The local fundamental group of a Kawamata log terminal singularity is finite

We prove a conjecture of Koll\'ar stating that the local fundamental group of a klt singularity $x$ is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of $x$ is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.


Introduction
We work over the field C of complex numbers. A log pair is an algebraic variety X together with a boundary divisor 0 ≤ ∆ < 1 of the form ∆ = ∆ ′ + ∆ ′′ , with 0 ≤ ∆ ′′ and ∆ ′ = (1 − 1/m i )∆ i is a sum of prime divisors ∆ i , whose coefficients satisfy m i ∈ Z >1 .
A Kawamata log terminal or klt singularity is a point x ∈ (X, ∆), such that for a log resolution f : Y → X, locally around x, the discrepancies a i , namely the coefficients of the exceptional divisors E i in the formula We call a log pair (X, ∆) weakly Fano, if it has only klt singularities and −(K X + ∆) is big and nef. The local fundamental group of a normal singularity x ∈ X is π loc 1 (X, x) := π 1 (B \ x) = π 1 (Link(x)), where B is the intersection of X with a small euclidean ball around x and the link Link(x) is the boundary ∂B. It is a deformation retract of B \ x and so π loc 1 is well defined. The following conjecture is due to Kollár [Kol11;Kol13]. π reg 1 is the more natural notion for non-isolated x. In particular, the proof of our two main theorems would not be possible considering only π loc 1 . In [Sti18, Thm. 2.2.6], Theorem 1 was proven for the profinite completionπ reg 1 . Stibitz also gave an example [Sti17, Ex. 2] of a non-isolated (non-klt) singularity, where allπ loc 1 are finite, butπ reg smooth locus of X: Cl(X) fin ∼ = H 1 (X sm , Z).
In [Bra19], it was shown that for weakly Fano varieties and klt quasicones an iteration of Cox rings is finite. That is, one takes the Cox ring ofX, which is possible since in both cases, this space is a Gorenstein canonical quasicone [Bra19, Thm. 3] -and iterates this procedure. After finitely many steps, one gets a quasietale quotient Z → X by a solvable reductive group, that is a group of the form G = (C * ) m ⋊ S. Here the iterated characteristic space Z is factorial and S is a finite solvable group. The iteration of Cox rings is reflected by the derived series of G. The quotient of Z by the normal subgroup (C * ) m yields a universal finite solvable cover Z/(C * ) m → X. Thus Theorems 1 and 2 yield the following solvable version of Corollary 4.
Corollary 5. Let X be a weakly Fano variety or a klt quasicone. Let Z → X be the iterated characteristic space with general fiber of the form (C * ) m ⋊ S. Then the finite part S is isomorphic to the solubilization of π 1 (X sm ).
A corollary merely of the definition of the regional fundamental group is inspired by a result of Serre [Ser58,Prop. 15], saying that any finite group is the fundamental group of a smooth projective variety.
Corollary 6. Let G be a finite group. Then G has a complex linear representation with no pseudoreflections. In particular, there exists a quotient singularity (X, x) = C n /G, such that π reg 1 (X, x) = G. Proof. Let G be a finite group and V a complex linear faithful representation. If V has a reflection, consider the sum V + V . If this representation of G has a reflection g, then consider the restricted representation V | g + V | g of the subgroup g , containing a pointwise fixed hyperplane H. Since this representation is reducible, by (the proof of) [JY12, Thm. 1], one of the copies of V | g is contained in H. This is a contradiction, since V was faithful. The quotient (X, x) := (V + V )/G is thus ramified only in codimension two, so π reg 1 (X, x) = G. The proof of Theorems 1 and 2. As done by [Xu14] in the case of theétale fundamental group, we will prove Theorems 1 and 2 simultaneously by a local-toglobal induction. One induction step is represented by the following two theorems.
The global-to-local part Theorem 8 has been proven by Tian and Xu in [TX17, Le. 3.1, 3.2] for π loc 1 . Then in [TX17,Le. 3.4], they deduce finiteness of π reg 1 of a klt singularity from finiteness of π loc 1 for all lower dimensional klt singularities. Unfortunately, there is a small gap in the proof, when the Seifert-van Kampen theorem is applied to certain tubular neighbourhoods of a Whitney stratification.
A careful analysis is taken out in Section 11 of the present paper. In fact, it turns out that this task is equally hard as trying to prove Theorem 7 with the same methods and assuming only finiteness of π loc 1 instead of π reg 1 . So in order for the induction to work, we really need the π reg 1 -version of Theorem 8. When we realized that we cannot use [TX17,Le. 3.4], Tian and Xu proposed to us to modify [TX17,Le. 3.1] for a direct proof avoiding their Lemma 3.4. After analyzing Lemma 3.1 in Section 11, we carry out this modification in Section 12 and thus are able to prove Theorem 8 in full generality.
The main part of the present paper is the proof of Theorem 7. So we have to prove finiteness of an orbifold fundamental group π 1 (Y sm , D ′ | Ysm ). In contrast to the proofs of simply connectedness of Fano manifolds using Atiyah's L 2 -index theorem, we encounter two main difficulties. Firstly, Y sm is not compact. Secondly, the orbifold version of the L 2 -index theorem is more subtle, since for a universal orbifold cover X → X , the L 2 -index on X is not equal to the Euler characteristic on X , as there are contributions from orbifold points, see [TX17,Sec. 4.1]. The problems can be seen in Tian and Xu's proof of Theorem 7 in the special case of 3-dimensional Fano varieties with canonical singularities [TX17,Thm. 4.1]. Thus we are mildly sceptical about the possibility of proving Theorem 7 in full generality using the orbifold L 2 -index theorem.
The solution is the following. As we already mentioned, the proof of simply connectedness of weakly Fano varieties X of Takayama [Tak00] manages to avoid the L 2 -index theorem and instead relies on the so called Γ-reduction or Shafarevich map, independently constructed by Campana and Kollár in [Cam94] and [Kol93] for compact Kähler manifolds and normal proper varieties respectively. Roughly said, it parameterizes maximal subvarieties of X with finite fundamental group. Takayama uses it to construct an L 2 -section of a certain line bundle on the universal cover of X. By the work of Gromov [Gro91], the existence of such a section means that if π 1 (X) is infinite, there are many sections of the corresponding line bundle on X.
Fortunately, the Γ-reduction is also available for orbifolds due to Claudon [Cla08]. But then we still have the problem that Y sm is not compact. This is where the hypothesis of Theorem 7 comes into play (and thus the very reason why we cannot directly prove Theorem 2 but have to carry out the induction). Consider a log resolution f : X → Y of the n-dimensional weakly Fano pair (Y, D ′ + D ′′ ) with exceptional prime divisors E i . Then a very small loop γ i around a general point e i of E i can be pushed forward to Y sm and there it lies in the smooth locus of a very small neighbourhood of the image of e i , which is a klt singularity. Thus by the hypothesis saying that the regional fundamental groups of klt singularities of dimension n are finite, we know that γ i is of finite order . So the normal subgroup of π 1 (f −1 (Y sm \ supp(D ′ ))) generated by all γ mi i is trivial. Thus π 1 (Y sm , D ′ ) is isomorphic to the orbifold fundamental group of the smooth compact orbifold (X, f −1 * D ′ + (1 − 1/m i )E i ). Then the remaining task in order to prove finiteness of the latter is to transfer the techniques of [Tak00] to the orbifold setting, which is done in Part 1 of the present paper.
Possible alternative ways of proof. We consider two alternative approaches to prove Theorems 1 and 2.
