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

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