# On the metric Kollár-Pardon problem

Vasily Rogov

###### Abstract.

Let $(M,g)$ be a compact real analytic Riemannian manifold and
$\pi\colon\widetilde{M}\to M$ its universal cover. Assume that $\widetilde{M}$
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
$\widetilde{g}:=\pi^{*}g$ is $\Sigma$-definable. For instance, this is the
case when $\widetilde{M}$ can be realised as a semi-algebraic submanifold in
$\mathbb{R}^{n}$ in such a way that the coefficients of the metric
$\widetilde{g}$ are semi-algebraic. We show that there exists a definable
smooth map $\widetilde{M}\to\widetilde{K}$ to a compact simply connected
$\Sigma$-definable space $\widetilde{K}$ such that its regular fibres are
Riemann locally homogeneous with respect to the metric $\widetilde{g}$. 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
$(\widetilde{M},\widetilde{g})$ is a homogeneous Riemannian manifold. A
similar result in the setting of complex algebraic geometry was earlier
conjectured by Kollár and Pardon ([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.

##  1\. Introduction

Let $\Gamma$ be a finitely generated group that acts by homeomorphisms
properly discontinuously and cocompactly on an $n$-dimensional ball $B$.
Choose any homeomorphism $\phi\colon B\xrightarrow{\sim}\mathbb{R}^{n}$. Let
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ be a polynomial map. Assume that $f$ is
$\Gamma$-invariant, that is $f(\phi(\gamma\cdot x))=f(\phi(x))$ for any $x\in
B,\ \gamma\in\Gamma$. Then $f$ is constant.

The aim of this note is to take a step towards a wide geometrical
generalisation of this elementary fact. The geometric meta-principle we are
aiming for can be formulated as the following:

<<If a geometric structure demonstrates _tame behaviour at infinity_ and is
invariant under a cocompact discrete group action, it is _close to being
homogenous_ >>.

The notion of tameness can be made precise using the language of o-minimal
geometry, see Section 2. The words <<_close to being homogenous_ >> should
most probably mean <<homogenous up to a compact factor>>, see Theorem A and
the discussion there.

The first result in this direction was proposed by Kollár and Pardon in their
paper [KP] and was motivated by the study of universal covers of complex
algebraic varieties. Let $X$ be a smooth complex projective variety and
$\pi\colon\widetilde{X}\to X$ its universal cover111By this we mean the
universal cover of the underlying complex analytic space $X^{\mathrm{an}}$..
The Shafarevich conjecture ([Shaf], p.407) predicts that $\widetilde{X}$ is
holomorphically convex, which is a deep complex-analytic property. It is now
known in many cases (see e.g. [EKPR]), while the general case is still widely
open. One of the difficulties of this conjecture is that the operation of
taking the universal cover is purely transcendental: the complex manifold
$\widetilde{X}$ usually does not possess an algebraic structure and lacks any
reasonable description in terms of polynomial equations, or even real
polynomial inequalities. The conjecture of Kollár and Pardon says that
$\widetilde{X}$ is indeed never semi-algebraic, except for several classical
cases.

######  Conjecture 1 (Kollár-Pardon, [KP]).

Let $X$ be a smooth complex projective variety and $\pi\colon\widetilde{X}\to
X$ its universal cover. Assume that $\widetilde{X}$ is biholomorphic to an
open real semi-algebraic subset $\Omega$ inside a complex algebraic variety
$Z$. Then $\widetilde{X}\simeq F\times\mathbb{C}^{n}\times D$, where $D$ is a
symmetric bounded domain and $F$ is a simply connected projective variety.

Recall that a subset $\Omega$ of a complex algebraic variety $Z$ is real semi-
algebraic if $Z$ admits a finite open cover by affine sets $U_{i}$ such that
$\Omega\cap U_{i}$ is given by a finite collection of real polynomial
equalities and inequalities for every $i$.

In the same paper, the authors proved their conjecture in some special case
(Theorem 3, [KP]). In the earlier work [CHK] some results were obtained when
$\Omega=Z$ is quasi-projective. See also [CH] for some results when $X$ is
compact Kähler and $\Omega$ is quasi-Kähler.

Let us clarify the relation between Conjecture 1 and the meta-principle that
we mentioned above. The complex structure on $\Omega$ is _semi-algebraic_.
This means that there exists a finite collection of real semi-algebraic charts
$\Omega=U_{1}\cup\ldots\cup U_{k}$ and real algebraic trivialisations of the
tangent bundle $T\Omega|_{U_{i}}$, in which the complex structure operator
$J_{\Omega}$ is given by semi-algebraic maps $\phi_{i}\colon
U_{i}\to\operatorname{Mat}_{n\times n}(\mathbb{R})$, where $n=\dim X$. On the
other hand, the holomorphic action of $\pi_{1}(X)$ on $\widetilde{X}$
translates naturally to $\Omega$ and $J_{\Omega}$ is $\pi_{1}(X)$-invariant.
Conjecture 1 therefore predicts that up to a compact factor $F$ (which is, in
fact, irrelevant to the $\pi_{1}(X)$-structure on $\widetilde{X}$), the
complex manifold $\widetilde{X}$ is biholomorphic to the complex homogeneous
manifold $\mathbb{C}^{n}\times D$. Moreover, $\widetilde{X}$ is Kähler since
it is locally biholomorphic to the Kähler manifold $X$, and every complete
Kähler homogenous simply connected manifold is of this form (see e.g. [Ghys],
Théorème 2.5).

A set-up for Kollár-Pardon conjecture which is more natural and general than
semi-algebraic geometry is _o-minimal geometry_ (see [KP], Remark 18). We
briefly recall the basics of o-minimal geometry in Section 2 and refer the
reader to [vdD] for the comprehensive introduction. The main result of this
paper is the following Riemannian version of Conjecture 1:

######  Theorem A.

[Theorem 3.2] Let $M$ be a compact real analytic manifold and $g$ a real-
analytic Riemannian metric on $M$. Let $\pi\colon\widetilde{M}\to M$ be the
universal cover and $\widetilde{g}:=\pi^{*}g$ the induced metric on it. Assume
that $\widetilde{M}$ admits a structure of a _manifold definable in an
o-minimal structure_ $\Sigma$ in such a way that $\widetilde{g}$ is
$\Sigma$-definable. Then there exists a $\Sigma$-definable simply connected
compact topological space $\widetilde{K}$ and a $\Sigma$-definable real
analytic map $\Phi\colon\widetilde{M}\to\widetilde{K}$, such that the regular
fibres of $\Phi$ are Riemannian locally homogeneous manifolds with respect to
the metric $\widetilde{g}$.

In fact, we prove a stronger result: for every point $x\in\widetilde{M}$ which
is not a critical point of $\Phi$, there exists a connected Lie group $G$ and
an open neighbourhood $U$ of $x$, such that $G$ locally acts on
$(U,\widetilde{g})$ by isometries and the fibres of $\Phi$ are given by the
orbits of this action.

If the map $\Phi$ has no critical points, Theorem A implies that
$(\widetilde{M},\widetilde{g})$ is fibred in locally homogeneous submanifolds
over a compact topological space. In this case, we get a Riemannian
counterpart to the Conjecture 1. We provide some examples where the map $\Phi$
has critical points, see Example 1. In this case, the best one can hope for is
to find a stratification of $\widetilde{M}$ by $\Sigma$-definable locally
closed subsets, each of which is fibred with locally homogeneous fibres. We
show that this is indeed the case (Proposition 3.3) and that $M$ can be
written as a union of closed locally homogeneous submanifolds which are
_strongly bi-definable_ in the sense of Definition 6, see Corollary 3.4.

We also deduce that a compact manifold $M$ satisfies the assumptions of
Theorem A, then $\pi_{1}(M)$ is quasi-isometric to a locally homogeneous
space, which seems to be a subtle geometric-group-theoretic property
(Corollary 3.5).

If $M$ is aspherical, Theorem A can be strengthened as the following.

######  Theorem B.

[Theorem 3.9] Assume $M,\ g$ and $\widetilde{M}$ are as in Theorem A. Assume
moreover that $\widetilde{M}$ is contractible. Then $\widetilde{K}$ is a point
and $(\widetilde{M},\widetilde{g})$ is a Riemannian homogeneous manifold.

The proof of Theorem B is based on a topological lemma (Lemma 3.7) that gives
control over submanifolds in aspherical manifolds with ’’topologically tame’’
preimage in the universal cover. One of the general consequences of this lemma
in o-minimal geometry is that compact aspherical manifolds with definable
universal covers do not admit strict submanifolds that are _strongly be-
definable_ (see Corollary 3.8; Definition 6 for the notion of strongly be-
definable submanifolds).

From this, we deduce an application to the original Kollár-Pardon conjecture.

######  Theorem C (Corollary 3.11).

Let $X$ be a smooth complex projective variety with universal cover
$\pi\colon\widetilde{X}\to X$. Assume that $\widetilde{X}$ admits a structure
of a definable complex manifold in some o-minimal structure. Assume moreover
that $X$ is aspherical and admits a _bi-definable Kähler metric_ , that is, a
Kähler Hermitian metric $h=g+\sqrt{-1}\omega$ such that $\pi^{*}h$ is
definable on $\widetilde{X}$. Then $(\widetilde{X},\pi^{*}h)$ is a Kähler
homogeneous manifold. In particular, the Kollár-Pardon Conjecture holds for
$X$.

