---
abstract: 'We prove that a shrinking gradient Ricci soliton which is asymptotic to a Kähler cone along some end is itself Kähler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally Kähler.'
address: 'School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA'
author:
- Brett Kotschwar
date: November 2016
title: Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones
---

Introduction
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Let $M$ be a smooth manifold of dimension $n$. A *shrinking gradient Ricci soliton* structure on $M$ consists of a Riemannian metric $g$ and a smooth function $f$ which together satisfy the equations $$\label{eq:shrinker}
  {\operatorname{Rc}}(g) + \nabla\nabla f= \frac{1}{2} g, \quad R + |\nabla f|^2 = f.$$ The latter equation is a normalization which can be achieved by adding an appropriate constant to $f$ on any connected component of $M$.

Shrinking gradient Ricci solitons (or *shrinkers*) correspond naturally to shrinking self-similar solutions to the Ricci flow, which are both the generalized fixed points of the equation and the prototypical models for the behavior of a solution in the vicinity of a developing singularity. They have been fully classified in dimensions two [@HamiltonSingularities] and three [@CaoChenZhu], 