Reductive quotients of klt singularities

Braun, Lukas; Greb, Daniel; Langlois, Kevin; Moraga, Joaquín

doi:10.48550/arxiv.2111.02812

Cited by 4 publications

(5 citation statements)

References 50 publications

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“…2, p.358]. We also note that one can conclude from this that certain moduli spaces of K-stable Fano manifolds are klt; this is again a special case of much deeper results of Braun et al [BGLM21] and [LWX18].…”

Section: Setupsupporting

confidence: 69%

Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

Casalaina-Martin¹,

Grushevsky²,

Hulek³

et al. 2022

Preprint

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The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient M GIT , as a Baily-Borel compactification of a ball quotient (B 4 /Γ) * , and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup M K → M GIT , while from the ball quotient point of view it is natural to consider the toroidal compactification B 4 /Γ → (B 4 /Γ) * . The spaces M K and B 4 /Γ have the same cohomology and and it is therefore natural to ask whether they are isomorphic.Here we show that this is in fact not the case. Indeed, we show the more refined statement that M K and B 4 /Γ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

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Section: Setupsupporting

confidence: 69%

Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

Casalaina-Martin¹,

Grushevsky²,

Hulek³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We also have an example of an AC divisor D, such that Supp 𝐷 ≠ Supp 𝑛𝐷 for an integer 𝑛 1. If we set 𝑋 := Spec C[𝑥, 𝑦]/(𝑦 2 − 𝑥 2 + 𝑥 3 ) and 𝐷 := (𝑦/𝑥)O 𝑋 , then the origin (0, 0) ∈ 𝑋 is contained in the support of D but not in that of 2𝐷 = (𝑥 − 1)O 𝑋 .…”

Section: A2 Differents Of Ac Divisorsmentioning

confidence: 99%

“…In this subsection, we recall the definition of the different of a Q-AC divisor and prove some basic results used in subsection 2. 3.…”

Section: A2 Differents Of Ac Divisorsmentioning

confidence: 99%

Deformations of Log Terminal and Semi Log Canonical Singularities

Sato

Takagi

2023

Forum of Mathematics, Sigma

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In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$ -Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann.271 (1985), 439–449] and S. Ishii [Math. Ann.275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].

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“…45]), and so canonical classes K X ss and K Y are defined. Note that while K X ss is Cartier, recall that there are elementary examples of quotients of smooth varieties by reductive groups that are not Q-Gorenstein (e.g., [BGL+,Exam.7.1]); in other words, K Y need not be Q-Cartier.…”

Section: Canonical Classes For Git Quotientsmentioning

confidence: 99%

Non-Isomorphic Smooth Compactifications of the Moduli Space of Cubic Surfaces

CASALAINA-MARTIN,

GRUSHEVSKY,

HULEK

et al. 2023

Nagoya Math. J.

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The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$ , as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$ , and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$ , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ . The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

show abstract

Reductive quotients of klt singularities

Cited by 4 publications

References 50 publications

Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

Deformations of Log Terminal and Semi Log Canonical Singularities

Non-Isomorphic Smooth Compactifications of the Moduli Space of Cubic Surfaces

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