The paper is organised as follows. In Section 2 we briefly recall the
necessary preliminaries. We discuss o-minimal structures and o-minimal
geometry in subsection 2.1 and 2.2 and translate the classical differential-
geometric notions to the context of o-minimal geometry in subsection 2.3. In
subsection 2.4 we recall the relation between cohomogeneity and additive Weyl
invariants after Console-Olmos ([CO]), a blackbox from Riemannian geometry
which plays a key role in the proof. In Section 3 we introduce the concept of
(strong) bi-definability (subsection 3.1). We prove Theorem A in subsection
3.2, Lemma 3.7 and Theorem B in subsection 3.3, and Theorem C in subsection
3.4. We finish with a discussion on some open questions (subsection 3.5).

Acknowledgements. I am thankful to Bruno Klingler for turning my attention to
the Kollár-Pardon Conjecture and several fruitful discussions. I am also
thankful to Paul Brommer-Wierig for interesting discussions and to Semyon
Podkorytov for sharing with me the main ideas behind Lemma 3.7.

##  2\. Preliminaries

###  2.1. o-minimal geometry

The notion of an o-minimal structure originated in model theory and found many
applications in other areas of mathematics during the last decade,
specifically in complex algebraic geometry ([BKT], [BBT], etc.) In this
section, we briefly recall the basics of o-minimal geometry and mention all
the needed geometric properties of o-minimally definable sets. The primary
reference for this part is [vdD].

######  Definition 1.

A _structure_ $\Sigma$ is a collection $(\Sigma_{n})_{n\in\mathbb{N}}$, where
each $\Sigma_{n}$ is a boolean algebra of subsets of $\mathbb{R}^{n}$ and the
following holds:

  * (i)

if $A\in\Sigma_{n}$ and $B\in\Sigma_{m}$, then $A\times B\in\Sigma_{n+m}$;

  * (ii)

if $A\in\Sigma_{n}$ and $p_{i}\colon\mathbb{R}^{n}\to\mathbb{R}^{n-1}$ is the
standard projection

$(x_{1},\ldots,x_{n})\mapsto(x_{1},\ldots x_{i-1},x_{i+1}\ldots,x_{n-1}),$

then $p_{i}(A)\in\Sigma_{n-1}$;

  * (iii)

every semi-algebraic subset $A\subseteq\mathbb{R}^{n}$ is in $\Sigma_{n}$.
Recall, that $A$ is semi-algebraic if it can be defined using a finite system
of real polynomial inequalities.

The elements of $\Sigma_{n}$ are called ($\Sigma$-)_definable sets_. If
$A\in\Sigma_{n}$ and $B\in\Sigma_{m}$ are two definable sets, then a map
$f\colon A\to B$ is called _definable_ if its graph $\Gamma_{f}\subseteq
A\times B$ is definable.

The most elementary example is given by the structure of semi-algebraic sets
$\mathbb{R}_{\mathrm{alg}}$. Notice that the property (ii) for
$\mathbb{R}_{\mathrm{alg}}$ is a consequence of the Tarski-Seidenberg
Quantifier Elimination Theorem.

If $\Sigma$ is a structure, images and preimages of $\Sigma$-definable sets
under $\Sigma$-definable maps are $\Sigma$-definable. Moreover, if
$A\subseteq\mathbb{R}^{n}$ is $\Sigma$-definable then its topological closure
$\overline{A}$, interior $A^{\circ}$, boundary $\partial
A=\overline{A}\setminus A$ and frontier $\overline{A}\setminus A^{\circ}$ are
$\Sigma$-definable (see e.g. [vdD], Chapters 1 and 2).

######  Definition 2.

A structure $\Sigma$ is called _o-minimal_ , if
$\Sigma_{1}=(\mathbb{R}_{\mathrm{alg}})_{1}$. Equivalently, the only one-
dimensional $\Sigma$-definable sets are finite collections of points,
intervals and rays.

The sets definable in $\mathbb{R}_{alg}$ share many nice topological
properties, and restricting oneself to working with semi-algebraic sets is a
safe way to escape pathological counterexamples from point-set topology. What
is spectacular, is that the same feature remains for any o-minimal structure.
Below we list the main geometric properties of sets definable in o-minimal
structures.

######  Theorem 2.1.

Let $\Sigma$ be an o-minimal structure.

  * (i)

A $\Sigma$-definable set is connected if and only if it is path-connected;

  * (ii)

definable sets admit _finite cell decomposition_ : every definable
$A\subset\mathbb{R}^{n}$ can be written as $A=A_{1}\sqcup\ldots\sqcup A_{k}$,
where each $A_{j}\subseteq A$ is a locally closed definable subset definably
homeomorphic to $\mathbb{R}^{n_{j}}$. In particular, every definable set has
finitely many connected components and is homotopically equivalent to a finite
CW-complex;

  * (iii)

definable functions are piecewise-continuous; moreover, if $A$ is a definable
set and $f\colon A\to\mathbb{R}$ is a definable map, for every $k\geq 0$ there
exists a finite decomposition $A=A_{1}\sqcup\ldots\sqcup A_{m}$ into definable
locally closed sets $A_{j}$ such that $f|_{A_{j}}$ is of class
$\mathcal{C}^{k}$;

  * (iv)

if $A\subseteq\mathbb{R}$ is a definable subset of $\mathbb{R}$ and $f\colon
A\to\mathbb{R}$ is a definable function, then its derivative $f^{\prime}$ is
definable whenever it exists.

Item (i) shows that working with sets definable in o-minimal structure indeed
allows one to exclude pathological examples that appear for arbitrary subsets
of $\mathbb{R}^{n}$. All the results of Theorem 2.1 can be found in [vdD]. The
item (ii) is [vdD], Chapter 3, Theorem 2.11. The items (iii) and (iv) are the
content of Chapter 7 loc. cit..

######  Remark 1.

Item (ii) of Theorem 2.1 can be strengthened as the following. Let $\Sigma$ be
an o-minimal structure and $A\subseteq\mathbb{R}^{n}$ a definable
subspace.Then for every $k\geq 0$ there exists a decomposition of $A$ into a
disjoint union of locally closed subsets

$A=A_{1}\sqcup\ldots\sqcup A_{k}$

and a collection of maps $\theta_{j}\colon A_{j}\to\mathbb{R}^{n_{j}}$, where
each $\theta_{j}$ is a definable homeomorphism of class $\mathcal{C}^{k}$. In
particular, every $A_{j}$ is a $\mathcal{C}^{k}$-submanifold of
$\mathbb{R}^{n}$. For the proof see Theorem 6.6 in [Cos]. In [RSW] the authors
showed that one can replace $k$ with $\infty$ if $\Sigma$ is a so-called
_structure generated by a quasianalytic Denjoy-Carleman class_ (in what
follows we will call it _of QA type_ , following the abbreviation in [RSW]).
The structures $\mathbb{R}_{alg},\mathbb{R}_{an},\mathbb{R}_{exp}$ and
$\mathbb{R}_{an,exp}$ are all of QA type.

An example of a set which is not definable in any o-minimal structure is the
graph of the sine function $\\{(x,\sin x)\\}\subset\mathbb{R}^{2}$. Indeed,
the sine function can not be definable since
$\sin^{-1}(0)=\\{n\pi|n\in\mathbb{Z}\\}\subset\mathbb{R}$ is not in
$(\mathbb{R}_{alg})_{1}$. Another example which is relevant to our goals is
the following. Let $M_{1}$ and $M_{2}$ be definable subsets of
$\mathbb{R}^{n}$ and $f\colon M_{1}\to M_{2}$ an infinite-folded cover. Then
the map $f$ is not definable in any o-minimal structure since a fibre over a
point is a discrete countable set and can not be definable by Theorem 2.1,
(ii).

O-minimal structures are in general very hard to construct. A structure
$\Sigma^{\prime}$ is called an _expansion_ of a structure $\Sigma$ if every
$\Sigma$-definable set is $\Sigma^{\prime}$-definable. If $\Sigma$ is a
structure and $\mathcal{T}$ is a collection of subsets of $\mathbb{R}^{n}$, by
Zorn’s lemma there exists a minimal structure
$\Sigma\langle\mathcal{T}\rangle$ which is an expansion of $\Sigma$ and
contains $\mathcal{T}$. Of course, even if we start with an o-minimal
structure $\Sigma$, its expansion $\Sigma\langle\mathcal{T}\rangle$ does not
need to be o-minimal. Here is the list of main known o-minimal structures:

  * •

$\mathbb{R}_{\mathrm{alg}}$;

  * •

$\mathbb{R}_{\mathrm{an}}$. This structure is defined as
$\mathbb{R}_{\mathrm{alg}}\langle\mathcal{T}\rangle$, where $\mathcal{T}$ is
the set of graphs of functions $f\colon\mathbb{R}^{n}\to\mathbb{R}$, where
$f\equiv 0$ outside a compact $K\subset\mathbb{R}^{n}$ and $f|_{K}$ is
analytic. The o-minimality of this structure was proved by van den Dries based
on the result of Gabrielov [Gab];

  * •

$\mathbb{R}_{\exp}$. This is structure is defined as
$\mathbb{R}_{\mathrm{alg}}\langle\Gamma_{\operatorname{exp}}\rangle$, where
$\Gamma_{\operatorname{exp}}=\\{(t,e^{t})\\}\subset\mathbb{R}^{2}$ is the
graph of real exponent. The o-minimality is due to Wilkie [Wil].

  * •

$\mathbb{R}_{\mathrm{an},\exp}$. This is the minimal structure which is an
expansion of both $\mathbb{R}_{\mathrm{an}}$ and $\mathbb{R}_{\exp}$. Its
o-minimality is proved in [vdDM].

  * •

The _Pfaffian closure_ of an o-minimal structure is an o-minimal structure,
see [Speis].

