this in mind, the following is an immediate consequence of the same arguments as in [22,14].…”

“…Then ω is a shrinking gradient Kähler-Ricci soliton with respect to the fundamental vector field W on M associated to b W . This is well-known, see [9,8,14,31], among others. Thus, any AK Kähler-Ricci soliton ω on a toric manifold M is a v-soliton with exponentially decaying weights with derivatives as in Definition 3.2.…”

“…In [14], it was observed that the boundary conditions classically known to be satisfied by the symplectic potential in the compact setting [2] also hold here: Proposition 2.12 ([14, Proposition 2.17], [2, Proposition 1]). Let µ ω ∶ M → t * be the moment map associated to an AK metric ω ∈ α, whose image is equal to Delzant polyhedron P by Lemma 2.8.…”

“…The soliton vector field W of a gradient Kähler-Ricci soliton always has the property that JW is Killing, and hence generates an action of a finite-dimensional real torus T on M . In fact, at least when the Ricci curvature of ω is bounded, the action always complexifies to an effective and holomorphic action of (C * ) k on M ([16, Theorem 5.1], [14,Theorem 4.2]). In this way, shrinking gradient Kähler-Ricci solitons are a natural starting place for the study of complete toric or "partially" toric geometry, as we will see below.…”

“…This, by definition, is a Kähler metric on M with respect to which the T-action is hamiltonian, and which admits a proper moment map µ ∶ M → P ⊂ t * = Lie(T) * , where P is now an unbounded polyhedron. Crucially, the image P of an AK metric depends only (up to translation in t * ) on the cohomology class ω ∈ α [14]. A new feature in this context is that many results depend intimately on the behavior of the weights (v, w) at infinity.…”