Item

Abstract

There is a sequence of positive numbers $\delta_{2n}$, such that for any
connected $2n$-dimensional Riemannian manifold $M$, there are two mutually
exclusive possibilities: $1)$ There is a complex structure on $M$ making it
into a K\"ahler manifold, or $2)$ For any almost complex structure $J$
compatible with the metric, at every point $p\in M$, there is a smooth loop
$\gamma$ at $p$ such that $dist(J_p, hol_\gamma^{-1}J_phol_\gamma)>
\delta_{2n}$.