###  2.2. o-minimal spaces

One would like to consider definable geometrical objects without fixing their
embedding into $\mathbb{R}^{n}$. This motivates the following definition:

######  Definition 3.

Let $\Sigma$ be an o-minimal structure and $X$ a topological space. A
$\Sigma$-_definable atlas_ $\xi=(U_{\alpha},\phi_{\alpha})_{\alpha\in A}$ is a
finite collection of open charts $U_{\alpha},\ X=\bigcup_{\alpha}U_{\alpha}$
and homeomorphic embeddings $\phi_{\alpha}\colon U_{\alpha}\to\mathbb{R}^{n}$,
such that

  * (i)

$\phi_{\alpha}(U_{\alpha}\cap U_{\beta})$ are $\Sigma$-definable in
$\mathbb{R}^{n}$ for every $\alpha$ and $\beta$;

  * (ii)

the transition functions
$\phi_{\beta}\circ\phi_{\alpha}^{-1}\colon\phi_{\alpha}(U_{\alpha}\cap
U_{\beta})\to\phi_{\beta}(U_{\alpha}\cap U_{\beta})$ are $\Sigma$-definable.

A $\Sigma$-definable topological space is a topological space $X$ with a fixed
equivalence class of $\Sigma$-definable atlases.

The finiteness of the atlas $\xi$ is crucial in Definition 3 since it allows
one to extend geometrical and topological finiteness properties of definable
sets to a broader context of definable spaces.

In what follows we will always assume that $\Sigma$ is an o-minimal structure
which is an expansion of $\mathbb{R}_{an}$ and we will often not specify
$\Sigma$ when talking about definable spaces. We all also abusively refer to a
pair $(X,\xi)$ as _definable space_ , although one topological space a priori
can admit non-equivalent definable structures.

If $X$ is a definable space, we say that a subspace $Y\subseteq X$ is
definable, if there exists a finite definable atlas
$(U_{\alpha},\phi_{\alpha})$, such that $\phi_{\alpha}(Y\cap U_{\alpha})$ is
definable for every $\alpha$. A product of two definable spaces carries a
natural definable space structure, which allows us to speak about definable
maps between definable spaces. All the properties listed in Theorem 2.1
transfer directly to the context of definable spaces. The Cellular
Decomposition Theorem (Theorem 2.1, (ii)) implies that a definable topological
space is homeomorphic to a finite CW-complex.

Notice that in Definition 3 we do not require the sets
$\phi_{\alpha}(U_{\alpha})$ to be smooth, therefore a definable topological
space is not necessarily a manifold. One says that $M$ is a $\Sigma$-definable
smooth (respectively $\mathcal{C}^{k}$-) manifold if it is a definable
topological space and the definable atlas defines a smooth (respectively
$\mathcal{C}^{k}$-) manifold structure on $M$. Again, a definable manifold is
not a property of a smooth manifold, but a structure on it. Since in this
paper we are not considering different definable structures on the same
manifold, we abbreviate _’’definable manifold’’_ for ’’a manifold admitting a
structure of a manifold definable in an o-minimal structure $\Sigma$’’.

In spite of the fact that not every definable topological space is a definable
manifold, it admits a stratification by definable $\mathcal{C}^{k}$\- (or
sometimes even $\mathcal{C}^{\infty}$-) manifolds, as we explain below.

By Theorem 2.1, (ii) there exists a decomposition $X=X_{1}\sqcup\ldots\sqcup
X_{k}$, where each $X_{i}$ is a locally closed definable subset of $X$
definably homeomorphic to $\mathbb{R}^{n_{i}}$. The _dimension_ of $X$ is
defined as $\dim X:=\max\\{n_{i}\\}$ (by definition, the dimension of an empty
set is $-\infty$). This notion does not depend on the choice of a cellular
decomposition and gives rise to a nice dimension theory for definable
topological spaces, see [vdD], Chapter 4. In particular, if $A\subseteq X$ is
an open definable subset of a definable topological space, $\dim\partial
A<\dim A$.

######  Lemma 2.2.

Let $\Sigma$ be an o-minimal structure and $X$ be $\Sigma$-definable space.
Then for every $k\geq 0$, there exists a stratification by definable locally
closed subsets

$\varnothing=S_{0}\subset\ldots\subset S_{r}=X,$

such that $\dim S_{i-1}<\dim S_{i}$ and each $V_{i}:=S_{i}\setminus S_{i-1}$
is a $\mathcal{C}^{k}$-definable manifold (with respect to the induced
definable structure). If, moreover, $\Sigma$ is of QA type (see Remark 1), one
can also take $k=\infty$.

######  Proof: .

We argue by induction on $\dim X$. If $\dim X=0$ there is nothing to prove. In
general, from Theorem 2.1, (ii) and Remark 1 it follows that we can find a
$\mathcal{C}^{k}$\- (and $\mathcal{C}^{\infty}$-, if $\Sigma$ is of QA type)
cellular decomposition

$X=X_{1}\sqcup\ldots\sqcup X_{p}.$

Let $I=\\{i_{1},\ldots,i_{q}\\}$ be the set of indices of cells of non-maximal
dimension, that is $\dim X_{i_{j}}<\dim X$ if and only if $i_{j}\in I$. Put
$S_{r-1}:=\bigcup_{i\in I}X_{i}$. Then $V_{r}:=X\setminus
S_{r-1}=\bigsqcup_{j\notin I}X_{j}$ is a union of non-interesecting definable
$\mathcal{C}^{k}$\- (respectively $\mathcal{C}^{\infty}$-) manifolds. At the
same time, $\dim S_{r-1}<\dim X$, so the induction assumption applies to it. ∎

In this text, we are concerned with universal covers of various manifolds and
spaces. Usually, the universal cover of a definable space does not possess any
natural definable structure. An important exception is the case of a finite
fundamental group.

Let $X$ be a topological space homeomorphic to a finite CW-complex and
$p\colon\widetilde{X}\to X$ be its universal cover. Let $f\colon Y\to X$ be a
continuous map from a simply connected topological space $Y$. Recall that $f$
uniquely lifts to a continuous map $\widetilde{f}\colon Y\to\widetilde{X}$
such that $f=p\circ\widetilde{f}$.

######  Lemma 2.3.

Let $X$ be a topological space and $\xi$ a $\Sigma$-definable atlas on $X$ for
an o-minimal structure $\Sigma$. Assume that $\pi_{1}(X)$ is finite. Then the
following holds:

  * (i)

The universal cover $\widetilde{X}$ admits a unique $\Sigma$-definable atlas
$\widetilde{\xi}$ such that the covering map $p\colon\widetilde{X}\to X$ is
definable;

  * (ii)

if $Y$ is a simply connected $\Sigma$-definable space and $f\colon Y\to X$ is
continuous and $\Sigma$-definable, the canonical lifting $\widetilde{f}\colon
Y\to\widetilde{X}$ is $\Sigma$-definable.

######  Proof: .

Let $U_{\alpha}\in\xi$. By Theorem 2.1, (ii) there exists a finite covering of
$U_{\alpha}$ by open $\Sigma$-definable simply connected $V^{\alpha}_{\beta},\
U_{\alpha}=\bigcup_{\beta}V^{\alpha}_{\beta}$. The cover
$p\colon\widetilde{X}\to X$ splits over $V^{\alpha}_{\beta}$, that is,
$p^{-1}(V^{\alpha}_{\beta})\simeq V^{\alpha}_{\beta}\times\\{1,\ldots,k\\}$.
Here we use that $p^{-1}(V^{\alpha}_{\beta})$ has finitely many connected
components since $\pi_{1}(X)$ is finite and $p$ is a finite cover. Each of
these connected components is homeomorphic to $V^{\alpha}_{\beta}$. The
connected components of preimages of $V^{\alpha}_{\beta}$ for all $\alpha$ and
$\beta$ give rise to a definable atlas $\widetilde{\xi}$ on $\widetilde{X}$
with local coordinates $\phi_{\alpha}\circ p$. The projection $p$ is definable
since in a finite collection of definable charts $V^{\alpha}_{\beta}$ it
coincides with the projection $V^{\alpha}_{\beta}\times\\{1,\ldots,k\\}\to
V^{\alpha}_{\beta}$.

To prove (ii) notice that the graph $\Gamma_{\widetilde{f}}\subset
Y\times\widetilde{X}$ is nothing but the preimage of the graph
$\Gamma_{f}\subset Y\times X$ under the (definable) map $id\times p\colon
Y\times\widetilde{X}\to Y\times X$. Therefore, $\widetilde{f}$ is definable
whenever $f$ is. ∎

###  2.3. o-minimal differential geometry

In this subsection we transfer the main notions of differential geometry into
the context of definable manifolds. These results are well-known to the
experts, although folklore and, up to our knowledge, are not written in full
generality anywhere in the literature. We always work in the smooth category,
although all the results remain valid in $\mathcal{C}^{k}$-category for $k\geq
2$.

######  Definition 4.

Let $M$ be a definable smooth manifold and $E\to M$ a smooth rank $d$ vector
bundle. _A definable structure on_ $E$ is a finite definable cover $M=\bigcup
U_{\alpha}$ (i.e., each $U_{\alpha}$ is open definable) and a collection of
trivialisations $\psi_{\alpha}\colon
E|_{U_{\alpha}}\xrightarrow{\sim}U_{\alpha}\times\mathbb{R}^{d}$, such that
the transition functions $\psi_{\beta}\circ\psi_{\alpha}^{-1}$ are definable.
We say that a section $s\in\mathcal{C}^{\infty}(M,E)$ is definable if it is
definable in the local definable charts. A definable vector bundle over a
definable manifold is a vector bundle with a chosen definable structure.

