The Jordan property for local fundamental groups

Braun, Lukas; Filipazzi, Stefano; Moraga, Joaquín; Svaldi, Roberto

doi:10.2140/gt.2022.26.283

Cited by 7 publications

(15 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is the exact sequence associated to a plt blow-up (see, e.g., [20,Proposition 4.9]). As usual, Δ 𝑈 𝑌 (resp.…”

Section: Regional Fundamental Group Of a Relative Fano-type Varietymentioning

confidence: 99%

“…Then, we have an exact sequence 1 → 𝐴 𝐹,0 → 𝐴 0 → 𝐴 𝑍 ′ ,0 → 1, so that 𝐴 𝐹,0 acts on (𝐹, Δ 𝐹 ) and 𝐴 𝑍 ′ ,0 acts on 𝑍 ′ . By [20,Theorem 3], there exists an abelian subgroup 𝐴 𝐹 ⩽ 𝐴 𝐹,0 of rank at most dim(𝐹) and index at most 𝑘(dim(𝐹)). By [20, Theorem 2], there exists an abelian subgroup 𝐴 𝑍 ′ of rank at most dim(𝑍) and index at most 𝑘(dim(𝑍)).…”

Section: Regional Fundamental Group Of a Relative Fano-type Varietymentioning

confidence: 99%

“…The following corollary should be compared to [38,Satz (6.1)], which together with the universal coefficient theorem [46, Corollary 3.3] -stating, in particular, that the torsion of second cohomology equals the torsion of first homology -is used in the proof of [20,Corollary 3] to show a similar statement in the restricted setting of Zariski and holomorphic local ring of klt singularities. However, the following proof does not depend on these statements (at least not directly).…”

Section: This Completes the Proof □mentioning

confidence: 99%

“…Furthermore, we have a surjection 𝜋 reg 1 ( X, Γ) → 𝜋 reg 1 (𝑋, Δ; 𝑥). Indeed, the birational map is a purely log terminal blow-up (see, e.g., [20,Proof of Theorem 4.14]). Observe that 𝑌 has dimension 𝑘.…”

Section: Jordan Property For 𝕋-Varietiesmentioning

confidence: 99%

“…The image of ℤ 𝑛−1 on 𝜋 reg 1 (𝑋, 𝑥) is a normal abelian subgroup 𝐴 0 of rank at most 𝑛 − 1. The elements of 𝐴 0 lie in the center of 𝜋 reg 1 (𝑋, 𝑥) (see, e.g., [20,Corollary 4.13]). By [57,Example 5.1], we know that there is an exact sequence…”

Section: Jordan Property For 𝕋-Varietiesmentioning

confidence: 99%

See 4 more Smart Citations

Iteration of Cox rings of klt singularities

Braun,

Moraga

2024

Journal of Topology

Self Cite

View full text Add to dashboard Cite

In this article, we study the iteration of Cox rings of klt singularities (and Fano varieties) from a topological perspective. Given a klt singularity , we define the iteration of Cox rings of . The first result of this article is that the iteration of Cox rings of a klt singularity stabilizes for large enough. The second result is a boundedness one, we prove that for an ‐dimensional klt singularity , the iteration of Cox rings stabilizes for , where only depends on . Then, we use Cox rings to establish the existence of a simply connected factorial canonical (or scfc) cover of a klt singularity, with general fiber being an extension of a finite group by an algebraic torus. The scfc cover generalizes both the universal cover and the iteration of Cox rings. We prove that the scfc cover dominates any sequence of quasi‐étale finite covers and reductive abelian quasi‐torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano‐type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano‐type morphisms.

show abstract

“…This is the exact sequence associated to a plt blow-up (see, e.g., [20,Proposition 4.9]). As usual, Δ 𝑈 𝑌 (resp.…”

Section: Regional Fundamental Group Of a Relative Fano-type Varietymentioning

confidence: 99%

Section: Regional Fundamental Group Of a Relative Fano-type Varietymentioning

confidence: 99%

Section: This Completes the Proof □mentioning

confidence: 99%

Section: Jordan Property For 𝕋-Varietiesmentioning

confidence: 99%

Section: Jordan Property For 𝕋-Varietiesmentioning

confidence: 99%

See 3 more Smart Citations

Iteration of Cox rings of klt singularities

Braun,

Moraga

2024

Journal of Topology

Self Cite

View full text Add to dashboard Cite

show abstract

Reductive quotients of klt singularities

Braun,

Greb,

Langlois

et al. 2024

Invent. math.

View full text Add to dashboard Cite

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety $X$ X endowed with the action of a reductive group $G$ G and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$ X → X / / G , we can find a boundary $B$ B on $X/\!/G$ X / / G so that the pair $(X/\!/G,B)$ ( X / / G , B ) is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$ G -manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$ n -dimensional K-polystable smooth Fano varieties of volume $v$ v has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

show abstract

A geometric characterization of toric singularities

Moraga,

Svaldi

2024

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

The Jordan property for local fundamental groups

Cited by 7 publications

References 33 publications

Iteration of Cox rings of klt singularities

Iteration of Cox rings of klt singularities

Reductive quotients of klt singularities

A geometric characterization of toric singularities

Contact Info

Product

Resources

About