As we mentioned before, simply connectedness of Fano manifolds can be proven by showing that they are rationally connected, from which follows that their fundamental group is finite. The notion of rational connectedness can also be formulated for orbifolds, and also here, from rational connectedness of a smooth orbifold X = (X, ∆) (in the sense of Campana) follows finiteness of the orbifold fundamental group of X [Cam11a, Cor. 12.25]. So rational connectedness of the orbifold (X, f −1 * D ′ + (1 − 1/m i )E i ) supported on a log resolution of a weakly Fano pair (Y, D) would yield an alternative proof of Theorem 7. But the definition of rational connectedness for orbifolds is subtle [Cam11a, Déf. 6.11, Rem. 6.12] and we have no idea how to prove it for the orbifold (X, f −1 . A different approach -which would yield a direct induction-free proof of Theorem 2 in any dimension -is the following. In Proposition 9.2, we prove finiteness of the orbifold fundamental group of (X, f −1 * D ′ + (1 − 1/m i )E i ) for any choice of m i , supported on a log resolution X of a weakly Fano pair (Y, D ′ + D ′′ ). Instead of arguing with the induction hypothesis of finiteness of the regional fundamental group of klt singularities, one also could try to show that if is already finite. It is known that there are finitely presented infinite groups with trivial profinite completion, but our situation is slightly different.
By passing to some ramified finite cover of (Y, D ′′ + D ′′ ) (which is still weakly Fano), we can assume thatπ 1 (Y sm , D ′ | Ysm ) is trivial, which means thatπ 1 (Y sm , D ′ | Ysm ) has no proper normal subgroups of finite index. So the normal subgroup γ m1 1 , . . . , γ m1 1 is the whole group π 1 (X \ supp(f −1 * D ′ + E i )) for any choice of m i . This seems to be a strong property and one could ask if infinite finitely presented groups of this kind even exist.
But they do. Mark Sapir sent us an example: the Thompson group T , which is simple, finitely presented, and infinite. It is generated by three elements, and two of them have infinite order, so all normal subgroups generated by any choice of powers of these elements are the whole group T .
We want to remark that π 1 (X \supp(f −1 * D ′ + E i )) is a so-called quasiprojective group, that is the fundamental group of a smooth quasiprojective variety. These groups satisfy some strong properties, see e.g. [DPS09, Sec. 1.5]. We do not know if it is possible to show that the negation of the above property is among them.
Structure of the paper. In Part 1 of the paper, we prove Theorem 7. While the proof itself happens in Section 9, in Sections 1 to 8, we review the definitions of complex orbifolds and basic related notions -e.g. of orbibundles, orbisheaves, and orbimetrics -but transfer also more sophisticated concepts for complex manifolds to the orbifold case.
In Part 2, we prove Theorem 8. After shortly recalling the notion of Whitney stratifications in Section 10, we carefully analyze Lemmata 3.1 and 3.4 of [TX17] in Section 11. In the last Section 12, we prove Theorem 8 by modifying Lemma 3.1 appropriately.
Thanks go also to Mark Sapir for providing the example of the Thompson group T , to Andreas Demleitner for a discussion on the same topic, and to Mirko Mauri for pointing out a mistake in the definition of E 0 in the proof of Theorem 8.

Complex orbifolds and orbimaps
The definition of an orbifold -under the name of V -manifold -goes back to Satake [Sat56] in the real and Baily [Bai56] in the complex case. The notion was then rediscovered by Thurston [Thu79], who finally gave it the name orbifold. Complex orbifolds are locally -but not necessarily globally -quotients of smooth complex manifolds, which makes them complex analytic spaces with an additional local quotient structure. We will use the following definition, see e.g. [CM13, Sec. 2.1].
Definition 1.1. Let X be a complex analytic space of dimension n. An orbifold chart on X is a tuple (U ′ , G, ϕ, U ), where U ′ ⊆ C n is a connected open complex analytic subspace, G is a finite subgroup of the automorphism group of U ′ , and ϕ : U ′ → U ⊆ X is a proper and finite holomorphic map to the open subspace U ⊆ X, such that ϕ • g = ϕ for every g ∈ G. We require the induced map U ′ /G → U to be a homeomorphism.
An injection betweeen two orbifold charts (U ′ , G, ϕ, U ) and (V ′ , H, ψ, V ) is a holomorphic embedding λ : An orbifold atlas on X is a family U = {(U ′ i , G i , ϕ i , U i )}, such that X = i U i , and, for two charts (U i , G i , ϕ i , U i ) and (U j , G j , ϕ j , U j ), and any x ∈ U i ∩U j , there is a third chart (U k , G k , ϕ k , U k ), such that x ∈ U k ⊆ U i ∩ U j , and there are injections An atlas U is a refinement of another atlas V, if for every chart V ′ of V, there is an injection U ′ → V ′ from a chart from U. An atlas U is maximal, if it has no nontrivial refinement.
Let U be a maximal orbifold atlas on X. Then we call the pair X := (X, U) a (complex) orbifold.
We sometimes will call U ′ → U a local uniformization and G the local uniformizing group. By the slice theorem, there is always an atlas consisting of linear charts (C n , G, ϕ, U ), such that G is a subgroup of the unitary group U (n) [MP97,Rem. (5)].
The actions of the local uniformizing groups G ⊂ Aut(U ′ ) and the injections λ : U ′ → V ′ behave well with respect to each other. Consider for example a chart (U ′ , G, ϕ, U ) and an element g ∈ G, then since ϕ • g = ϕ holds, G : U ′ → U ′ is an injection. Moreover, the following holds [MP97, Rem. In particular, for g ∈ G the composition µ : The following is a direct consequence of Lemma 1.2, which we haven't found in the literature. Corollary 1.3. Let X := (X, U) be a complex orbifold and (U ′ , G, ϕ, U ) and orbifold chart on X . Let λ, µ : U ′ → U ′ an injection from (U ′ , G, ϕ, U ) to itself. Then there is a g ∈ G, such that λ = g.
Let (U ′ , G, ϕ, U ) be an orbifold chart around x ∈ U ⊆ X, and p ∈ ϕ −1 (x). Up to conjugacy, the isotropy subgroup G p is determined by x. Moreover, according to Lemma 1.2, if (V ′ , H, ψ, V ) is another chart around x and q ∈ ψ −1 (x), then G p ∼ = H q , so the following is well defined up to isomorphy [BG08, Def. 4.1.2]. Definition 1.4. Let X = (X, U) be an orbifold and x ∈ X. For an orbifold chart (U ′ , G, ϕ, U ) around x and p ∈ ϕ −1 (x), we call G x := G p the isotropy group of x. We call those x ∈ X with G x = {e G } orbifold regular points, and all points with G x = {e G } orbifold singular points.
Note that the singular points (in the usual sense) of the complex analytic space X are a subset of the orbifold singular points of X = (X, U). In particular, an orbifold singular point x is a smooth point of X if and only if G x is a reflection group (for some and in consequence for all local uniformizations around x). A direct consequence of the well-definedness of the isotropy group of points of X is the following stricter version of the already mentioned [MP97, Rem. (5)], which again we haven't found in the literature. Lemma 1.5. Let X := (X, U) be a complex orbifold and x ∈ X with isotropy group G x . Then there is an orbifold chart (C n , G x , φ, U ) around x, such that φ −1 (x) = 0 ∈ C n and G x acts as a subgroup of U (n).
Campana in [Cam04] introduced another notion of orbifold for pairs (X, ∆), where ∆ is a certain divisor on a complex analytic space X. We will see that under certain conditions -which we will encounter in our setting -, his notion is equivalent to that of a complex orbifold we gave in Definition 1.1. In order to distinguish between the two notions, we will call such pairs (X, ∆) geometric orbifolds, following [Cam11b]. Definition 1.6. A geometric orbifold is a pair (X, ∆), where X is a complex analytic space and ∆ a divisor of the form where we assume that the m i are integers greater than zero and the δ i are prime divisors.