A definable structure on a vector bundle $E$ naturally gives rise to a
structure of a definable manifold on its total space such that the projection
$p\colon\operatorname{Tot}(E)\to M$ is definable, and vice versa. A section of
a definable vector bundle $E\to M$ is definable if and only if it is definable
as a map from $M$ to $\operatorname{Tot}(E)$. A trivial vector bundle has a
canonical definable structure that comes from the product definable structure
on its total space $M\times\mathbb{R}^{d}$.

A morphism $f\colon E\to F$ of definable vector bundles is definable if it
sends definable sections of $E$ to definable sections of $F$.

The following proposition is straightforward and is left to the reader as an
exercise.

######  Proposition 2.4.

Let $M$ be a definable smooth manifold, $E\to M$ a definable vector bundle.
Then:

  * (i)

$E^{*},\ \Lambda^{k}E,\ \operatorname{Sym}^{k}E$ carry natural definable
structures;

  * (ii)

if $F$ is another definable vector bundle, then $E\otimes F,\ Hom(E,F),\
E\oplus F$ carry natural definable structures;

  * (iii)

the trace morphism $\operatorname{tr}\colon E\otimes E^{*}\to\mathbb{R}_{M}$
is definable;

  * (iv)

a morphism $f\colon F\to E$ is definable if and only if it is definable as a
section of $Hom(E,F)$;

  * (v)

a morphism $f\colon E\to F$ is definable if and only if it determines a
definable map $\operatorname{Tot}(E)\to\operatorname{Tot}(F)$. If
$\operatorname{ker}f$ (resp. $\operatorname{coker}f$) is a vector bundle, it
carries a natural definable structure;

  * (vi)

if $N$ is a definable manifold and $f\colon N\to M$ is a definable smooth map,
then $f^{*}E$ carries a natural definable structure;

######  Proposition 2.5.

Let $M$ be a definable manifold. Let $(U_{\alpha},\phi_{\alpha})$ be a
definable atlas on $M$. Then

$d\phi_{\alpha}\colon TM|_{U_{\alpha}}\to
T\phi_{\alpha}(U_{\alpha})=\phi_{\alpha}(U_{\alpha})\times\mathbb{R}^{n}$

defines a definable structure on $TX$, called _the canonical definable
structure_ ;

######  Proof: .

Without loss of generality, we may assume that $\phi_{\alpha}$ are all open
embeddings. Then $d\phi_{\alpha}^{-1}(\partial/\partial x_{i})$ defines
trivialisations of $TM|_{U_{\alpha}}$. The rest of the Proposition follows
from the fact that the derivatives of definable smooth functions are definable
and computations in the local coordinates. ∎

Using Proposition 2.4 one can canonically extend the definable structure on
$TM$ to any tensor bundle $(TM)^{\otimes p}\otimes(T^{*}M)^{\otimes q}$.
Without specifying we say that a tensor on a definable manifold is definable
if it is definable in the canonical definable structure. In other words, if
$E:=(TM)^{\otimes p}\otimes(T^{*}M)^{\otimes q}$, a tensor
$\tau\in\mathcal{C}^{\infty}(M,E)$ is definable if and only if there exists a
definable finite open covering $\\{U_{\alpha}\\}$ on $M$ such that for every
$\alpha$ the components of $\tau|_{U_{\alpha}}$ with respect to a definable
trivialisation of $E|_{U_{\alpha}}$ can be written as definable functions
$\tau^{i_{1},\ldots,i_{p}}_{j_{1},\ldots,j_{q}}\colon
U_{\alpha}\to\mathbb{R}$.

All the notions of differential geometry easily transfer to the definable
context. For example, one says that a Riemannian metric $g$ on a definable
manifold $M$ is definable if $g$ is definable as a section of
$\operatorname{Sym}^{2}T^{*}M$.

We say that a connection $\nabla$ on $E$ is definable if, on a finite covering
by definable open charts $U_{\alpha}$ in which both $E$ and $TM$ can be
simultaneously definably trivialised, $\nabla$ can be written as
$\nabla=d+P_{\alpha}$ for definable local sections $P_{\alpha}$ of
$End(E)\otimes T^{*}M$.

######  Proposition 2.6.

Let $M$ be a smooth definable manifold and $E\to M$ a definable smooth vector
bundle.

  * (i)

if $\nabla$ is a definable connection on $E$, then for every definable section
$s\in\mathcal{C}^{\infty}(M,E)$ the section $\nabla
s\in\mathcal{C}^{\infty}(M,E\otimes T^{*}M)$ is definable;

  * (ii)

if $g$ is a definable metric on $M$, its Levi-Civita connection is definable.

######  Proof: .

(i). Let $s_{\alpha}=s|_{U_{\alpha}}$. Then
$\nabla(s_{\alpha})=ds_{\alpha}+P_{\alpha}s_{\alpha}$. Both summands are
definable: the second one because of the definability of $P_{\alpha}$, and the
first one because the derivative of a definable function is definable (Theorem
2.1, (iv)) and $ds_{\alpha}$ can be expressed via partial derivatives of the
components of $s_{\alpha}$ with respect to local definable trivialisations.

(ii). Choose a definable chart $U$ with definable coordinates
$(x^{1},\ldots,x^{n})$. On this chart, $\nabla$ can be written as $d+\Gamma$,
where the elements of the tensor $\Gamma$ are Christoffel symbols
$\Gamma_{i,j}^{k}$. Those are expressed in terms of the partial derivatives of
$g$ along the basis vector fields:

$\Gamma_{i,j}^{k}=\frac{1}{2}g^{kl}\left(\frac{\partial g_{li}}{\partial
x^{j}}+\frac{\partial g_{lj}}{\partial x^{i}}-\frac{\partial g_{ij}}{\partial
x^{l}}\right).$

The functions $g^{a,b}$ are definable with respect to $x^{c}$ and a derivation
of a definable function is definable. Hence $\nabla$ is definable.

∎

######  Definition 5.

Let $(M,g)$ be a Riemannian manifold. We define the class of _Weyl invariants_
among the tensors on $M$ as the following:

  * •

$g$ is a Weyl invariant;

  * •

the curvature tensor $R_{g}$ is a Weyl invariant;

  * •

if $\tau$ is a Weyl invariant and $\nabla$ is the Levi-Civita connection,
$\nabla\tau$ is a Weyl invariant;

  * •

a linear combination of Weyl invariants is a Weyl invariant;

  * •

if $\tau\in(TM)^{\otimes p}\otimes(T^{*}M)^{\otimes q}$ is a Weyl invariant,
then its image under any tensorial convolution

$(TM)^{\otimes p}\otimes(T^{*}M)^{\otimes
q}\to(TM)^{\otimes(p-k)}\otimes(T^{*}M)^{\otimes(q-k)}$

is a Weyl invariant.

Typical examples of Weyl invariants are the Ricci curvature, the Weyl tensor,
the scalar curvature, iterated covariant derivatives of the Riemannian
curvature $\nabla^{\circ s}R_{g}$ et caetera. Combining Propositions 2.4 and
2.6 we get the following result:

######  Corollary 2.7.

Let $M$ be a definable smooth manifold and $g$ a definable Riemannian metric
on it. Then every Weyl invariant on $(M,g)$ is definable.

###  2.4. Additive Weyl invariants and cohomogeneity

Recall that a Riemannian manifold $(M,g)$ is called locally homogeneous if for
any two points $x,y\in M$ there exist open neighbourhoods $U,\ V$ of $x$ and
$y$ respectively such that $(U,g|_{U})$ and $(V,g|_{V})$ admit a smooth
isometry. Equivalently, this means that for every point $x\in M$ there is an
open neighbourhood $U$ of $x$ and a collection of Killing vector fields
$\xi_{1},\ldots,\xi_{n}\in\mathcal{C}^{\infty}(U,TM|_{U})$ that generate
$TM|_{U}$ as a $\mathcal{C}^{\infty}(U)$-module. A locally homogeneous
Riemannian manifold is homogeneous, that is, admits a transitive Lie group
action by isometries, if it is complete and simply connected.

Let $(M,g)$ be a Riemannian manifold. Let $\operatorname{Fr}_{g}(M)$ be the
bundle of orthonormal frames on $M$. This is a principal
$\operatorname{O}(TM)$-bundle endowed with a projection
$p\colon\operatorname{Fr}_{g}(M)\to M$ and $p^{*}TM$ canonically trivialises.
In particular, every Weyl invariant $\tau\in(TM)^{\otimes
p}\otimes(T^{*}M)^{\otimes q}$ defines a vector-valued function
$p^{*}\tau\colon\operatorname{Fr}_{g}\to\mathbb{R}^{n^{2}pq}$, where $n=\dim
M$. In 1960 Singer proved ([Sing]) that there exists a constant $k$ depending
only on the dimension of $M$, such that $(M,g)$ is locally homogeneous if and
only if the maps $p^{*}(\nabla^{s}R_{g})$ are constant for every $s,0\leq
s\leq k$. Later, Prüfer, Tricerri and Vanhecke reduced Singer’s criterion to
the following elegant form ([PTV], Theorem 2.3):

######  Theorem 2.8 (Prüfer-Tricerri-Vanhecke).

Let $(M,g)$ be a Riemannian manifold. There exists a constant $k$ depending
only on $\dim M$ and a collection of smooth functions

$w_{i}\colon M\to\mathbb{R},\ i=1,\ldots,k,$

such that $(M,g)$ is locally homogeneous if and only if each $w_{i}$ is
constant. Moreover, every $w_{i}$ is a Weyl invariant in the sense of
definition 5.

