Collapsing K3 surfaces and Moduli compactification

Odaka, Yuji; Oshima, Yoshiki

doi:10.3792/pjaa.94.81

Cited by 14 publications

(12 citation statements)

References 32 publications

(40 reference statements)

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“…In a similar vein, as L. Foscolo, S. Sun and J. Viaclovsky kindly pointed out to us in June of 2018 after our [OO18], the continuity of Φ around the boundary component M K3 (b 2 ) (resp., M K3 (c 2 )) should fit with collapsing of the K3 surfaces constructed in Chen-Chen [CC16] by gluing along cylindrical metrics, to the segment (resp., the collapsing of glued K3 surfaces of very recent Hein-Sun-Viaclovsky-Zhang [HSVZ18] to the segment.) We thank them for the discussions, which seem to provide more evidences to above Conjecture 6.2.…”

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confidence: 67%

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Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type

Odaka¹,

Oshima²

2018

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We study certain Morgan-Shalen type compactifications for locally Hermitian symmetric spaces and identify them with Satake compactifications for the adjoint representation.Then, via such compactification theory, we provide a moduli-theoretic framework for the collapsing of Ricci-flat-Kähler metrics. In other words, we give geometric meaning to the Satake compactifications for the adjoint representations or the Morgan-Shalen type compactifications, for certain cases.More precisely, we apply the compactification to moduli spaces of compact hy-perKähler manifolds, and state our main conjectures that they parametrize the Gromov-Hausdorff limits of the rescaled hyperKähler metrics with fixed diameters. Before partially proving the conjectures mainly for K3 surfaces, we first establish abelian varieties version, refining the previous work of the first author. A benefit of our conjecture is that, once it is confirmed, then for quite general sequences which are not even necessarily "maximally degenerating", we can determine the Gromov-Hausdorff limits.Our partial confirmation of the conjectures provides, for example, a proof of [KS06, Conjecture 1] and [GroWil00, Conjecture 6.2] at least for K3 surfaces, when applied to one parameter maximally degenerating family. In this case, the obtained Gromov-Hausdorff limits are metrized spheres S 2 , studied in [GroWil00, KS06], which have natural integral affine structures with singularities and are often regarded as tropical analogue of K3 surfaces. We also identify such limits along one parameter holomorphic families from the monodromy information.Trying to prove our conjectures for higher dimensional hyperKähler manifolds, in this paper, we solve the non-collapsing part and make certain progress on algebrogeometric preparations for the collapsing part. More precise form of the former part states that, for a fixed deformation class of polarized irreducible symplectic manifolds, the set of all Q-Gorenstein degenerations as symplectic varieties with ample Q-line bundles are bounded, and further the corresponding partial compactification of the moduli space becomes an orthogonal locally symmetric variety as in K3 surfaces case. Finally, we discuss possible extension of our collapsing picture for general Ricci-flat Kähler metrics of K-trivial varieties.Since the first author presented the main conjecture 4.3 at Oxford in the Clay conference "Algebraic geometry -new and old-" in September 2016 (resp., Conjecture 6.2 in the Mirror symmetry international conference at Kyoto university in December 2016), we have talked and discussed on the topic at Singapore, Levico Terme, Ann Arbor, Shanghai, Xiamen and various cities in Japan. We thank the organizers and the hosts for making such enjoyable chances which encouraged us.We also would like to thank our colleagues and friends, who have given helpful comments during our project;

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confidence: 67%

“…1.2 Outline of this paper [OO18] is the announcement of this paper. Most of the contents in this paper are stated there, while we add some improvements especially in §2, §8.…”

Section: Introductionmentioning

confidence: 99%

Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type

Odaka¹,

Oshima²

2018

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“…This paper originally stems out as a part of the series for ongoing joint work with Y.Oshima on collapsing of hyperKähler metrics, with recent focus on K3 surfaces to segments, with great inspirations input from [HSZ19] and [ABE20] as well. Our whole framework depends on the one initiated in our previous joint paper [OO18b] (its short summary is [OO18a]), whose particular focus of the latter part was on type III degenerations and associated collapsing to spheres.…”

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confidence: 99%

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

Odaka

2020

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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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“…Hence we get a new coordinate system (u 1 k , • • • , u n k ) and we discard the original coordinate system, and we use the same notation for tensors written in both coordinate systems, so in the new coordinate system we have (29) g k,ij − δ ij C m+1,β (B + L+3 ) → 0. Now we want to improve the convergence of g k,ij from elliptic equations with Neumann boundary conditions.…”

Section: 3mentioning

confidence: 99%

“…A closed hyperkähler 4-manifold is diffeomorphic to either a torus or the K3 manifold, and the moduli space of all hyperkähler metrics are described by Torelli theorems. There have been extensive recent studies on the Gromov-Hausdorff compactification of these moduli spaces, see for example [29] [32].…”

Section: Introductionmentioning

confidence: 99%

A compactness theorem for hyperkaehler 4-manifolds with boundary

Liu¹

2022

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In this paper, we study the compactness of a boundary value problem for hyperkähler 4-manifolds. We show that under certain topological conditions and the positive mean curvature condition on the boundary, a sequence of hyperkähler triples converges smoothly up to diffeomorphisms if and only if their restrictions to the boundary converge smoothly up to diffeomorphisms. We also generalize this result to torsion-free hypersymplectic triples.

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Collapsing K3 surfaces and Moduli compactification

Cited by 14 publications

References 32 publications

Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type

Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type

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

A compactness theorem for hyperkaehler 4-manifolds with boundary

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