Kählerity along $V$. The assumptions are entirely local to the end $V$; in particular, $(M, g)$ need not be complete nor have only one end. However, when $(M, g)$ is complete and connected, its universal cover $(\tilde{M}, \tilde{g})$ will be globally Kähler and, consequently, connected at infinity [@MunteanuWangKaehler]. It follows then that $(M, g)$ is also connected at infinity and that the Kähler structure on $W$ must extend to all of $M$.

Our proof of Theorem \[thm:kaehlerity\] uses a combination of the methods in [@KotschwarRFHolonomy] and [@KotschwarWangConical]. In [@KotschwarWangConical], it is shown that a shrinker which is asymptotic to a cone $(E_0, g_c)$ gives rise to a smooth solution $g(t)$ to the Ricci flow on some end $W\subset V$ for $t\in [-1, 0)$ which begins at $g(-1) = g$, evolves self-similarly for $-1 \leq t < 0$, and converges smoothly as $t\nearrow 0$ to a limit metric isometric to $g_c$. By this construction, recalled in Proposition \[prop:brf\] below, the first assertion in Theorem \[thm:kaehlerity\] is reduced to a problem of “backward propagation of Kählerity,” that is, to the problem of showing that a solution to the Ricci flow which becomes Kähler after some time must have actually been Kähler all 