Local homogeneity of $(M,g)$ in Theorem 2.8 can be replaced by global,
provided that $(M,g)$ is complete and simply connected.

The authors in [PTV] call the functions $w_{i}$ _additive Weyl invariants_. In
fact they all are of the form $\operatorname{tr}(\nabla^{\circ
s_{1}}R_{g}\otimes\ldots\nabla^{\circ s_{p}}R_{g})$. If $\dim M=n$, one can
take $k=\frac{n(n-1)}{2}$ in both Singer’s and Prüfer-Tricerri-Vanhecke
theorems. It is natural to gather all additive Weyl invariants into a single
map $W_{M}=(w_{1},\ldots,w_{k})\colon M\to\mathbb{R}^{k}$. The meaning of this
map was explained by Console and Olmos in [CO].

######  Theorem 2.9 (Console-Olmos, [CO]).

Let $(M,g)$ be a Riemannian manifold and let

$W_{M}=(w_{1},\ldots,w_{k})\colon M\to\mathbb{R}^{k}$

be as above. Then the following holds:

  * (i)

every Killing vector field on $(M,g)$ is tangent to the level sets of $W_{M}$;

  * (ii)

if $N\subseteq M$ is a non-critical level set of $W_{M}$, then $(N,g|_{N})$ is
a locally homogeneous Riemann manifold;

  * (iii)

for every $W_{M}$-non-critical point $x\in M$ there exists a neighbourhood $U$
of $x$ and a collection of Killing vector fields
$\xi_{1},\ldots,\xi_{r}\in\mathcal{C}^{\infty}(U,TM|_{U})$ that generate
$\operatorname{Ker}dW_{M}$ as an $\mathcal{C}^{\infty}(U)$-module.

In what follows we refer to the map $W_{M}$ as _Console-Olmos map_. We point
out, that if $(M,g)$ is a definable Riemannian manifold, the map $W_{M}$ is
automatically definable, as follows from Corollary 2.7.

######  Remark 2.

For technical reasons, we will need to apply Console-Olmos theorem not only to
smooth manifolds but also to manifolds of $\mathcal{C}^{m}$-class with $m$
arbitrary large. This difficulty arises only if we do not assume our o-minimal
structure $\Sigma$ to be of QA type (see Remark 1). This causes no harm since
Theorem 2.9 is still valid for $\mathcal{C}^{m}$-manifolds with
$\mathcal{C}^{m}$-Riemannian metric if $m\gg k(n)=\frac{n(n-1)}{2}$, where
$n=\dim M$ (it is sufficient that $p$-th partial derivatives of the
coefficients of the metric in local coordinates exist and are continuous for
$p<k(n)$).

##  3\. Riemannian manifolds with definable universal covers

###  3.1. Strongly bi-definable sets and bi-definable functions

######  Definition 6.

Assume that $\Sigma_{1}$ and $\Sigma_{2}$ are o-minimal structures and
$\Sigma_{2}$ is an expansion of $\Sigma_{1}$. Let $M_{1}$ (respectively
$M_{2}$) be a $\Sigma_{1}$-definable (respectively $\Sigma_{2}$-definable)
topological space. Let $\phi\colon M_{2}\to M_{1}$ be a continuous map. We say
that a subset $S\subseteq M_{1}$ is _strongly bi-definable_ if $S$ is
definable in $M_{1}$ and $\phi^{-1}(S)$ is definable in $M_{2}$. We say that a
map $f\colon M_{1}\to T$ to a $\Sigma_{1}$-definable space $T$ is _bi-
definable_ if $f$ is $\Sigma_{1}$-definable and $\phi^{*}f\colon M_{2}\to T$
is $\Sigma_{2}$-definable.

Our definition of strongly bi-definable subsets is stronger than the notion of
_bi-definable subsets_ introduced by Klingler, Ullmo and Yafaev in [KUY].
Roughly speaking, the authors in [KUY] only require a single connected
component of $\phi^{-1}(S)$ to be definable.

From now we will always assume that $M_{1}=M$ is a compact smooth manifold and
$M_{2}=\widetilde{M}$ is its universal cover. The map
$\phi=\pi\colon\widetilde{M}\to M$ is the covering map. We will not mention
explicitly the structures $\Sigma_{1}$ and $\Sigma_{2}$. In order to make the
discussion more intrinsic, one can assume that
$\Sigma_{1}=\mathbb{R}_{\mathrm{an}}$, since every compact smooth manifold is
naturally endowed with a structure of a $\mathbb{R}_{\mathrm{an}}$-definable
manifold. Therefore, our only assumption is that its universal cover
$\widetilde{M}$ is endowed with a structure of a $\Sigma$-definable smooth
manifold, where $\Sigma=\Sigma_{2}$ is an o-minimal expansion of
$\mathbb{R}_{\mathrm{an}}$.

It is easy to check that finite unions, finite intersections and complements
of strongly bi-definable sets are strongly bi-definable.

The following proposition shows that strongly be-definable subsets share very
non-trivial topological properties.

######  Proposition 3.1.

Let $i\colon N\hookrightarrow M$ be a strongly bi-definable subset. Let
$\widetilde{N}:=\pi^{-1}(N)$. The following holds:

  * (1)

for any ring R the cohomology algebra $H^{\bullet}(\widetilde{N},R)$ is
finitely generated;

  * (2)

The image of $i_{*}\colon\pi_{1}(N)\to\pi_{1}(M)$ is of finite index in
$\pi_{1}(M)$.

######  Proof: .

The item (i) follows from the finite cell decomposition for $\widetilde{N}$
(Theorem 2.1, (iii)). To see (ii) note that (i) implies that $\widetilde{N}$
has finitely many connected components and the connected components of
$\widetilde{N}$ are in bijection with $\pi_{1}(M)/i_{*}(\pi_{1}(N))$. ∎

###  3.2. Proof of Theorem A

######  Theorem 3.2.

Let $M$ be a compact real analytic manifold and $g$ a real analytic Riemann
metric on $M$. Let $\pi\colon\widetilde{M}\to M$ be the universal cover and
$\widetilde{g}:=\pi^{*}g$. Assume that $\widetilde{M}$ is a definable manifold
and that $\pi^{*}g$ is a definable Riemannian metric on it. Then there exists
a compact simply-connected definable space $\widetilde{K}$ and a smooth
definable map $\Phi\colon\widetilde{M}\to\widetilde{K}$ such that every non-
critical level set of $\Phi$ is a locally homogeneous Riemannian manifold with
respect to the metric $\widetilde{g}$. Moreover, if $x\in\widetilde{M}$ is a
non-critical point, the level sets of $\Phi$ in a neighbourhood $U$ of $x$ are
given by orbits of a connected Lie group locally acting by isometries on $U$.

######  Remark 3.

One should clarify what is meant by non-critical points and non-critical level
sets of $\Phi$, since a priori $\Phi$ is just a map to a topological space. In
fact, the target $\widetilde{K}$ of $\Phi$ is definable and by Remark 1 there
exists a dense open definable subset $U\subset\widetilde{K}$ which is a
definable $\mathcal{C}^{2}$ manifold. Therefore the differential $d\Phi$ is
well-defined on $\Phi^{-1}(U)$. We say that $x\in M$ is critical (respectively
that a level set $N=\Phi^{-1}(y),\ y\in\widetilde{K},$ is critical), if
$\Phi(x)\in U$ and $d\Phi|_{x}$ is not surjective (respectively if $y\in U$
and $d\Phi|_{x}$ is not surjective for some $x\in N$).

###### Proof of Theorem 3.2.

Let $W_{\widetilde{M}}$ to be the Console-Olmos map of
$(\widetilde{M},\widetilde{g})$. Let $K\subseteq\mathbb{R}^{k}$ be the image
of $W_{\widetilde{M}}$. By Corollary 2.7 the map $W_{\widetilde{M}}$ is
definable. In particular, $K$ is definable in $\mathbb{R}^{k}$ and the fibres
of $W_{\widetilde{M}}$ are definable subspaces of $\widetilde{M}$.

Notice that $W_{\widetilde{M}}$ coincides with the pullback $\pi^{*}W_{M}$ of
the Console-Olmos map of $(M,g)$. Indeed,
$\pi\colon(\widetilde{M},\widetilde{g})\to(M,g)$ is a local isometry,
therefore each point $x\in\widetilde{M}$ has a neighbourhood $U$, such that
$W_{\widetilde{M}}|_{U}=(\pi^{*}W_{M})|_{U}$. Hence $K=W_{M}(M)$ is compact
since $M$ is compact.

Let $p\colon\widetilde{K}\to K$ be the universal cover of $K$ and
$\Phi\colon\widetilde{M}\to\widetilde{K}$ be the canonical lifting of
$W_{\widetilde{M}}$. We get a diagram:

$\textstyle{\widetilde{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{W_{\widetilde{M}}}$$\scriptstyle{\pi}$$\textstyle{\widetilde{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{M}}$$\textstyle{K}$

Since $W_{\widetilde{M}}$ is definable, its fibres have finitely many
connected components. At the same time, $W_{\widetilde{M}}=p\circ\Phi$, so the
fibres of $p$ have finitely many connected components as well. This means that
$\pi_{1}(K)$ is finite and by Lemma 2.3 the topological space $\widetilde{K}$
can be endowed with a definable structure in such a way that
$\Phi\colon\widetilde{M}\to\widetilde{K}$ is definable. Since $\widetilde{K}$
is a finite cover of $K$ and $K$ is compact, $\widetilde{K}$ is compact as
well. The rest of the claim follows immediately from Console-Olmos theorem
(Theorem 2.9) ∎

The conclusion of Theorem 3.2 holds only for non-critical points and this
limitation cannot be ruled out. Below we present two examples when the map
$\Phi$ admits critical fibres.

