Collapsing hyperkähler manifolds

Tosatti, Valentino; Zhang, Yuguang

doi:10.48550/arxiv.1705.03299

Cited by 9 publications

(19 citation statements)

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“…Finally, the appendix has some results of independent interest, where we study the collapsing rate of Ricci-flat Kähler-Einstein metrics on general Abelian fibered Calabi-Yau manifolds. Here we improve on the previous results of [39,40,67].…”

Section: Introductionsupporting

confidence: 75%

“…ϕ t = (φ t + ξ t ) • T σ 0 . If we denote λ t : U × C m−n → U × C m−n the dilation given by λ t (w, z) = (w, t − 1 2 z), then λ * t it∂ ∂η = i∂ ∂η, andλ * t p * ω SF t = i∂ ∂η + f * ω.for constants C ℓ > 0, and by Lemma 4.7 in[39] (also Proposition 3.2 of[67]),λ * t p * T * σ 0 ω t → i∂ ∂η + f * ω when t → 0, in the locally C ∞ -sense. If we denote ψ t = ϕ t • T σ 0 • p • λ t , then ψ t is t where a + bZ = (a 1 + b 1 Z 1 , • • • , a m−n + b m−n Z m−n ) for any a j , b j ∈ Z.By the above we can writeλ * t p * T * σ 0 ω t = i∂ ∂η + ω + i∂∂ψ t ,and note that i∂∂ψ t C ℓ Denote ψ t,w k wl = ∂ 2 ψ t ∂w k ∂ wl , ψ t,z k zl = ∂ 2 ψ t ∂z k ∂ zl , and ψ t,z k wl = ∂ 2 ψ t ∂z k ∂ wl .For any ν ∈ N and ℓ ≥ 0, there is a constant C ′ ℓ,ν > 0 such that ψ t,w k wl − χ t,kl C 0…”

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confidence: 99%

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Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces

Datar

Jacob

Zhang

2018

Preprint

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We prove a convergence result for a family of Yang-Mills connections over an elliptic K3 surface M as the fibers collapse. In particular, assume M is projective, admits a section, and has singular fibers of Kodaira type I1 and type II. Let Ξt k be a sequence of SU (n) connections on a principal SU (n) bundle over M , that are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of M . Given certain non-degeneracy assumptions on the spectral covers induced by ∂Ξt k , we show that away from a finite number of fibers, the curvature FΞ t k is locally bounded in C 0 , the connections converge along a subsequence (and modulo unitary gauge change) in L p 1 to a limiting L p 1 connection Ξ0, and the restriction of Ξ0 to any fiber is C 1,α gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections Ξt k to a converging family of special Lagrangian multi-sections in the mirror HyperKähler structure, addressing a conjecture of Fukaya in this setting.

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Section: Introductionsupporting

confidence: 75%

mentioning

confidence: 99%

Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces

Datar

Jacob

Zhang

2018

Preprint

Self Cite

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“…See related Hodge-theoretic estimates in [GTZ13, Proposition 2.1, (Lemma 3.1) and its proof] (cf. also [Yskw10], [TZ17], [EMM16]).…”

Section: Estimate Of Mclean Metric Near Discriminant Pointsmentioning

confidence: 93%

“…Theorem 4.21 (cf. [Tos10], [GTZ13], [GTZ16], [TZ17]). Let S be an open subset of KΩ ∩ pr −1 (Ω e (Λ K3 )) such that S ⊂ KΩ e≥0 and the closure S in KΩ is a compact subset of KΩ e≥0 .…”

Section: Theorem 410 (A Weaker Version Of Theorem 419)mentioning

confidence: 99%

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

Odaka¹,

Oshima²

2018

Preprint

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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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“…An outline is as follows. Firstly, we use Theorem 1 to determine the asymptotics of the generalized Kähler-Einstein current on Σ; secondly, under the assumption that Ricci curvature is uniformly bounded from below, we modify discussions in Section 4 to obtain a uniform diameter upper bound for the Kähler-Ricci flow on X; finally we apply some results and arguments of Cheeger-Colding [5,6] and Gross-Tosatti-Zhang [13,14,29] to prove Gromov-Hausdorff convergence. 5.1.…”

Section: Proofs Of Theorems 3 Andmentioning

confidence: 99%

Collapsing limits of the Kähler–Ricci flow and the continuity method

Zhang

2018

Math. Ann.

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We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kähler metric on the total space of a Fano fibration with one dimensional base converges in Gromov-Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kähler-Einstein current on a Riemann surface is compact.

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Collapsing hyperkähler manifolds

Cited by 9 publications

References 50 publications

Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces

Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces

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

Collapsing limits of the Kähler–Ricci flow and the continuity method

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