Finiteness of algebraic fundamental groups

Compositio Mathematica, 2012

95 citations recorded by the source corpus.

Abstract We show that the algebraic local fundamental group of any Kawamata log terminal singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c3e371209adc18316a368f8920cba7e889aceda6

Citations in the corpus

• A Guided Tour to Normalized Volume (2018, methodology): “Definition 2.12 (Kollár component, [Xu14]).”
• Uniform K-stability and plt blowups of log Fano pairs (2017, methodology): “Plt blowups The following theorem is inspired by [Xu14, Lemma 1].”
• Uniqueness of the minimizer of the normalized volume function (2020, background): “For the germ of a klt singularity (X,∆), by [Xu14, Bra20] (see also [TX17]), the fundamental group π1(x,X sm) of the smooth locus Xsm of is finite.”
• Stability of Valuations: Higher Rational Rank (2017, methodology): “Our study of the degeneration heavily relies on recent developments in the minimal model program (MMP) based on [BCHM10] (cf. e.g. [LX14, Xu14] etc.).”
• K-stability of cubic threefolds (2017, background): “We will show that |π̂loc1 (X, x)|·v̂ol(x,X) 324 (note that this gives a new proof that the algebraic fundamental group of a three dimensional klt singularity is finite (see [SW94, Xu14]).”
• Rational points on log Fano threefolds over a finite field (2015, methodology): “For the case X = Z in Proposition 2.15, Y is very similarly constructed as the Kollár component in [Xu14, Lemma 1].”
• The local fundamental group of a Kawamata log terminal singularity is finite (2020, methodology): “We already mentioned that building on the results of [Xu14], in [GKP16, Thm. 1.5] it was shown that a quasiprojective klt pair (X,∆) allows a finite quasi-étale cover Y → X such that π̂1(Ysm) is isomorphic to π̂1(Y ).”
• A minimizing valuation is quasi-monomial (2019, methodology): “ = Sand if we write (KY +µ−1 ∗ ∆+S)|S = KS +∆S, then (S,∆S) is a log Fano pair. Such a morphism µis called a plt blow up (see [Pro00]), as (Y,µ−1 ∗ ∆+S) is a plt pair. Inspired by the construction in [Xu14] and the special test configuration construction in [LX14] in the global setting, Koll´ar compo”
• Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties (2013, methodology): “he literature), the material of Section 4 is new to the best of our knowledge and might be of independent interest. The main result is proven in Part II. Starting with a recent result of Chenyang Xu, [Xu14],Sections 6–8discuss andprove variantsof the main resultin a number of special cases, with inc”
• Stability of valuations and Kollár components (2016, methodology): “minimal model program. The following statement is the local analogue of [LX14, Theorem 1.6], which can be obtained by following the proof of the existence of Kollar component. (See e.g. the proof of [Xu14].) Proposition 2.10. Let o2(X;D) be a klt singularlty. Let : Z!Xbe a model, such that is an iso”
• The dual complex of Calabi–Yau pairs (2015, unknown): “The finiteness of ˆ π 1 X sm follows from [Xu14, GKP13]; it is conjectured that π 1 X sm is finite.”