Center of mass and K\"ahler structures

TL;DR

This paper establishes a dichotomy for connected even-dimensional Riemannian manifolds, showing they are either K"ahler or have a uniform lower bound on the deviation of compatible almost complex structures from holonomy transformations.

Contribution

It introduces a sequence of positive constants that distinguish manifolds admitting K"ahler structures from those with almost complex structures far from holonomy transformations.

Findings

Manifolds are either K"ahler or have a uniform deviation from holonomy.

Existence of a positive sequence _{2n} separating the two cases.

Provides a quantitative criterion for the K"ahler condition.

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 $dist(J_p, hol_\gamma^{-1}J_phol_\gamma)> \delta_{2n}$.