# K\"ahlerity of shrinking gradient Ricci solitons asymptotic to K\"ahler cones

## TL;DR

This paper proves that certain shrinking gradient Ricci solitons that resemble K"ahler cones at infinity are themselves K"ahler near infinity, and globally if complete, extending the understanding of their geometric structure.

## Contribution

It establishes the K"ahler property for shrinking gradient Ricci solitons asymptotic to K"ahler cones, both locally near infinity and globally in the complete case.

## Key findings

- Shrinking gradient Ricci solitons asymptotic to K"ahler cones are K"ahler near infinity.
- Complete shrinkers asymptotic to K"ahler cones are globally K"ahler.
- The result links asymptotic cone behavior to the global geometric structure.

## Abstract

We prove that a shrinking gradient Ricci soliton which is asymptotic to a K\"ahler cone along some end is itself K\"ahler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally K\"ahler.