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

Invent. math. https://doi.org/10.1007/s00222-021-01062-0 The local fundamental group of a Kawamata log terminal singularity is finite Lukas Braun1 Received: 23 June 2020 / Accepted: 1 July 2021 © The Author(s) 2021 Abstract We prove a conjecture of Kollár 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. Mathematics Subject Classification 14F35 · 14B05 · 14J45 · 32S50 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental groups of the whole space . . . . . . . . . . . . . . . . . . . . . . . . . . . Étale fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regional fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consequences (and non-consequences) of our main theorems . . . . . . . . . . . . . . . The proof of Theorems 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible alternative ways of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The author was partially supported by the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg GK1821 “Cohomological Methods in Geometry” at the University of Freiburg. B Lukas Braun 1 Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Strasse 1, 79104 Freiburg im Breisgau, Germany 123 L. Braun Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local to global . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Complex orbifolds and orbimaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Orbibundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Orbisheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Orbimetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The orbifold universal cover and the Γ -reduction . . . . . . . . . . . . . . . . . . . . . . 7 Dolbeault and L 2 -cohomology for Kähler orbifolds . . . . . . . . . . . . . . . . . . . . 8 L 2 -vanishing for orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Maximal compact subspaces of orbifold universal covers . . . . . . . . . . . . . . . . . . 10 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact orbifolds supported on a log resolution . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global to local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Whitney stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The work of Tian and Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tian and Xu’s Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tian and Xu’s Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Finiteness of the regional fundamental group . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction We work over the field C of complex numbers. A log pair is an algebraic variety   X together with a boundary  divisor 0 ≤ Δ < 1 of the form Δ = Δ + Δ ,   with 0 ≤ Δ and Δ = (1 − 1/m i )Δi is a sum of prime divisors Δi , whose coefficients satisfy m i ∈ Z>1 . A Kawamata log terminal or klt singularity is a point x ∈ (X, Δ), such that for a log resolution f : Y → X , locally around x, the discrepancies ai , namely the coefficients of the exceptional divisors E i in the formula K Y + f ∗−1 Δ ∼Q f ∗ (K X + Δ) +  ai E i satisfy ai > −1. We call a log pair (X, Δ) weakly Fano, if it has only klt singularities and −(K X + Δ) is big and nef. The local fundamental group of a normal singularity x ∈ X is π1loc (X, x) := π1 (B\x) = π1 (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 π1loc is well defined. The following conjecture is due to Kollár [40,43]. 123 The local fundamental group Conjecture 1 Let x ∈ (X, Δ) be a klt singularity. Then the local fundamental group π1loc (X, x) is finite. In the case of a weakly Fano pair (X, Δ), one can consider the smooth locus of X and state the following global conjecture [1,60]. Conjecture 2 Let (X, Δ) be a weakly Fano pair. Then the fundamental group π1 (X sm ) of the smooth locus is finite. This conjecture has been proven for log del Pezzo surfaces [28,36,37] and log Fano varieties of high Fano index [60]. We prove generalized versions of both conjectures in the present paper. Firstly, for a log pair (X, Δ) with decomposition Δ = Δ + Δ as above, we can consider the complex orbifold X = (X, Δ ), see Sect. 2. Then one can consider the orbifold fundamental group of the smooth locus, denoted by π1 (X sm , Δ ). This group is defined to be π1 (X sm \supp(Δ ))/N , where N is the normal subgroup generated by the γim i , where γi is a small loop around Δi . Similarly to the global case, in the local setting we can consider the fundamental group π1 (Bsm ) = π1 (B\X sing ) of the smooth locus of B, instead of the local fundamental group. We call this group the regional fundamental reg group and denote it by π1 (X, x). It is also possible to define the orbifold reg fundamental group of (Bsm , Δ  B ), which we denote by π1 (X, Δ , x). Our sm two main theorems are the following. Theorem 1 Let x ∈ (X, Δ + Δ ) be a klt singularity. Then the regional reg reg fundamental groups π1 (X, x) and π1 (X, Δ , x) are finite. Theorem 2 Let (X, Δ + Δ ) be a weakly Fano pair. Then the orbifold fundamental group π1 (X sm , Δ ) of the smooth locus is finite. Before sketching the structure of the (simultaneous) proof of these theorems, we give a short overview of related results and state some consequences. Fundamental groups of the whole space Fano manifolds are known to be simply connected, and there are several proofs of this fact, relying for example on Atiyah’s L 2 -index theorem or rational connectedness. Generalizing the smooth case, it was shown b (...truncated)