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K\"ahler Geometry of non-Compact Toric Manifolds


Toric manifolds and toric varieties have played an important and particularly illuminating role in algebraic, symplectic, and K\"ahler geometry going back at least to the 1970's. The high degree of symmetry in many cases allows one to reduce the complexity of a given geometric question, giving the impression that we can really ``see'' the structure. A notable example of this in the K\"ahler setting is Donaldson's classic paper [27].

Until relatively recently, this has been mostly confined to the setting of compact manifolds (complete varieties). Both the algebraic and the symplectic perspectives have come to be fairly well understood individually in the non-compact case, but there has been little application of the intersection of the two theories, which naturally comprises toric K\"ahler geometry. The notable exception to this rule is the case of toric K\"ahler cones [56, 55, 39, 17, 16]. These non-compact manifolds inherit a great deal of structure from their compact \emph{link}, which are interesting spaces in their own right. However, these manifolds are (typically) highly singular at precisely one point.

In this dissertation, we describe a procedure for the study of K\"ahler geometry on smooth non-compact toric manifolds. This involves tying together the algebraic and symplectic perspectives. We study a class of toric manifolds and show that here we can apply a non-compact version of the classical Delzant classification together with its application to K\"ahler geometry [4, 41, 24, 42]. We apply this to non-compact shrinking K\"ahler-Ricci solitons, where we propose a useful class of complete K\"ahler metrics, and ultimately prove a general uniqueness theorem for shrinking K\"ahler-Ricci solitons on non-compact toric manifolds.

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