Polystable log Calabi-Yau varieties and Gravitational instantons

Odaka, Yuji

doi:10.48550/arxiv.2009.13876

Cited by 3 publications

(13 citation statements)

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“…However, in general, such approach certainly gives either non-uniqueness of the compactification due to the additional data of choosing the divisors and change the framework of discussion somewhat. In this paper, together with [Od20a], we still seek for canonical "algebro-" geometric compactifications without adding divisors or canonical "algebro-"geometric limits of degenerations.…”

Section: Proposition 15 (Cf §4 For Precise Meanings) Over the Minimal...mentioning

confidence: 99%

“…Basic of dlt minimal models. This section discusses dlt minimal models which do not necessarily satisfy open K-polystability (of [Od20a]), and their certain limits.…”

Section: -Gmentioning

confidence: 99%

“…From now, we consider finite ramified base change of degenerating Calabi-Yau varieties and its "subdividing" effects, following [Od20a,§4]. More precisely, we obtain a projective system of reductions as follows.…”

Section: Subdivision By Base Change and Galaxiesmentioning

confidence: 99%

“…Therefore, we can assume that ∆ ′ is the one constructed in Step 2 of the proof of Theorem 4.4 in [Od20a], as used in (2.8). From the construction, over simple normal crossing locus of (X , X 0 ) which contains all lc centers, the birational morphism X ′ → X (N) = X × ∆ ∆ ′ is toroidal and the exceptional divisors corresponds to 1/N-integral points of the maximal simplices of ∆ alg (X 0 ), from the definition of the affine structures in [MN12, §3.2] (cf., also [BFJ15, §2.1]).…”

Section: Subdivision By Base Change and Galaxiesmentioning

confidence: 99%

“…Also, if k is a valuation field, we put X an i the complex analytification of X i and X an ∞ := lim ← −i X an i . (ii) A quasi-limit dlt model or its reduction i.e., quasi-limit sdlt reduction is said to be open K-polystable if one can take dlt approximation models sequence as all open K-polystable ones (cf., [Od20a]). (iii) We call a quasi-limit dlt model over Puiseux series in the above sense is complete or limit dlt model if it is represented by a sequence of dlt approximation models X i over k[[t 1/N i ]] satisfying: for any positive integer N, there is i 0 with ∀i > i 0 , N | N i .…”

Section: Subdivision By Base Change and Galaxiesmentioning

confidence: 99%

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Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Odaka¹

2020

Preprint

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YUJI ODAKA A. For any degenerating Calabi-Yau family, we introduce new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi-Yau varieties, parametrized by the rational points of the Kontsevich-Soibelman's essential skeleton, while dominated by the Huber adification over the Puiseux series field.Other topics include: projective limits of toroidal compactifications ( §3), locally modelled on limit toric varieties ( §2.4), the way to attach tropicalized family to given Calabi-Yau family ( §4), which are weakly related to each other.Here, B denotes the essential skeleton (as 1.2 again) and B(Q) means its rational points with respect to the Q-affine structure, which does not depend on i by Theorem 2.1 (iii) and, where NKLT stands for the non-klt closed loci.Contents of §3 (Limit toroidal compactifications). In §3 over k = C in turn, we discuss on the projective limit of toroidal compactifications of [AMRT] and its analogue for more general (moduli) varieties. It sounds somewhat independent from §2 but here we observe an analogous phenomenon of the above theorem 1.2, especially the natural continuous maps to their tropical versions.More precisely, what we do there is as follows. Fix a locally Hermitian symmetric space M of non-compact type, i.e., of the ubiquitous form M = Γ\G/K where G is a real valued points of a simple algebraic group over Q, K its maximal compact subgroup with one dimensional center, Γ an arithmetic discrete subgroup of G. Some renowned examples are the moduli space of g-dimensional principally polarized abelian varieties or that of primitively polarized K3 surfaces with possibly ADE singularities (of fixed genera).Recall that for certain combinatorial data i.e., admissible collection of fans Σ = {Σ(F)}, there is an associate toroidal compactification M tor,Σ of M constructed in[AMRT], and its complex analytification M tor,Σ,an . Now, we consider and introduce the projective limit of all of its toroidal compactifications as a locally ringed space and call the limit toroidal compactification: M tor,∞ an := lim ← − Σ

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Section: Proposition 15 (Cf §4 For Precise Meanings) Over the Minimal...mentioning

confidence: 99%

“…Basic of dlt minimal models. This section discusses dlt minimal models which do not necessarily satisfy open K-polystability (of [Od20a]), and their certain limits.…”

Section: -Gmentioning

confidence: 99%

Section: Subdivision By Base Change and Galaxiesmentioning

confidence: 99%

Section: Subdivision By Base Change and Galaxiesmentioning

confidence: 99%

Section: Subdivision By Base Change and Galaxiesmentioning

confidence: 99%

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Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Odaka¹

2020

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Pl Density Invariant for Type Ii Degenerating K3 Surfaces, Moduli Compactification and Hyper-Kähler Metric

Odaka¹

2021

Nagoya Math. J.

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A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit piecewise linear convex function from the interval with at most $18$ nonlinear points. Forgetting its actual function behavior, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples. From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyper-Kähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [AB19], [ABE20], [Brun15] in a more elementary manner, and analyze the cusps more explicitly. We also interpret the glued hyper-Kähler fibration of [HSVZ18] as a special case from our viewpoint, and discuss other cases, and possible relations with Landau–Ginzburg models in the mirror symmetry context.

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PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics

Odaka

2020

Preprint

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A. A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit PL (piecewise linear) convex function from the interval with at most 18 non-linear points. Forgetting its actual function behaviour, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples.From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyperKähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [Brun15, AB19, ABE20] in a more elementary manner, analyze the cusps more explicitly.We also interpret the glued hyperKähler fibration of [HSVZ18] as a special case from our viewpoint, discuss other cases, and possible relations with Landau-Ginzburg models in the mirror symmetry context.

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Polystable log Calabi-Yau varieties and Gravitational instantons

Cited by 3 publications

References 93 publications

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

Pl Density Invariant for Type Ii Degenerating K3 Surfaces, Moduli Compactification and Hyper-Kähler Metric

PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics

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