Definition 1.7. We say that the geometric orbifold (X, ∆) is smooth, if X is a smooth complex manifold and supp(∆) is a simple normal crossing divisor.
Remark 1.8. A smooth geometric n-dimensional orbifold (X, ∆) always can be represented by a complex orbifold in the sense of Definition 1.1. Consider a local chart C n → V ⊂ X of X as an analytic space. Then after suitable adjustment, in this chart, ∆ is given by . . , x n ), which is nothing but the quotient of the action of Z/m 1 Z × . . . × Z/m k Z acting diagonally on C n by roots of unity. If a local analytic chart of X does not intersect ∆, we can take it as orbifold chart. The compatibility of these charts is straightforward. We call this the canonical orbifold structure of a smooth geometric orbifold.
The local uniformizing subgroups of the canonical orbifold structure of a smooth geometric orbifold are reflection groups. This can be seen from the fact that the analytic space X is smooth or directly from the explicit orbifold charts in Remark 1.8. The analogy between geometric orbifolds (X, ∆) and complex orbifolds X actually goes further [BGK05, Sec. 2], but we are only interested in the particular case of smooth geometric orbifolds here.
We close this section with the definition of orbimaps. The original definitions from [Sat56; Bai56] do not in general induce morphisms of orbibundles and orbisheaves -which we will define in Sections 2 and 3 respectively. This has been realized in [MP97] and additional compatibility criteria have been introduced to remedy this problem. This led to the equivalent notions of 'strong' [MP97] and 'good' [CR04] orbifold maps. Since we will work with orbibundles and -sheaves, for us the definition of a holomorphic orbimap includes the additional compatibility criteria. That is to say, our maps are always 'strong'/'good', compare [BG08, Def. 4.1.8].
Definition 1.9. Let X = (X, U) and Y = (Y, V) be two complex orbifolds. A map f : X → Y is called a holomorphic orbimap if the following hold: (1) For any x ∈ X, there are orbifold charts ( of charts on Y in the sense of item (1), and any injection λ ji : Remark 1.10. In the setting of Definition 1.9, consider an injection λ ji : So the µ ji are determined only up to multiplication with elements of H j . Now let i = j and consider an injection λ ji = g : U ′ i → U ′ i given by an element g ∈ G i . In contrary to the second statement of Lemma 1.2, now there is not necessarily But if we fix an assignment λ ji → µ ji between injections on X and Y fulfilling the requirements of Definition 1.9, then for each i, we get group homomorphisms A system of charts on X and Y fulfilling Item (1) of Definition 1.9 together with an assignment λ ji → µ ji between injections of such charts is called a compatible system in [CR02,Sec. 4.4]. If a map between orbifolds allows a compatible system, it is called 'good' [CR02,Def. 4.4.1]. The problem is that one map may allow different compatible systems, as the following easy example shows [CR02, Ex. 4.4.2b].
As [CR02, Le. 4.4.3] states, for any compatible system, there is a unique pullback of orbibundles. But for different compatible systems as in Example 1.11, these pullbacks may differ.
Nevertheless, the only holomorphic orbimaps we encounter are orbifold (universal) covers. These always have a unique compatible system, since they are locally trivial in the orbifold sense.

Orbibundles
In this section, we define orbifold vector bundles or orbibundles as a reasonable generalization of (complex) vector bundles over (complex) manifolds. The probably most important notion is that of the orbifold tangent space T X and related constructions.
Definition 2.1. Let X = (X, U) be a complex orbifold. An orbifold vector bundle or orbibundle of rank k over X is a collection of vector bundles π ′ i : i by (ordinary) vector bundle maps, such that: (1) Each π ′ i is G i -invariant, so that the following diagram is commutative for any g ∈ G i : Choosing small enough orbifold charts on X , we can assume that E ′ i ∼ = U ′ i ×C k and the action of G i on U ′ i ×C k is diagonal and acting as a subgroup of GL(k) on the second factor. Then since π ′ i is equivariant, we have a unique 'projection' π i , so that the following diagram commutes: Now we can glue the sets E i in the following way, stemming from the gluing condition on X : let x ∈ U i ∩ U j = ∅. Then according to Definition 1.1 there is a chart . Gluing E i and E j acccording to this data results in an orbifold E with underlying space E and an orbimap π : E → X , which is locally given by the equivariant Probably the easiest but still important example of an orbibundle is the trivial line bundle, given by trivial line bundles together with a trivial action of G i on the second factor. Then clearly E i ∼ = U i × C and the total space E is just X × C.
Example 2.4. Another very important example is that of the tangent orbibundle T X . It can be constructed in the following natural way [BG08, Ex. 4.2.10]. On a chart U ′ i , take the tangent bundle T U ′ i ∼ = U ′ i ×C n and for any injection λ ji : is given by λ ji on the first factor and the Jacobian Jac(λ ji ) on the second one. This construction obviously generalizes to the cotangent orbibundle T * X , (symmetric, antisymmetric) tensor orbibundles et cetera [BBFMnT17].

Remark 2.5.
Locally around x ∈ X, the fiber π −1 (x) ⊆ T X is not isomorphic to C n , but is holomorphic to a small neighbourhood of x ∈ X, because in a local chart, the actions of g ∈ G i on U ′ i and of Jac(g) on On the other hand, even if (X, ∆) is a smooth geometric orbifold with canonical orbifold structure X , the underlying space of T X is not necessarily the ordinary tangent space T X. Now having defined orbibundles, we have to ask ourselves what is a reasonable definition of (holomorphic) sections of these. Obviously for an orbibundle π : E → X , a section of E should be a holomorphic orbimap s : Of course the local sections must be compatible with injections as well, so that we arrive at the following definition [BG08, Def. 4.2.9].
Definition 2.6. Let π : E → X be an orbibundle. Then a holomorphic section of E is given by any of the two equivalent definitions: (1) s : X → E is a holomorphic orbimap satisfying π • s = id X .
(2) A collection of holomorphic sections s i : U ′ i → E ′ i of the local bundles over charts of X , such that for any injection λ ji : U i → U j , the following diagram commutes: Remark 2.7. Equivariance of the local sections s i obviously is the right requirement, otherwise they would not glue to a global section s : X → E. When (locally) the action of G i on the fiber is trivial, then of course equivariance means nothing else than invariance, as it is the case for the trivial line bundle from Example 2.3. Sections of this bundle clearly are in a one-to-one-correspondence with holomorphic orbimaps from X to C endowed with the trivial orbifold structure. So they are a good candidate for a structure orbisheaf on X , see Section 3.
In order to get coherent sheaves on the underlying space X, nonetheless, we have to deal with invariant sections of line bundles The other way round works quite as well. If the underlying space X of a complex orbifold is a manifold, then line bundles and Weil divisors coincide and we can pull them back to the local uniformizations, so they give orbibundles on X . Now for example, we can ask ourselves which divisor on X gives the canonical orbibundle K X .
Example 2.8. To answer this question, we just have to pull back a top differential form in a local uniformization We clearly have Thus we have to multiply with functions that along a ramification divisor x i = 0 are allowed to have poles of order at most m i − 1. On X, this means we have to multiply with functions that on the branch divisors z i = 0 have poles of order

Orbisheaves
We first introduce the notion of an orbisheaf following [MP97] and [BG08, which is functorial. We are mainly interested in sheaves of modules over a reasonable structure sheaf, so first, we have to define such structure sheaf, see [BG08,Def. 4 On a complex orbifold X , by O X we will always denote the structure sheaf of holomorphic functions.