######  Example 1.

Let $g$ be a Riemannian metric on the two-dimensional sphere $S^{2}$ such that

$\operatorname{Isom}(S^{2},g)\simeq\operatorname{U}(1).$

More explicitly, one can take $f\colon[-1;1]\to\mathbb{R}$ to be an analytic
function with

$\begin{cases}f(-1)=f(1)=0\\\ f(t)>0,\ -1<t<1.\end{cases}$

Let $(M,g)$ be the surface of revolution of the graph of $f$ inside
$\mathbb{R}^{3}$ with the induced metric. Then $M$ is diffeomorphic to $S^{2}$
and if $f$ is generic enough, the only isometries of $(M,g)$ are given by
rotations around the $z$-axis.

The Console-Olmos map in this case is the projection onto the $z$-axis

$W_{M}\colon M\to[-1;1].$

The non-critical fibres $W_{M}^{-1}(t),t\neq\pm 1$ are isometric to $S^{1}$
with the standard homogeneous metric (up to scaling), while the critical
fibres $W_{M}^{-1}(-1)$ and $W_{M}^{-1}(1)$ are just points.

Notice that $M$ is simply connected, therefore
$(\widetilde{M},\widetilde{g})=(M,g)$ and $(M,g)$ admits a natural structure
of a definable Riemannian manifold in $\mathbb{R}_{an}$.

######  Example 2.

The previous example formally satisfies the assumptions of Theorem A, although
not very interesting from our perspective, since in this case
$\pi_{1}(M)=\\{1\\}$. Here is a way to modify this example in order to get
examples with interesting fundamental groups.

Let $S$ be a two-dimensional sphere and $g_{S}$ be a metric on it as in
Example 1. Let $(M_{0},g_{0})$ be a compact locally homogeneous Riemannian
manifold. Take $M:=M_{0}\times S$ with the product metric $g=(g_{0}\times
g_{S})$.

The universal cover $\widetilde{M_{0}}$ of $M_{0}$ can be identified with a
homogeneous space $G/K$. The proof of [BKT], Lemma 2.1 can be easily adapted
to show that $\widetilde{M_{0}}=G/K$ carries canonical
$\mathbb{R}_{\mathrm{alg}}$-definable structure. The metric $g_{0}$ pullbacks
to a semi-algebraic homogeneous metric $\widetilde{g_{0}}$ on
$\widetilde{M_{0}}$. This implies that the universal cover of $(M,g)$ is
isometric to $(\widetilde{M_{0}}\times S,\widetilde{g_{0}}\times g_{S})$ and
is an $\mathbb{R}_{an}$-definable Riemannian manifold.

The Console-Olmos map for $(M,g)$ is the composition of the projection on $S$
and the Console-Olmos map for $S$:

$\textstyle{M=M_{0}\times
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{S}}$$\scriptstyle{W_{M}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{S}}$$\textstyle{[-1;1].}$

In particular the non-critical fibres of $W_{M}$ are isometric to locally
homogeneous manifolds $M_{0}\times S^{1}$, while the critical fibres
$W_{M}^{-1}(\\{-1\\})$ and $W_{M}^{-1}(\\{1\\})$ are isometric to
$(M_{0},g_{0})$. One can further modify the geometry of $M$ such that it does
not split as a direct product anymore: instead of taking a product metric
$g_{0}\times g_{S}$ one can take a skew-product metric
$(p^{*}_{S}e^{\phi})g_{0}\times g_{S}$ where $\phi\colon S\to\mathbb{R}$ is a
real-analytic function.

Examples 1 and 2 suggest that there is no way to get rid of the singularities
of the map $\Phi$. Nevertheless, one can find a stratification of $M$ by
strongly bi-definable locally closed subsets each of which admits a smooth
fibration by locally homogeneous manifolds. This is the content of the
following Proposition.

######  Proposition 3.3.

Let $\Sigma$ be an o-minimal structure and $(M,g)$ a $\Sigma$-definable
Riemannian manifold of class $\mathcal{C}^{m}$ for $m\gg\frac{n(n-1)}{2}$,
where $n=\dim M$. Let $\pi\colon\widetilde{M}\to M$ be a cover of $M$ and
$\widetilde{g}:=\pi^{*}g$. Assume that $(\widetilde{M},\widetilde{g})$ is a
definable Riemannian manifold. Then for every $k,\ m>k>0$ there exists a
stratification of $M$ by locally closed strongly bi-definable subsets

$\varnothing=S_{0}\subset\ldots\subset S_{r}=M,$

such that each $V_{i}:=S_{i}\setminus S_{i-1}$ is a
$\mathcal{C}^{k}$-submanifold of $M$ that admits definable locally trivial
fibrations $f_{i}\colon V_{i}\to K_{i}$ with locally homogeneous fibres onto a
definable topological space $K_{i}$. If, moreover, $\Sigma$ is of QA type, the
same holds for $k=\infty$.

Notice that we do not require $(M,g)$ to be compact anymore. We also do not
ask that $\widetilde{M}\to M$ be universal.

###### Proof of Proposition 3.3.

The proof is done by induction on $\dim M$. If $\dim M=0$, the statement is
clear. Otherwise, let $W_{M}\colon M\to\mathbb{R}^{n}$ be the Console-Olmos
map of $(M,g)$ and $K=W_{M}(M)$. This map exists (see Remark 2) and its
regular fibres are locally homogeneous. It is moreover bi-definable with
respect to the map $\widetilde{M}\to M$ since the same argument as in the
proof of Theorem 3.2 applies.

Let $K^{\circ}\subset K$ be the set of non-critical values of $W_{M}$.
Clearly, $K^{\circ}$ is an open dense definable subset and
$M^{\circ}:=W_{M}^{-1}(K^{\circ})$ is an open dense definable submanifold of
$M$. In fact, $M^{\circ}$ is strongly bi-definable since
$\pi^{-1}(M^{\circ})=W_{\widetilde{M}}^{-1}(K^{\circ})$ is definable in
$\widetilde{M}$.

Let $N:=M\setminus M^{\circ}$. This is a closed strongly bi-definable subset
of $M$. Notice that $\dim N<\dim M$. In fact, $N=S_{r-1}$ will be the first
step of our stratification.

By Lemma 2.2 we can find a stratification of $N$ by definable locally closed
subsets $N_{i},\ \varnothing=N_{0}\subset\ldots\subset N_{s}=N$.

We claim that in fact, we can choose all $N_{i}$ to be strongly bi-definable.
Indeed, one can choose $N_{s-1}$ to be the singular locus of $N$. Clearly
$\pi^{-1}(N_{s-1})$ is the singular locus of the definable subset
$\pi^{-1}(N)\subset\widetilde{M}$, hence $N_{s-1}$ is strongly bi-definable
and we proceed by induction on $\dim N$.

Now for every $i$ the set $U_{i}:=N_{i}\setminus N_{i-1}$ is a definable
Riemannian $\mathcal{C}^{k}$-manifold with definable cover
$\pi_{i}\colon\widetilde{U_{i}}=\pi^{-1}(U_{i})\to U_{i}$ and
$\pi_{i}^{*}g=\widetilde{g}|_{\widetilde{U_{i}}}$ is definable. Finally,
notice that $\dim U_{i}\leq\dim N<\dim M$. Therefore, we may apply the
induction assumption to $U_{i}$ and for each $U_{i}$ get stratifications

$\varnothing=S^{i}_{0}\subset\ldots\subset S^{i}_{r_{i}}$

with the required properties. It is left to gather all nested stratifications
$S^{i}_{j}$ into a single stratification $S_{p}$ of $M$. For this one can put
$S_{p}:=\bigcup_{\dim S^{i}_{j}=n_{p}}S^{i}_{j}$, where $\\{n_{p}\\}$ is the
set of natural numbers appearing as dimensions of the strata of $S^{i}_{j}$.

For each $(V_{i}:=S_{i}\setminus S_{i-1},g|_{V_{i}})$ the Console-Olmos map
$f_{i}:=W_{V_{i}}\colon V_{i}\to K_{i}$ has no critical fibres, therefore
defines a locally trivial fibration with locally homogeneous fibres. ∎

######  Corollary 3.4.

In the assumptions of Theorem 3.2 the manifold $M$ can be written as a
disjoint union of strongly bi-definable locally homogeneous manifolds.

######  Proof: .

Take the union of all the fibres of the maps $f_{i}$, where $f_{i}$ are as in
Proposition 3.3. ∎

Another Corollary of Theorem 3.2 is that the definability of the Riemannian
universal cover of a Riemannian manifold $(M,g)$ puts certain topological
restrictions on $M$, namely on its fundamental group.

######  Corollary 3.5.

Let $\Sigma$ be an o-minimal structure and $(U,g)$ a $\Sigma$-definable
Riemannian manifold. Let $\Gamma$ be a finitely generated group. Assume that
$\Gamma$ acts on $(U,g)$ by isometries and the action is free, cocompact and
properly discontinuous. Then $\Gamma$ is quasi-isometric to a locally
homogeneous space.

######  Proof: .

Let $M:=U/\Gamma$ and $p\colon U\to M$ the projection. The metric $g$ descends
to a Riemannian metric $\overline{g}$ on $M$. Since $U\to M$ is a
$\Gamma$-Galois cover, we have a surjective homomorphism
$\pi_{1}(M)\to\Gamma$.

