Abstract：

In recent years, it has been shown that various problems in Kähler geometry can be unified under the framework of the weighted cscK equation, introduced by Lahdili. One feature of this setup is that, if a given manifold Y admits a special kind of fibration structure with toric fiber M and base B, then many interesting equations for the metric on Y can be reduced to the weighted cscK equation on M. This can be thought of as a generalized form of the Calabi Ansatz, which one recovers by taking M = \mathbb{C}. I will present on recent work extending some of this picture to the non-compact setting, with particular attention on (shrinking) Kähler-Ricci solitons. In particular, if a non-compact toric manifold M admits a weighted cscK metric (for suitable choices of weights), then it must be weighted K-stable. I will explain the relationship with Kähler-Ricci solitons, and time permitting I will explain how this leads to a simple proof of a well-known result of Futaki-Wang on the existence of shrinking Kähler-Ricci solitons on the total space of certain line bundles over a compact Kähler-Einstein Fano base.