It is clear that this definition neither will give us a sheaf on the underlying space X nor it coincides with the holomorphic sections in the sense of Definition 2.6 of the trivial orbibundle, see Remark 2.2. We have to use local G i -invariant sections of such sheaves and glue them together over X [BG08, Lemma 4.2.4]. We will often work with these invariant sections of orbisheaves (or invariant local sections of orbibundles, which do not in general coincide with the equivariant sections from Definition 2.6). We will always denote sheaves on X coming from invariant local sections of orbisheaves F by F X . In particular (O X ) X ∼ = O X holds for the structure sheaves. Now recall that the functor V → V G taking a vector space with an action of a finite group G to its G-invariant subspace is exact. This means in particular that for a coherent orbisheaf F of O X -modules, the sheaf F X made up of (locally) G i -invariant sections is a coherent sheaf of O X modules. As exact sequences are preserved, it also makes sense to formulate orbisheaf cohomology, orbifold Dolbeault cohomology et cetera, see Section 6.

Orbimetrics
In this section, we consider metrics on orbifolds, or orbimetrics. By the preceding considerations, it is clear that these should be (invariant) metrics on the local uniformizations U ′ i of an orbifold X = (X, U). Definition 4.1. Let X = (X, U) be a complex orbifold and E → X an orbibundle. A Hermitian orbimetric on E is a collection of Hermitian metrics h ′ i on the local On the other hand, if the underlying space X of a complex orbifold X = (X, U) is smooth, then (usual) divisors or line bundles on X can be pulled back to the local uniformizations and thus define orbibundles as we have seen in Example 2.8 in the case of the canonical divisor. Now when the underlying space X is even a Kähler manifold (X, ω) with Kähler form ω (in the usual sense), then Claudon [Cla08, Prop. 2.1] has constructed a Kähler orbiform ω ′ out of ω in the following way.
we can assume that on U i with coordinates z 1 , . . . , z n , the Kähler form ω is given by Analogous to Example 2.8, the pullback under ϕ is where m j = 1 if x j = 0 is not the restriction of a divisor ∆ j . This form is clearly degenerate at the origin if ∆ ∩ U i = ∅. In particular, it is no Kähler orbiform. Now consider the global (1, 1)-form ω ∆ with values in O X (2∆) given by Locally we can assume that s j is just given by z j , so the pullback by ϕ is In general, k < n, so this form is denenerate as well. Now combine these two to a form ω ′ = ω + ω ∆ . Then on the one hand, ω ′ is smooth on X \ supp(∆), and for c ∈ R >0 small enough, ω ′ ≥ cω as currents. On the other hand, the pullback What we need here is a stronger result. Consider the following situation: X = (X, ∆) is a complex orbifold with X a manifold. Let L be an ample line bundle on the manifold X. Then according to [BG08,Thm. 4.3.14] and the preceding paragraph therein, the first orbifold Chern class of L is just the usual first Chern class with respect to X. Thus L (or the pullback to local uniformizations) defines an ample (or positive) line orbibundle.  In particular, the form ω ′ given by Finally note that we can integrate n-forms by a partition of unity and by setting 5. The orbifold universal cover and the Γ-reduction Definition 5.1. The orbifold fundamental group of a geometric orbifold X = (X, ∆) is the quotient π 1 (X, ∆) := π 1 (X \ supp(∆))/ γ mi i , i ∈ I , where for each i ∈ I, γ i is a small loop around a general point of the divisor ∆ i . Associated to the orbifold fundamental group, there is the notion of orbifold universal cover π : X → X . It is a ramified Galois cover between complex analytic spaces,étale over X \ supp(∆).
Remark 5.2. Let X = (X, ∆) be a smooth geometric orbifold. Then over a (sufficiently small) orbifold chart (U ′ i , G ′ i , ϕ i , U i ) of X as in Remark 1.8, the preimage under the orbifold universal cover π : X → X has connected components V i , such that V i has a local uniformization ( In particular, since G i is abelian, H i is so as well and V i only has toric singularities. Rem. 1.2]. So in a sense, the universal cover is locally trivial as we expect from a cover. As we mentioned before, the analogy between geometric and classical orbifolds not only holds if the underlying space is smooth. In particular, on the underlying space X of a classical complex orbifold X = (X, U) one always can define a divisor ∆, such that the geometric orbifold (X, ∆) has the canonical orbifold structure X = (X, U), see [BGK05,p. 561]. In particular, this holds for the orbifold universal cover X . But we do not need the structure of a geometric orbifold on X here.
An important observation for us will be that if X is a complex analytic space, ∆ 1 , . . . , ∆ m are smooth prime divisors on X with normal crossings, and small loops γ i around general points of ∆ i are of finite order m i in π 1 (X \ (∆ 1 ∪ . . . ∪ ∆ m )), then Note that by the Hopf-Rinow-Theorem for orbifolds [Car19, Thm. 4.2.2], the orbifold covers of a complete orbifold (with respect to an orbimetric ω ′ , cf. Section 4), are complete with respect to the pullback metric (since orbifold geodesics can be lifted). In particular, the orbifold universal cover of a compact orbifold with a Hermitian orbimetric is complete with respect to the pullback orbimetric.
An important ingredient for us is the Γ-reduction or Shafarevich map. This construction has been introduced by Kollár for proper normal projective varieties [Kol93, Def. 1.4] and independently by Campana for compact Kähler manifolds [Cam94, Thm. 3.5, Def. 3.8]. Formulated on the universal cover X of a compact Kähler manifold X, it says that there is a unique almost holomorphic fibration γ : X Γ( X), such that any compact irreducible subvariety of X through a very general point x ∈ X is contained in the fiber γ −1 ( γ(x)). The general fibers of γ are exactly the maximal compact subvarieties of X. The action of π 1 (X) on X descends to Γ( X) and thus by quotienting induces an almost holomorphic fibration γ : X Γ(X), of which the fibers are the maximal subvarieties with finite fundamental group. In turn, the connected components of the preimages of such fibers are exactly the fibers of γ.
Theorem 5.3. Let X = (X, ∆) be a compact smooth geometric Kähler orbifold and π : X → X its orbifold universal cover. There are almost holomorphic fibrations γ : X Γ( X ) and γ : X Γ(X ) , such that the diagram commutes and the following hold: (1) If V ⊆ X is a smooth subvariety meeting ∆ transversally, such that the image of π 1 (V, ∆| V ) in π 1 (X, ∆) is finite, and V meets the fiber of γ through a very general point, then V is contained in this fiber.
(2) Every compact irreducible subvariety of X through a very general point x ∈ X is contained in the fiber γ −1 ( γ(x)).
Remark 5.4. Theorem 0.2 of [Cla08] is formulated only on the universal cover, while [Cam11a, Thm. 12.23] is formulated on the orbifold X itself. The connection between the both is [Cla08, Le. 2.2]. The third item has not been formulated in the orbifold case, but the argument at the end of the proof of Proposition 2.4 in [Kol93] works here as well.

Dolbeault and L 2 -cohomology for Kähler orbifolds
Following [BBFMnT17, Sec. 5], we can define orbifold Dolbeault cohomology for complete Kähler orbifolds (X = (X, U), ω) in the following way. Denote by Ω p,q X the sheaf of (p, q)-orbiforms, defined by the usual (p, q)-forms on the local uniformizations. The locally invariant sections define the sheaf Ω p,q X on the udnerlying space X and we denote the space of global sections by Ω p,q X (X). The exterior derivative and the Dolbeault operators d = ∂ + ∂ are well defined, with ∂ : Ω p,q X → Ω p+1,q X , ∂ : Ω p,q X → Ω p,q+1 X . Definition 6.1. The (p, q)-th orbifold Dolbeault cohomology group is defined by .
If E → X is a holomorphic orbibundle, then one can similarly define the Dolbeault complex (Ω p,q X (X, E), ∂ E ) of (p, q)-orbiforms with values in E as well as Dol- .

