of this theorem is that any non-trivial isometry of the cross-section of the cone must be reflected in a non-trivial isometry of the end of the soliton. In this paper, we show that the Kählerity of the cone is another feature which is necessarily inherited by an asymptotic shrinker.

\[thm:kaehlerity\] Suppose $(M, g, f)$ is a shrinking gradient Ricci soliton asymptotic to the regular cone $({E}_0, g_c)$ along the end $V\subset M$. If $({E}_0, g_c)$ is Kähler with complex structure $J_c$, then there is an end $W\subset V$ of $M$ and a complex structure $J$ defined on $W$ relative to which $g|_W$ is Kähler and $(W, J)$ is biholomorphic to $({E}_R, J_c)$ for some $R > 0$. If $(M, g)$ is complete, the Kähler structure extends to all of $M$.

If it is already known that there is a Kähler shrinker $(\hat{M}, \hat{g}, \hat{f})$ asymptotic to $({E}_0, g_c)$ along some end of $(\hat{M}, \hat{g})$, the first assertion follows from the uniqueness result in [@KotschwarWangConical]. Here, however, the existence of such a Kähler competitor $(\hat{M}, \hat{g}, \hat{f})$ is not presumed in advance and the Kählerity of the soliton $(M, g, f)$ on the end $W$ is established only from its asymptotic 