Moduli of Polarized Calabi—Yau Pairs

Kollár, János; Xu, Chen

doi:10.1007/s10114-020-9300-x

Cited by 6 publications

(5 citation statements)

References 16 publications

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“…Now, turning back to weak K-moduli, the mentioned explicit examples of (e.g., [AN99, Ale02, Nak04, Sha80, Loo03, Zhu17, AET19, ABE20]) are all weak Kmoduli, giving the affirmative confirmation of the full conjecture 1.8, hence has been its supporting evidences. Most of the above are obtained as special case of logarithmic version (i.e., a version with boundary divisors) of KSBA construction, which is also discussed in recent [Bir20,KX20]. However, we should keep in mind and emphasize that a priori these could be moduli of the pairs of varieties and their divisors, hence some locus could possibly preserve the ambient varieties while the divisors change.…”

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

confidence: 97%

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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: 97%

“…Hence the recent deep results on birational geometry of "toroidalization" [AK00, ATW20] may well be effective as in the proof of Theorem 1.9 or [KX20].…”

Section: Now Let Us Observe the Following Examples Of Degenerations O...mentioning

confidence: 99%

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

Odaka¹

2020

Preprint

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“…In particular, a compactification of the space of pairs using similar machinery is described in [DH21]. In general, Kollár and Xu show that one can make a coarse moduli space of polarized log Calabi Yau pairs where each irreducible component is projective [KX20].…”

Section: H-stable Pairsmentioning

confidence: 99%

Moduli of surfaces in

DeVleming¹

2022

Compositio Math.

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The main goal of this paper is to construct a compactification of the moduli space of degree $d \geqslant 5$ surfaces in $\mathbb {P}^{3}_{{{\mathbb {C}}}}$ , i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $\mathbb {P}^{3}$ and whose boundary points correspond to degenerations of such surfaces. We consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = {{\mathbb {P}}}^{3}$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.

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“…In our notations, R is a polarizing divisor. By [KX20], in any dimension the irreducible components of the moduli of Calabi-Yau pairs with a polarizing divisor are projective.…”

Section: Recognizable Divisorsmentioning

confidence: 99%

Compact moduli of K3 surfaces

Alexeev¹,

Engel²

2021

Preprint

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Let F be a moduli space of lattice-polarized K3 surfaces and suppose that one has chosen a canonical effective ample divisor on a general fiber. We call this divisor "recognizable" if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification F R for a recognizable divisor R is semitoroidal.We also prove that for polarized K3 surfaces (X, L) of degree 2d the sum R of rational curves in the linear system |L| is a recognizable divisor, giving a modular semitoroidal compactification of F 2d for all d.

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Moduli of Polarized Calabi—Yau Pairs

Cited by 6 publications

References 16 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

Moduli of surfaces in

Compact moduli of K3 surfaces

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