L 2 -vanishing for orbifolds
The singular Hermitian metrics from Section 4 will be more useful to us than just for constructing Kähler orbiforms from positive line bundles on the underlying space. Note that the functions φ may only be given locally, so in this notation φ can rather be seen as a collection of locally defined functions. On the other hand, it may still be possible to express φ globally by certain sections as e.g. in Proposition 4.4.
A plurisubharmonic or shortly psh function is defined by certain semicontinuity properties, see e.g. On the other hand, we have the Nadel coherence theorem [Dem96, Prop. (5.7)], stating that I(φ) is a coherent sheaf if φ is psh. This can be easily transformed to the orbifold setting. First let us define the analogue of the multiplier ideal sheaf following [BG08, Def. 5.2.9].
Definition 7.2. Let X = (X, ∆) be a complex orbifold and let (L, H = he −φ ) be a singular Hermitian orbibundle on X . The multiplier ideal orbisheaf I X (φ) is the orbisheaf defined on local uniformizations U ′ i → U i by The orbifold version of Nadel's coherence theorem follows from the standard version since the functor taking G i -invariant sections is exact by finiteness of G i . Thus we have: Theorem 7.3 (Nadel's coherence theorem for orbifolds). Let X = (X, ∆) be a complex orbifold and (L, H = he −φ ) be a singular Hermitian orbibundle on X . Then the (pushforward of the) multiplier ideal orbisheaf I X (φ) is a coherent sheaf of O X -modules on X.
The next step to go now is the Nadel vanishing theorem. The orbifold version is the following [DK01, Thm. 6.5].
Theorem 7.4 (Nadel's vanishing theorem for orbifolds). Let (X , ω ′ ) be a Kähler orbifold, that is a complex orbifold X = (X, ∆) with a Kähler orbiform ω ′ . Let (L, H = he −φ ) be a singular Hermitian orbibundle on X , where h is a smooth Hermitian orbimetric on L. Assume that there exists a constant c ∈ R >0 , such that iΘ(L, H) ≥ cω ′ . If K X ⊗ L is an invertible sheaf on X, then H q (X, K X ⊗ L ⊗ I X (φ)) = 0 for q ≥ 1.
Several things have to be noted. First, as Kollár and Demailly stress in the paragraph after [DK01, Thm. 6.5], for the orbifold version, it is really necessary that K X ⊗ L is an invertible sheaf on X. This is because on the one hand, the above tensor product K X ⊗ L ⊗ I X (φ) means first taking the usual tensor product on local uniformizations U ′ i , then taking G i -invariant sections, and finally the direct image sheaves on U i . On the other hand, the statement is obtained by L 2 -estimates -with respect to the weight e −φ -of sections of K X ⊗ L on X \ supp(∆).
It turns out that for us these L 2 -estimates are even more important than the statement of Theorem 7.4. As they are not explicitly stated in [DK01], we refer to [Dem96, Cor.

Maximal compact subspaces of orbifold universal covers
This section is merely a translation of [Tak99,Sec. 4] to the orbifold case. Following [Tak99, Sec. 3B], for any complex analytic space X, by a subvariety W ⊆ X, we mean an irreducible reduced complex subspace. By a maximal compact subspace Z ⊆ X, we mean a not necessarily reduced nor irreducible compact subspace, such that every subvariety W ⊆ X with Z ∩ W = ∅ is contained in Z.
We have the following (compare [Tak99, Prop. 4.1]). To prove the proposition, we need the following three lemmata.
Proof. This is basically the proof of [Tak99, Le. 4.2] translated to the orbifold setting. Since {x i } i∈N has no accumulation point, we can take a subsequence, which by abuse of notation we again denote by {x i } i∈N , such that there exists By the bounded geometry of X as orbifold cover of the compact orbifold X , compare [Cla08, Le. 2.1], there is a constant c ∈ R >0 such that for any i and the standard metric It is obvious that φ is U (n)-and thus G i -invariant for all i ∈ N. The multiplier ideal orbisheaf I X (φ) defines a complex subspace of X, which is exactly {x i } i∈N . Now since ( L, h) is positive, there is a 0 ∈ Z >1 , such that i∂∂ log ω n + a 0 ω is positive, that is K ⊗(−1) X ⊗ L ⊗a0 is positive. Moreover, due to the definition of φ and the bounded geometry property from above, there is b 0 ∈ Z >1 , such that vanishes for every m > m 0 := a 0 + b 0 due to Proposition 7.5. This means that the map Proof. Since Y is non-compact, we can take a sequence of points {x i } i∈N in Y with no accumulation points in X (since Y is closed). By Lemma 8.2, we can take a subsequence, which again we denote by {x i } i∈N , such that there exists m ∈ N with the map H 0 (2) X, L ⊗m → ℓ i=1 O X /M X,xi being surjective for all ℓ ∈ N. We consider now the exact sequence The last term has dimension d ∈ Z ≥0 , since Z is compact. Using Lemma 8.2, we . Also s| Y is not the zero section over Y due to the choice of the s i 's.
Let α ∈ Q >0 . Following [Tak99], by a multivalued L 2 -section of L ⊗α , we denote a section s of L ⊗α , such that there is p ∈ Z >0 with pα ∈ Z and s p ∈ H 0 (2) ( X, L ⊗pα ). We can then define the pointwise length (2) ( X, L ⊗k ) tends to zero at infinity. In particular, in the above setting, if s p ∈ H 0 (2) ( X, L ⊗pα ), then s q ∈ H 0 (2) ( X, L ⊗qα ) as well for q ∈ Z ≥p . Proof. First, note that there exists a positive integer q, such that, for every m ∈ Z >0 , we have , where redZ is the reduced structure of Z.
Fix such an integer q. By Lemma 8.3, there is m 1 ∈ Z >0 and a nonzero L 2 -section s ′ 1 ∈ H 0 (2) (X, L ⊗m1 ⊗I m1qN redZ ). We set s 1 := s ′ 1 1/(m1N ) , which is a multivalued section of L ⊗1/N ⊗ I q redZ . Now if there is no non-compact irreducible component Y of (s 1 ) 0 intersecting U , set s := {s 1 }. If there is a non-compact irreducible component Y of (s 1 ) 0 intersecting U , then apply Lemma 8.3 for Z and this Y . It follows that there is an L 2 -section s 2 of L ⊗1/N such that s m2N 2 ∈ H 0 (2) (X, L ⊗m2 ⊗ I m2qN redZ ) for a positive integer m 2 , such that s 2 | Y is not the zero section. Now we pass to (s 1 ) 0 ∩ (s 2 ) 0 and check if there is a non-compact irreducible component intersecting U . If yes, proceed again with Lemma 8.3 to obtain a section s 3 et cetera. Since U is relatively compact, after a finite number of steps, we have L 2 -sections s 1 , . . . , s k ∈ L ⊗1/N satisfying the requirements of the lemma. In this section we prove Theorem 7, which makes up the local-to-global part of the induction in the proof of our main theorems. First we recall the necessary definitions.
Definitions. We call a pair (Y, D) of a normal complex algebraic variety Y and an effective Q-divisor D = d j D j on Y with K Y + D being Q-Cartier a log pair. For a log pair (Y, D), in the following we will often decompose D = D ′ + D ′′ , where D ′′ ≥ 0, and D ′ = (1 − 1/m i )D i is a sum of prime divisors D i , whose coefficients satisfy m i ∈ Z >1 .
We say that a birational divisorial contraction f : X → Y is a log resolution of the pair (Y, D), if X is smooth and f −1 We call a log pair (Y, D) Kawamata log terminal or klt shortly, if 0 < d i < 1 and there exists a log resolution f : X → Y , such that we can write where the a i , which we call log-discrepancies, are greater than zero. Note that We call a projective variety Y weakly Fano, if there exists an effective Q-divisor D = d j D j on Y , such that (Y, D) is klt and −(K Y + D) is big and nef.