Let $w_{M}\colon M\to K$ be the Console-Olmos map on $(M,\overline{g})$ and
$F\subset M$ a connected component of a non-critical fibre of $w_{M}$. Denote
by $\Gamma_{F}$ the image of the composition homomorphism
$\pi_{1}(F)\to\pi_{1}(M)\to\Gamma$. The connected components of $p^{-1}(F)$
are in bijection with $\Gamma/\Gamma_{F}$. On the other hand, $p^{-1}(F)$
consists of several connected components of a fibre of the Console-Olmos map
$w_{U}\colon(U,g)\to K^{\prime}$, hence definable. It follows that
$\pi_{0}(p^{-1}(F))=\Gamma/\Gamma_{F}$ is finite. Therefore, $\Gamma$ is
quasi-isometric to $\Gamma_{F}$.

Let $\widetilde{F}$ be a connected component of $p^{-1}(F)$. The group
$\Gamma_{F}$ acts on $(\widetilde{F},g|_{\widetilde{F}})$ by isometries and
the action is free and properly discontinuous with a compact quotient
$F=\widetilde{F}/\Gamma_{F}$. By Milnor-Švarc Lemma ([BH], Proposition 8.19)
this implies $\Gamma_{F}$ is quasi-isometric to
$(\widetilde{F},g|_{\widetilde{F}})$. By the Console-Olmos Theorem (Theorem
2.9) $(\widetilde{F},g|_{\widetilde{F}})$ is locally homogeneous. ∎

Notice that being quasi-isometric to a locally homogeneous space seems to be a
subtle property of a finitely generated group. For instance a group $\Gamma$
as in Corollary 3.5 can not be of intermediate growth.

###  3.3. The case of aspherical $M$

Now we want to prove Theorem B, which strengthens Theorem A in the case of
aspherical $M$. The main idea behind the proof is that if $M$ is aspherical,
it can have no strict closed submanifolds which are strongly bi-definable. On
the other hand, the smooth fibres of the Console-Olmos map should provide such
submanifolds, which forces the Console-Olmos map to be constant.

######  Proposition 3.6.

Let $M$ be a finite-dimensional compact connected CW-complex and $N\subseteq
M$ a closed connected subcomplex. Let $\pi\colon\widetilde{M}\to M$ be the
universal cover and $\widetilde{N}:=\pi^{-1}(N)$ the preimage of $N$ in it.
Assume that the cohomology algebra
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is finitely generated.
Then there exists a finite connected cover $q\colon M^{\prime}\to M$ and a
connected subcomplex $N^{\prime}\subseteq M^{\prime},\ q(N^{\prime})=N$, such
that the action of $\pi_{1}(N^{\prime})$ on
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is trivial.

######  Proof: .

Note that we do not assume $\widetilde{N}$ to be connected. Nevertheless,
$\widetilde{N}$ has only finitely many connected components since its
cohomology algebra is finitely generated, and

$\pi_{0}(\widetilde{N})=\pi_{1}(M)/i_{*}\pi_{1}(N)$

is finite (here $i\colon N\hookrightarrow M$ is the tautological embedding).

Let $\Gamma:=\pi_{1}(M)$. Since $\widetilde{N}$ is a $\Gamma$-Galois cover of
$N$, the action of $\pi_{1}(N)$ on
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ factors through the
natural homomorphism $\pi_{1}(N)\to\pi_{1}(M)=\Gamma$. At the same time, if
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is finitely generated, it
is a finite-dimensional $\mathbb{Z}/2\mathbb{Z}$-algebra and the group
$\operatorname{Aut}(H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is
finite. Let $G$ be the image of $\pi_{1}(N)$ in
$\operatorname{Aut}(H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}))$.
Consider the composition of homomorphisms of groups

$\pi_{1}(N)\to\Gamma\to
G\hookrightarrow\operatorname{Aut}(H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})).$

Let $q\colon M^{\prime}\to M$ be the finite $G$-Galois cover of $M$ associated
to the group homomorphism $\pi_{1}(M)=\Gamma\to
G\subseteq\operatorname{Aut}(H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}))$.
Let $N^{\prime}:=q^{-1}(N)$. Notice that $N^{\prime}$ is connected since both
$\pi_{1}(N)$ and $\pi_{1}(M)$ map to $G$ surjectively. Of course the universal
cover $\widetilde{M}\xrightarrow{\pi}M$ factors through
$M^{\prime}\xrightarrow{q}M$ and $\widetilde{N}\to N$ factors through
$N^{\prime}\to N$. At the same time, the action of $\pi_{1}(N^{\prime})$ on
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is trivial. ∎

######  Lemma 3.7.

Let $\Gamma$ be a finitely generated group and $M$ a compact manifold which is
a $K(\Gamma,1)$. Let $N\subseteq M$ be a closed submanifold. Let
$\pi\colon\widetilde{M}\to M$ be the universal cover and
$\widetilde{N}:=\pi^{-1}(N)$. Assume that
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is finitely generated.
Then $N=M$.

######  Proof: .

First of all, notice that we can replace $M$ and $N$ with $M^{\prime}$ and
$N^{\prime}$ as in Proposition 3.6 so that the action of $\pi_{1}(N)$ on
$H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is trivial. Indeed,
$M^{\prime}$ will be still aspherical, will have the same universal cover, and
$N^{\prime}=M^{\prime}$ if and only if $N=M$. Therefore, starting from now we
assume that $\pi_{1}(N)$ acts trivially on the cohomology algebra of
$\widetilde{N}$.

Since $\widetilde{N}$ is a $\Gamma$-Galois cover of $N$, the
$\mathbb{Z}/2\mathbb{Z}$-cohomology groups of $N$ can be computed using the
Cartan-Leray spectral sequence ([McCl], Theorem $8^{bis}.9$):

$E_{2}^{i,j}=H^{i}(\Gamma,H^{j}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}))\implies
H^{i+j}(N,\mathbb{Z}/2\mathbb{Z}).$

Since we assume that the $H^{j}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$’s are
trivial $\pi_{1}(N)$-modules and the action of $\pi_{1}(N)$ on $\widetilde{N}$
factors through $\Gamma$, each $H^{j}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$
is a finite-dimensional $\mathbb{Z}/2\mathbb{Z}$-vector space with trivial
$\Gamma$-action and
$H^{i}(\Gamma,H^{j}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}))=H^{i}(\Gamma,\mathbb{Z}/2\mathbb{Z})\otimes_{\mathbb{Z}/2\mathbb{Z}}H^{j}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$.
Let $n:=\dim M$. The compact manifold $M$ is a $K(\Gamma,1)$ space, thus
$H^{n}(\Gamma,\mathbb{Z}/2\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$ and
$H^{n+i}(\Gamma,\mathbb{Z}/2\mathbb{Z})=0$ for $i>0$.

Let $r$ be maximal integer such that
$H^{r}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is non-zero. Consider

$E_{2}^{n,r}=H^{n}(\Gamma,H^{r}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}).$

It is non-zero, since it equals

$H^{n}(\Gamma,\mathbb{Z}/2\mathbb{Z})\otimes
H^{r}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})=H^{r}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}).$

At the same time

$E_{2}^{n-2,r+1}=H^{n-2}(\Gamma,H^{r+1}(\widetilde{N},\mathbb{Z}/2\mathbb{Z}))=0$

and

$E_{2}^{n+2,r-1}=H^{n+2}(\Gamma,H^{r-1}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})=0.$

Therefore $E_{2}^{n,r}$ survives in the spectral sequence and
$H^{n+r}(N,\mathbb{Z}/2\mathbb{Z})$ is non-zero. We assumed that $N$ is a
closed manifold, so $H^{p}(N,\mathbb{Z}/2\mathbb{Z})$ vanishes for $p>\dim N$.
It follows that $n+r\leq\dim N$. Of course $\dim N\leq\dim M=n$ and it follows
that $n+r=\dim N$, that is, $\dim N=\dim M$ and $r=0$. Since $M$ was assumed
to be connected and $N$ is a closed submanifold, we deduce that $N=M$. ∎

######  Corollary 3.8.

Let $M$ be a compact aspherical real analytic manifold and
$\pi\colon\widetilde{M}\to M$ its universal cover. Assume that $\widetilde{M}$
admits a structure of a manifold definable in an o-minimal structure. Then $M$
has no strongly bi-definable strict submanifolds.

######  Proof: .

If $N\subseteq M$ is strongly be-definable,
$\pi^{-1}(N)=:\widetilde{N}\subseteq\widetilde{M}$ is definable. In
particular, $H^{\bullet}(\widetilde{N},\mathbb{Z}/2\mathbb{Z})$ is finitely
generated by Theorem 2.1, (ii). The Corollary then follows from Lemma 3.7. ∎

######  Theorem 3.9.

Let $(M,g)$ be a compact analytic Riemannian manifold and
$\pi\colon\widetilde{M}\to M$ its universal cover. Assume that $\widetilde{M}$
admits a structure of a manifold definable in an o-minimal structure and the
metric $\widetilde{g}:=\pi^{*}g$ is definable. Assume further that
$\widetilde{M}$ is contractible, that is, $M$ is aspherical. Then
$(\widetilde{M},\widetilde{g})$ is a Riemannian homogenous space.

