On the metric Kollár-Pardon problem

Abstract

Let $(M,g)$ be a compact real analytic Riemannian manifold and $\pi :\tilde{M}\to M$ its universal cover. Assume that $\tilde{M}$ can be realised as a manifold definable in an o-minimal structure $\Sigma$ expanding ${\mathbb{R}}_{\mathrm{an}}$ in such a way that the pullback metric $\tilde{g}:={\pi }^{\ast }g$ is $\Sigma$-definable. For instance, this is the case when $\tilde{M}$ can be realised as a semi-algebraic submanifold in ${\mathbb{R}}^n$ in such a way that the coefficients of the metric $\tilde{g}$ are semi-algebraic. We show that there exists a definable smooth map $\tilde{M}\to \tilde{K}$ to a compact simply connected $\Sigma$-definable space $\tilde{K}$ such that its regular fibres are Riemann locally homogeneous with respect to the metric $\tilde{g}$. We deduce that under these assumptions ${\pi }_1(M)$ is quasi-isometric to a locally homogeneous space. In the case when $M$ is aspherical we show that $(\tilde{M},\tilde{g})$ is a homogeneous Riemannian manifold. A similar result in the setting of complex algebraic geometry was earlier conjectured by Kollár and Pardon (\cite{KP}). Using our results, we prove the conjecture of Kollár-Pardon in the special case of smooth aspherical varieties admitting a bi-definable Kähler metric and discuss the analogues of this conjecture in other branches of geometry.