The statement of Theorem 7 to prove now is the following: assume that ndimensional klt-singularities have finite regional fundamental group, then n-dimensional weakly Fano pairs (Y, D ′ + D ′′ ) have finite orbifold fundamental group π 1 (Y sm , D ′ ).
Compact orbifolds supported on a log resolution. The proof of the above statement relies on the following two propositions, which essentially say that for a log resolution f : X → Y of a weakly Fano pair (Y, D ′ + D ′′ ) with exceptional divisor E, for any admissible Q-divisor ∆ supported on supp(E) ∪ supp(f −1 * D ′ ), the smooth geometric orbifold (X, ∆) has finite fundamental group.
Proposition 9.1. Let (Y, D ′ +D ′′ ) be a weakly Fano pair, with D ′ = (1−1/e j )D j for some e j ∈ Z >1 and D ′′ ≥ 0. Let f : X → Y be a log resolution with exceptional prime divisors E 1 , . . . , E k . For arbitrary m i ∈ Z >0 , consider the smooth geometric orbifold X := (X, ∆ : where c i := a i − 1/m i and the a i > 0 are the log-discrepancies. Consider L as an orbibundle on X . Then the orbifold universal cover π : X → X has a nontrivial L 2 -section ν ∈ H 0 (2) ( X , K X ⊗ π * L). Proof. Consider a pair (Y, D ′ + D ′′ ), a log resolution f : X → Y and arbitrary m i ∈ Z >0 as in the proposition. Set ∆ : Consider the smooth geometric projective orbifold X = (X, ∆). Then the orbifold canonical divisor of X is defined by K X := K X + ∆, see Section 2. By the above ramification formula, we can write where c i := a i − 1/m i > −1 and −f * (K Y + D) is big and nef. Now define and L := −f * (K Y + D) + ∆ ′ . With these definitions, the ramification formula becomes K X + L = E. Now since L is the sum of a big and nef and a simple normal crossing Q-divisor with coefficients strictly between 0 and 1, L = A + ∆ ′′ , where A is an ample Q-divisor and ∆ ′′ = ∆ ′ + N , where N is a very small effective Q-divisor. This means in particular that the pair (X, ∆ ′′ ) is klt.
Since A is an ample Q-divisor, there is a positive integer a, such that, by Proposition 4.4, A ⊗a is a positive line orbibundle with orbimetric h A . Denote by ω := iΘ(A ⊗a , h A ⊗a ) the corresponding Kähler orbiform. Then (X , ω) is a compact Kähler orbifold. Now following [Tak99, Sec. 3B], take a multivalued canonical section σ ∆ ′′ of O X (∆ ′′ ), that is m∆ ′′ is an integral effective divisor for some positive integer m, and div(σ m ∆ ′′ ) = m∆ ′′ . Take in addition a Hermitian metric h m∆ ′′ of O X (m∆ ′′ ) and define a function |∆ ′′ | := |σ m ∆ ′′ | Now consider the orbifold universal cover π : X → X of X = (X, ∆) and the Γ-reduction γ : X Γ(X ) from Theorem 5.3. Let F be a very general fiber of the restriction γ| X 0 : X 0 → Γ(X ) 0 . The preimage π −1 (F ) is a disjoint countable union of copies of a finite cover of F . So the restriction of π to a connected component F of this preimage is a finite cover π| F : F → F ,étale over F \ supp(∆).
Denote by L, A, and ∆ ′′ the pullbacks (as orbibundles) of L, A, and ∆ ′′ by π respectively. In the same manner, denote the pullback of functions, orbimetrics, and orbiforms by a tilde. In particular, ( X , ω := π * ω) is a complete Kähler orbifold.
If γ(F ) ⊂ U ⊂ Γ(X ) 0 is a sufficiently small neighbourhood of γ(F ) biholomorphic to the unit ball B 1 (0) ⊆ C n , then the connected component U of π −1 • γ −1 (U ) containing F is a relatively compact open neighbourhood of F biholomorphic to F × U .
Take a positive integer N > a. By Lemma 8.4, we have k ∈ Z >0 and multivalued L 2 -sections s = {s i } i=1,...,k of A ⊗a/N , such that their set of zeros U ∩(s) 0 is compact and I(s) is contained in the ideal sheaf I F . By shrinking U if necessary, we can assume that U ∩ (s) 0 = F . Now we want to define a singular Hermitian metric on L = A ⊗ ∆ ′′ . Recall that we have a Hermitian metric h m∆ ′′ of the line bundle ∆ ′′⊗m . Define This is a singular Hermitian metric of L. Since N > a, the curvature iΘ( L, H s ) ≥ (1/a − 1/N ) ω is positive. By klt-ness of (X, ∆ ′′ ), we have so I(H s ) = I(s) as in the proof of Proposition 8.1. Now consider the (n, 0)-form (ρ • γ • π) · σ from above. The (n, 1)-form τ := ∂(ρ • γ • π) · σ = σ∂(ρ • γ • π) is ∂-closed and square-integrable with respect to H s and ω, because its support lies in the relatively compact U \ F and the poles of H s lie in F . By Proposition 7.5, there is a L-valued (n, 0)-form υ on X , with ∂υ = τ , again square-integrable with respect to H s and ω. Now set ν := (ρ•γ •π)· σ −υ. Applying ∂, we see that ν is holomorphic, and since υ is integrable with respect to H s , we have υ| F ≡ 0 and thus ν| F is not trivial.
As we know from the proof of Proposition 8.1, |s| 2 is bounded. So there is a positive constant c, such that Hs dV ω .
Moreover, since (ρ • γ • π) · σ is supported on U , it is square-integrable with respect to h A ⊗a and ω as well. So ν is a nontrivial section of H 0 (2) ( X , K X ⊗ L). Proof. Define the divisor L as in Proposition 9.1. Then by Proposition 9.1, there is a nontrivial section ν ∈ H 0 (2) ( X , K X ⊗ L). For any k ∈ Z >0 , the power ν ⊗2k is a global L 1 -section of (K X ⊗ L) ⊗2k . This is due to the fact that X = X /π 1 (X ) is compact, see [Gro91,p. 286]. Here it is not necessary that π 1 (X ) acts freely on the complex analytic space X. The Poincaré series converges and defines a holomorphic π 1 (X )-invariant section of (K X ⊗ L) ⊗2k . Now consider products ki P (ν ⊗2ki ) of these sections. Then [Gro91,Prop. 3.2.A] says that if π 1 (X ) is infinite there exists at least one κ and at least two partitions κ = k i and κ = k ′ i , such that is a nonconstant meromorphic π 1 (X )-invariant function on X . Thus ki P (f ⊗2ki ) and k ′ i P (k ⊗2k ′ i ) define two linearly independent sections of H 0 (X, (K X ⊗ L) ⊗2κ ). But on the other hand, since E and thus also 2κE is effective f -exceptional, we have dim H 0 (X, O X (2κE)) = 1. But we have seen that E is linearly equivalent to K X + L (seen as a divisor). This is a contradiction, so π 1 (X ) is finite.
Proof of Theorem 7. Now by the induction hypothesis, for an n-dimensional weakly Fano pair (Y, D ′ + D ′′ ), we can relate finiteness of π 1 (Y sm , D ′ ) to the finiteness of the fundamental group of a compact orbifold supported on a log resolution, which finishes our proof.
Proof of Theorem 7. Let (Y, D ′ + D ′′ ) be an n-dimensional weakly Fano pair and assume that n-dimensional klt singularities have finite regional fundamental group. Consider a log resolution f : . Now let γ i be a very small loop around a general point e i of E i . Then γ i can be pushed forward to Y sm and there it lies in the smooth locus of a very small neighbourhood of the image of e i , which is a klt singularity y i . So by the induction hypothesis, f * γ i has finite order m i in π reg 1 (Y, y i ). Therefore, it has finite order in Y sm ∼ = X \ E. Thus γ mi i , i ∈ I is trivial and by Definition 5.1 of the orbifold fundamental group But the latter is finite by Proposition 9.2. Thus π 1 (Y sm , D ′ ) is finite as well and we are done.

