We show that, up to biholomorphism, that 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 n -invariance 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.

We show that the underlying complex manifold of a complete non-compact twodimensional shrinking gradient Kähler-Ricci soliton (M, g, X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg → 0 is biholomorphic to either C × P 1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.(i) M is biholomorphic to either C × P 1 or to Bl p (C × P 1 ), that is, the blowup of C × P 1 at a fixed point p of the standard torus action on C × P 1 . (ii) There exists a biholomorphism γ : M → M such that γ −1 * (JX) lies in the Lie algebra of the real torus T acting on these spaces in the standard way and γ * g is T-invariant. (iii) γ −1 * (JX) is determined and its flow generates a holomorphic isometric S 1 -action of (M, J, γ * g). (iv) Assuming existence, γ * g is the unique T-invariant complete shrinking gradient Kähler-Ricci soliton on M .Conclusions (ii)-(iv) for M = C × P 1 have already been established in [Cif20] where it is shown that any complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on this manifold is isometric to the Cartesian product of the flat Gaussian soliton ω C on C and twice the Fubini-Study metric ω P 1 on P 1 . The new possibility arising is when M is the blowup of C × P 1 at one point, in which case γ −1 * (JX) is given by (2.16). In light of this, we make the following conjecture. Conjecture 1.1. There exists a complete shrinking gradient Kähler-Ricci soliton ω on Bl p (C × P 1 ), that is, the blowup of C × P 1 at a fixed point p of the standard torus action on C × P 1 , invariant under the real torus action induced by the standard real torus action on C × P 1 , with bounded scalar

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.

We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili [35] depending on weight functions (v, w), on certain non-compact semisimple toric fibrations. The latter notion is a generalization of the Calabi Ansatz originally defined by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman [2]. The this setup turns out to reduce the weighted cscK problem on the total space to a different weighted cscK problem on a fixed toric fiber M . We show that the natural analog of the weighted Futaki invariant of [35] can under reasonable assumptions be interpreted on an unbounded polyhedron P ⊂ R n associated to M . In particular, we fix a certain class W of weights (v, w), and prove that if M admits a weighted cscK metric, then P is K-stable, and we give examples of weights on C 2 for which the weighted Futaki invariant vanishes but do not admit (v, w)-cscK metrics. Following [34], we introduce a weighted Mabuchi energy Mv,w and show that the existence of a (v, w)-cscK metric implies that it Mv,w proper. The well-definedness of Mv,w in this setting also allows us to prove a uniqueness result using the method of [29]. We apply the theory in a few special cases and make connections with asymptotic geometry. In particular, we show that weighted K-stability of the abstract fiber C is sufficient for the existence of weighted cscK metrics on the total space of line bundles L → B over a compact Kähler base, extending the result in [35] in the P 1 -bundles case. The right choice of weights corresponds to the (shrinking) Kähler-Ricci soliton equation, and we give an interpretation of the asyptotic geometry in this case.