######  Proof: .

Let $W_{M}\colon M\to\mathbb{R}^{k}$ be the Console-Olmos map. Let $N\subseteq
M$ be a connected component of a non-critical level set of $W_{M}$. Then $N$
is a smooth submanifold of $M$ and in the same time, it is strongly bi-
definable: $\pi^{-1}(N)$ is a union of several connected components of a level
set of the Console-Olmos map $W_{\widetilde{M}}$ of
$(\widetilde{M},\widetilde{g})$. However, by Corollary 3.8 this implies $N=M$
and the map $W_{M}$ is constant. Since $(\widetilde{M},\widetilde{g})$ is
complete and simply connected, it is homogeneous by Prüfer-Tricerri-Vanhecke
theorem (Theorem 2.8). ∎

######  Corollary 3.10.

Let $(M,g)$ be a compact Riemannian manifold with a non-positive sectional
curvature. Let $\pi\colon\widetilde{M}\to M$ be the universal cover and
$\widetilde{g}:=\pi^{*}g$. Assume that $\widetilde{M}$ admits a structure of a
$\Sigma$-definable manifold for some o-minimal structure $\Sigma$ such that
$\widetilde{g}$ is $\Sigma$-definable. Then $(M,g)$ is locally homogeneous.

######  Proof: .

The universal cover $\widetilde{M}$ is contractible by Cartan-Hadamard Theorem
([BH], Ch. 4), so the statement follows from Theorem 3.9. ∎

###  3.4. Applications to the classical Kollár-Pardon problem

Using Theorem 3.9 we prove the original Kollár-Pardon Conjecture in the
special case where $X$ is aspherical and possesses a bi-definable Kähler
metric.

######  Corollary 3.11.

Let $X$ be a smooth algebraic variety and $\pi\colon\widetilde{X}\to X$ be its
universal cover. Assume that $\widetilde{X}$ admits a structure of a definable
complex manifold in some o-minimal structure. Assume moreover, that $X$ is
aspherical and admits a _bi-definable Kähler metric_ , that is, a Kähler
Hermitian metric $h=g+\sqrt{-1}\omega$ such that $\pi^{*}h$ is definable on
$\widetilde{X}$. Then $(\widetilde{X},\pi^{*}h)$ is a Kähler homogeneous
manifold. In particular, Conjecture 1 holds for $X$.

######  Proof: .

Let $\widetilde{h}=\pi^{*}h$ and
$\widetilde{h}=\widetilde{g}+\sqrt{-1}\widetilde{\omega}$. Here
$\widetilde{g}$ is a Riemannian metric on $\widetilde{X}$ which is the real
part of $\widetilde{h}$, and $\widetilde{\omega}$ is a symplectic form which
is the imaginary part of $\widetilde{h}$. Being the real part of a definable
tensor, $\widetilde{g}$ is definable. Theorem 3.9 implies that
$(\widetilde{X},\widetilde{g})$ is homogeneous as a Riemannian manifold. Let
$\xi$ be a Killing vector field on $\widetilde{X}$. Since the form
$\widetilde{\omega}$ is harmonic with respect to $\widetilde{g}$, its Lie
derivative along $\xi$ vanishes. This implies that any one-parameter subgroup
of diffeomorphisms of $\widetilde{X}$ that preserves $\widetilde{g}$,
preserves also $\widetilde{\omega}$. Hence it preserves the complex structure
operator $J_{\widetilde{X}}$ as well. We deduce that if $G$ is a Lie group
that acts by isometries and transitively on $\widetilde{X}$, its action
preserves the complex structure. Therefore, $\widetilde{X}$ is homogeneous not
only as Riemannian but also as a complex (and as a Kähler) manifold. ∎

###  3.5. Open questions

It seems that one can consider a Kollár-Pardon-type Conjecture with regard to
any geometric structure. Informally speaking, the corresponding conjecture
should sound as follows. Let $M$ be a compact smooth manifold and
$\pi\colon\widetilde{M}\to M$ be its universal cover. Assume that
$\widetilde{M}$ is definable in some o-minimal structure and $M$ is endowed
with a _geometric structure_ which gives rise to a definable geometric
structure on $\widetilde{M}$. What can be said about the geometry of
$\widetilde{M}$? Under which conditions is it true that there exists a Lie
group action on $\widetilde{M}$ with compact Hausdorff quotient which
preserves this geometric structure?

We do not make here the notion of a geometric structure precise, although
there are different ways of doing this, see e.g. [KN]. One can ask the
following naïve version of the same question:

######  Question 1.

Let $M$ be a compact smooth manifold and $\tau$ a tensor on $M$. Let
$\pi\colon\widetilde{M}\to M$ be its universal cover. Assume that
$\widetilde{M}$ admits a definable structure such that $\pi^{*}\tau$ is
definable. Is it true that there exists a definable simply connected space $K$
and a definable smooth map $\Phi\colon\widetilde{M}\to K$, such that
$\operatorname{Ker}d\Phi$ is locally generated by vector fields preserving
$\tau$?

######  Question 2.

In the assumptions of Question 1, is it true that if $\widetilde{M}$ is
contractible, there exists a Lie group $G$ acting transitively on
$\widetilde{M}$ and preserving $\tau$?

We do not expect this to hold for an arbitrary tensor $\tau$, although we are
not aware of counterexamples. If $\tau$ is a Riemannian metric, the
affirmative answer is given by our Theorems A and B.

If $\tau$ is an integrable complex structure, Kollár-Pardon conjecture is a
particular case of Question 1, but Question 1 is much more general in this
case. For example, a Hopf surface
$H=(\mathbb{C}^{2}\setminus\\{0\\})/\mathbb{Z}$ is a compact complex manifold
which is not algebraic, although its universal cover is definable. Notice that
the methods of [KP] and [CHK] are based on deep algebro-geometric and Hodge-
theoretic techniques and fail in the non-Kähler setting.

It would be interesting to study the same question in the cases when $\tau$ is
a symplectic structure or an integrable para-complex structure.

Notice that once the affirmative answer to Question 1 is established,
Corollary 3.8 together with the arguments of the proof of Theorem 3.9 imply
the affirmative answer to Question 2.

Another possible direction of generalisation is to consider a manifold $M$,
which is not necessarily compact, but definable in some o-minimal structure
(this motivates the generality that we chose in Definition 6). The first step
in this direction would be to find an analogue of the Claudon-Höring-Kollár
Conjecture [CHK] and to understand smooth complex quasi-projective varieties
with algebraic universal cover. We are going to address this question in our
subsequent work [Rog].

## Bibliography

  * [BBT] B. Bakker, Y. Brunebarbe, J. Tsimerman. o-minimal GAGA and a conjecture of Griffiths. Invent. Math., Volume 232, Issue 1 (2023)
  * [BH] M. R. Bridson, A. Haefliger. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer-Verlag, Berlin, 1999.
  * [BKT] B. Bakker, B. Klingler, J. Tsimerman. Tame topology of arithmetic quotients and algebraicity of Hodge loci. J. Amer. Math. Soc. 33 (2020), 917-939
  * [CH] B. Claudon, A. Höring. Compact Kähler manifolds with compactifiable universal cover. Bull. Soc. Math. France, 141 (2013), no. 2, 355-375
  * [CHK] B. Claudon, A. Höring, J. Kollár. Algebraic varieties with quasi-projective universal cover. J. Reine Angew. Math. 679: 207-221 (2013).
  * [CO] S. Console, C. Olmos. Level sets of scalar Weyl invariants and cohomogeneity. Transactions of the American Mathematical Society, Vol. 360, No. 2 (Feb. 2008), pp. 629-641
  * [Cos] M. Coste. An Introduction to O-minimal Geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000)
  * [vdD] L. van den Dries. Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.
  * [vdDM] L. van den Dries, C. Miller. Geometric categories and o-minimal structures. Duke Math. J. 84 no.2, (1996), 497-540.
  * [Gab] A.M Gabrielov. Projections of semi-analytic sets. Funkt. Ana. i Prilozen 2 (1968), 18-30
  * [EKPR] P. Eyssidieux, L. Katzarkov, T. Pantev, M. Ramachandran. Linear Shafarevich conjecture. Ann. Math. (2), 176(2012)(3):1545–1581.
  * [Ghys] E. Ghys. Feuilletages holomorphes de codimension un sur les espaces homogènes complexes. Annales de la faculté des sciences de Toulouse 6e série, tome 5, no 3 (1996), p. 493-519
  * [KN] S. Kobayashi, K. Nomizu. (1963) Foundations of Differential Geometry. Vol. 1, Wiley, New York.
  * [KP] J. Kollár, J. Pardon. Algebraic varieties with semialgebraic universal cover. J. Topol. 5 (2012), no. 1, 199–212.
  * [KUY] B. Klingler, E. Ullmo, A.Yafaev. Bi-algebraic geometry and the André-Oort conjecture. Proceedings of 2015 AMS Summer Institute in Algebraic Geometry, PSPMS 97-2, AMS, 2018, 319-360
  * [McCl] J. McCleary. A User’s Guide to Spectral Sequences.Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press (2000).
  * [PTV] F. Prüfer, F. Tricerri, L.Vanhecke. Curvature invariants, differential operators and locally homogeneity. Transactions Amer. Math. Soc. 348 No. 11 (1996), 4643-4652.
  * [Rog] V. Rogov. Quasi-projective varieties with algebraic universal cover. _to appear_.
  * [RSW] J.-P. Rolin, P. Speissegger, A.J.Wilkie. Quasianalytic Denjoy-Carleman classes and o-minimality. J.Amer.Math.Soc. 16 (2003), no.4, pp. 751-777
  * [Shaf] I. R. Shafarevich. Basic algebraic geometry. Springer Study Edition. Springer-Verlag, Berlin-New York (1977). Translated from the Russian by K. A. Hirsch, Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974.
  * [Sing] I.M. Singer. Infinitesimally homogeneous spaces. Comm. Pure Appl. Math., 13 (1960), 685–697.
  * [Speis] P. Speissegger.The Pfaffian closure of an o-minimal structure. vol. 1999, no. 508, 1999, pp. 189-211.
  * [Wil] A. J. Wilkie. Model completeness results for expansions of the ordered field or real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), no.4, 1051-1094