Part 2. Global to local
In order to complete the induction, we have to show in this part that if (n − 1)dimensional weakly Fano pairs (Y, D ′ + D ′′ ) have finite orbifold fundamental group π 1 (Y sm , D ′ | Ysm ), then n-dimensional klt singularities have finite regional fundamental group. We do this by modifying an argument of [TX17]. First let us briefly recall the notions related to Whitney stratifications and their systems of tubular neighbourhoods.

Whitney stratifications
We refer to [Gor81, Sec. 2] for the following definitions. Let X be a complex analytic space of dimension n embedded in a smooth complex manifold M . In our context, we can always assume M ∼ = P m (C) for some m ≥ n.
To a submanifold N of M , we associate a tubular neighbourhood T N in the following way: choose a Riemannian metric h on the normal bundle E → N and fix δ ∈ R >0 . Then T N is the image of a smooth embedding φ : E δ → M , where E δ := {v ∈ E; |v| h < δ} and φ takes the zero section of E identically to N . For 0 < ε < δ, we define T N (ε) := φ({v ∈ E; |v| < ε}) and its boundary A Whitney stratification of X ⊆ M is a filtration by closed subsets X 0 ⊂ X 1 ⊂ . . . ⊂ X n = X, such that the connected components of X i \ X i−1 are locally closed i-dimensional submanifolds of M , the i-dimensional strata. If A and B are strata with A ∩ B = ∅, then A ⊂ B and we write A < B. Any Whitney stratification allows a system of compatible tubular neighbourhoods of the strata, so called control data. For two strata A < B, the tubular distance functions and projections from above have to satisfy π A • π B = π A and ρ A • π B = ρ A . Moreover, for some 0 < ε, the boundaries S N (ε) have to satisfy certain transversality properties. Namely if A 1 , . . . , A µ and B 1 , . . . , B η are two disjoint collections of strata, then S A1 (ε) ∩ . . . ∩ S Aµ (ε) and S B1 (ε) ∩ . . . ∩ S Bη (ε) are transversal and they are also transversal to any other stratum C.
One can check these properties easily in the pictures of the next section.

The work of Tian and Xu
It was shown in Lemmata 3.1 and 3.2 of [TX17], that if (n−1)-dimensional weakly Fano pairs (Y, D ′ +D ′′ ) have finite orbifold fundamental group π 1 (Y sm , D ′ | Ysm ), then n-dimensional klt singularities have finite local fundamental group.

Tian and Xu's Lemma 3.4.
In [TX17,Le. 3.4], finiteness of the regional fundamental group of an n-dimensional klt singularity (X, x) is deduced from finiteness of the local fundamental group of k ≤ n-dimensional klt singularities. Unfortunately, there is a gap in the proof, as described in the following.
The proof uses a Whitney stratification of a neighbourhood of the singularity, together with a system of tubular neighbourhoods of the strata. Such a tubular neighbourhood minus the stratum itself is a fiber bundle over the stratum and the fiber over a point has finite fundamental group by assumption, since it is (homeomorphic to) a slice through a pointed neighbourhood of the point, which is klt.
Then the Seifert-van Kampen theorem is invoked to merge all these tubular neighbourhoods together to a neighbourhood of x with the whole singular locus removed. The way the tubular neighbourhoods fit together is depicted below.
x A Here the singular locus X sing = A ∪ {x} has the zero-dimensional stratum {x} and the one-dimensional stratum A, together with their tubular neighbourhoods T x and T A . Note that their boundaries S x and S A are transversal. The regional fundamental group of x is nothing else than the fundamental group of U x := T x \ X sing . We have T 0 are fiber bundles over A ∩ T x with the fiber having finite fundamental group due to the assumption and trivial fundamental group, respectively, h 2 has finite kernel. Now the Seifert-van Kampen theorem says that π 1 (T 0 x ) -which is finite by assumption -is the quotient of the free product π 1 (U x ) * π 1 (T A ∩ T x ) by the normal subgroup N generated by all elements h 1 (g)h 2 (g) −1 , where g ∈ T 0 A ∩ T x . The intersection of N with π 1 (U x ) ⊆ π 1 (U x ) * π 1 (T A ∩ T x ) is nothing but the image under h 1 of the kernel of h 2 . Now in the proof of [TX17, Le. 3.4], it is argued that since h 2 has finite kernel, π 1 (U x ) can not be infinite. This is not necessarily true, since h 1 does not have to be surjective, that is, h 1 (ker h 2 ) does not have to be a normal subgroup, and its normal closure can be infinite.
In fact, it is not hard to see that deducing finiteness of the regional fundamental group from finiteness of the local fundamental group is equally hard as deducing finiteness of π 1 (Y sm , D ′ ) for weakly Fano pairs (Y, D ′ + D ′′ ) from it. This is since Y has a Whitney stratification as well and since we know that Y is simply connected, the above arguments could be applied in the exact same manner.
But as Tian and Xu suggested, one could try to modify Lemma 3.1 of [TX17] to directly prove finiteness of the regional fundamental group. So let us have a close look at this lemma.
On the other hand, we can write ∆ E = D ′ + D ′′ , where D ′ is the 'different of zero' as defined in [Cor92,.5] by Then [Cor92,Prop. 16.6] says that on the one hand, D ′ is of the form (1 − 1/m i )D ′ i , and on the other hand, the D ′ i are singular strata in Y of maximal dimension, in particular, locally analytically at a general point of It is shown in [TX17, Le. 3.1], that the fundamental group of a certain open subset V 0 of a neighbourhood U of E surjects to the fundamental group of U 0 := U \ E, which is nothing else but the local fundamental group of x.
Remark 11.1. It is not explicitly mentioned in [TX] that (E, D ′ + D ′′ ) is log Fano but one has to consider the orbifold fundamental group π 1 (E sm , D ′ ) -where D ′ and D ′′ only coincide when the boundary ∆ on X is trivial. But by [Cor92,Prop. 16.6] this does not matter. This can be seen directly by observing that klt-ness of general points of singular strata of codimension two does not depend on the boundary, since two-dimensional klt singularities are log-terminal. Now if we could show that π 1 (V 0 ) even surjects to π 1 (U sm \ E) -which is nothing but the regional fundamental group of x, we would be done. So how does the proof of [TX17, Le. 3.1] look like and how can it be modified?
After the blowup f : Y → X, extracting the Kollár component E = f −1 (x), a Whitney stratification of E is chosen, with biggest stratum E 0 := E sm . Choose ε > 0. From the tubular neighbourhoods of the strata, a neighbourhood U (ε) of E in Y is defined as follows, after [Gor81, Def. 7.1]: Note that strictly speaking, we have to embed Y in a smooth manifold M and the tubular neighbourhoods are neighbourhoods in M , not in Y . So as Goresky points out, the closure U (ε) in M is a manifold with corners, the corners being the intersections S A1 (ε) ∪ . . . S A k (ε), where A 1 < . . . < A k are incident strata. Nevertheless, we will denote the intersections of all these objects with Y in the same way. We can draw a similar picture as before to depict the situation.
Here x ∈ E is the only stratum apart from E 0 . The closure U (ε) has boundary (S E0 (ε)∪S x (ε))\(T E0 (ε)∪T x (ε)) and corners S E0 (ε)∩S x (ε). Now following [Gor81, Sec. 7], we construct a deformation retraction ψ : U (ε) → E as follows. First for every stratum consider a retraction r A : T A (2ε) \ A → S A (2ε), such that the following hold whenever A < B are incident strata: These retractions have been constructed in [Gor78, Sec. 2] under the name families of lines. From these one can define homeomorphisms h A : ρ A (p)). Now fix a smooth nondecreasing function q with q(t) = 0 for t ≤ ε, q(t) > 0 for t > ε and q(t) = t for t ≥ 2ε. Now define • H AN , where A 1 , . . . , A N are the strata of E in any order. Let ψ :=ψ U(ε) . The restriction of ψ to E is homotopic to the identity [GM88, p. 220]. For each stratum A and η > 0 define the η-interior as in [GM83,p. 180]. With this definition, we see that ψ(π −1 A (A ε )) = A and ψ(A \ A ǫ ) ⊆ B<A B. In our picture, ψ collapses the (darkgray) T x (ε) to x and the (lightgray) π −1 E0 (E ε 0 ) to E 0 : Now what is shown in [TX17, Le. 3.1], is that the canonical group homomorphism surjective. This is done by adding to V 0 closures in U (ε) of the sets V 0 A := ψ −1 (A) \ E for all strata A, starting with those of highest dimension. Since all boundaries are collared, it is possible to invoke the Seifert-van Kampen theorem, in order to show that π 1 (V 0 ) → π 1 (V 0 ∪ V 0 A ) is surjective, and so on. This is done by successive fiber bundle decompositions of V 0 A . In order to really see what happens, we need a higher-dimensional picture with more strata.
Here, the horizontal plane depicts the divisor E, having three strata: the origin o, a one-dimensional stratum A, and the big open stratum E 0 . It holds {o} < A < E 0 . Also the ( boundaries of the) tubular neighbourhoods T N (ε) of these strata are depicted, and their union is the open neighbourhood U (ε) of E. Now we have V := ψ −1 (E 0 ) = π −1 E0 (E ε 0 ) = T E0 (ε) \ (T A (ε) ∪ T o (ε)), which is depicted below, and in order to get V 0 we have to subtract E.
In a first step, the closure of V 0 A := ψ −1 (A) \ E has to be added to V 0 . The Seifert-van Kampen theorem can be used to compute the fundamental group of the resulting space. Taking into account that all these spaces have collared boundaries, we can assume the intersection of V 0 and V 0 A is ∂V 0 A ∩ T E0 (ε), which is denoted L 2 in [TX17]. Then if π 1 (L 2 ) → π 1 (V 0 A ) is surjective, so is π 1 (V 0 ) → π 1 (ψ −1 (A ∪ E 0 ). But L 2 is a fiber bundle over A ε , with fiber L 2 homotopic to π −1 A (a) ∩ ∂V \ E for some a ∈ A ε , which is depicted in the cross-section through a below.
Compare also the map f in [GM83, Sec. 6.1] and [GM88, Part II, Sec. 6.13.1]. In our picture, we see that approximately the fibers Z A,t for t ∈ D are horizontal sections of V A .
So setting Z a,t := Z A,t ∩ π −1 A (a), we see that L is a Z a,t -bundle over D 0 and L 2 is a ∂Z a,t -bundle over D 0 . So we have to show that π 1 (∂Z a,t ) → π 1 (Z a,t ) is surjective. But Z a,t is homotopic to a collared affine analytic space of dimension c, where c is the codimension of A in E, see [GM88, Part II, Prop. 6.13.5]. Since c ≥ 2, it follows that π 0 (∂Z a,t ) → π 0 (Z a,t ) is an isomorphism and π 1 (∂Z a,t ) → π 1 (Z a,t ) is surjective.
Repeating this procedure for all strata of E, Lemma 3.1 of [TX17] is proven.
12. Finiteness of the regional fundamental group In this section, we prove Theorem 8, the global-to-local part of our induction, by modifying the proof of [TX17, Le. 3.1] appropriately.
Proof of Theorem 8. As in Lemma 3.1 of [TX17], we start with an n-dimensional singularity x ∈ X of a klt pair (X, ∆). We assume that the smooth locus of (n − 1)-dimensional weakly Fano pairs has finite orbifold fundamental group. Let f : Y → X be a plt blowup extracting the Kollár component E = f −1 (x). Consider a Whitney stratification of Y , such that the biggest stratum is Y sm and for k ≤ n−2, the k-dimensional strata are the relative interiors -with respect to Y sing -of the irreducible k-dimensional components of the singular locus Y sing . This induces a Whitney stratification of E by cutting each stratum with E. Fix this stratification.
Let 0 < ε << 1 and U (ε) be a neighbourhood of E as constructed in the previous section. Then π reg 1 (X, x) ∼ = π 1 (U (ε) \ (E ∪ Y sing )). Again as in the previous section, construct the retraction ψ : U (ε) → E. Note that for any stratum of Y sing there are two possibilities. Either it is of dimension (n − 2) and it is contained in E, and thus is of codimension one in E. Or it's intersection with E is of codimension greater or equal to two in E. Now define Then if we choose ε small enough, it is clear that Y sing ∩ U (ε) lies in U (ε) \ ψ −1 (E 0 ). The situation is depicted below.
Here Y sing has a 2-dimensional stratum Y A that meets E in the 1-dimensional stratum A and a 1-dimensional stratum Y o that meets E in the 0-dimensional stratum o (A and o as denoted in the last section). Now as in the proof of [TX17, Le. 3.1], start with V 0 E0 := ψ −1 (E 0 ) \ E. But instead of adding (the closures of) V 0 N := ψ −1 (N ) \ E to V 0 E0 for all strata N of E \ E 0 , now we have to add V sm N := V 0 N \ Y sing in order to arrive at U (ε) \ (E ∪ Y sing ). Now everything works the same way as in [TX17, Le. 3.1], untill we arrive at the Z N,t for some stratum N , compare the explanations in the previous section.
Here, now we have to show that π 1 (∂Z n,t \ Y sing ) → π 1 (Z n,t \ Y sing ) is surjective for an element n of the ε-interior N ε in order to finish the proof. In our picture, for N = A, the situation looks like this.
Note that in general, the singular locus Y sing can have nontrivial intersection with ∂Z N,t . This is the case for example for N := {o}, the zero-dimensional stratum, where ∂Z o,t has nontrivial intersection with the 2-dimensional stratum Y A of Y sing .
In [TX17, Proof of Le. 3.1], it was argued that Z a,t is homeomorphic to an affine complex analytic space with collared boundary. This is due to [GM88, Part II, Prop. 6.13.5]. Looking into the proof therein, we see that this statement is obtained by using Thom's first isotopy lemma to show that Z a,t is homeomorphic to the intersection of Z A,t with smooth submanifolds of M transversal to A and a small euclidean ball around a. But this is an even stronger statement. It means that by this homeomorphy, we can assume that {x a } := Z a,t ∩ Y A is a klt singularity in some c-dimensional variety Z, and Z a,t in turn is the intersection of Z with a small ball around x a . Note that x a does not have to be isolated, since the singular locus (Z a,t ) sing = Z a,t ∩ Y sing in general is bigger. Nevertheless, we know that ∂Z a,t ∩ Y sing is nothing but the regional link (i.e. Link(x a ) ∩ Z sm ) of x a and thus π 1 (∂Z a,t ∩ Y sing ) = π reg 1 (Z, x a ) = π 1 (Z a,t ∩ Y sing ). By repeating this procedure for every stratum N of E, we arrive at the surjection π 1 (V 0 E0 ) → π 1 (U (ε)\(E ∪Y sing )) as wanted. By Lemma 3.2 of [TX17], we know that π 1 (V 0 E0 ) is finite due to the induction hypothesis, so π reg 1 (X, x) is finite. By [TX17, Le. 3.5], also the regional orbifold fundamental group π reg 1 (X, ∆, x) is finite and the proof is finished.