Limit mixed Hodge structures of hyperkähler manifolds

Soldatenkov, Andrey

doi:10.48550/arxiv.1807.04030

Cited by 5 publications

(14 citation statements)

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“…Therefore if we choose a primitive class β ∈ Λ K3 so that β, β = 0 and β, α = 0, then β determines a cusp c β of M BB α . The following result follows from arguments of Soldatenkov, but our statement is slightly different from the those in [29]. Proof.…”

Section: P=w For Maximal Log Calabi-yau Pairs In Dimensionmentioning

confidence: 68%

“…Recall that P ∨ α is the subset of the quadric τ ∈ P(α ⊥ ⊗ C), τ, τ = 0 which can be obtained as exp(2iπtN β,ρ )x for some t ∈ C and x ∈ P α . In [29,Lemma 4.4]. For some fixed x ∈ P ∨ α , the subset given by {exp(2iπtN β,ρ )x : t ∈ C} is called a nilpotent orbit of N β,ρ .…”

Section: P=w For Maximal Log Calabi-yau Pairs In Dimensionmentioning

confidence: 99%

“…Thus we can construct a family of K3 surfaces S * → ∆ * whose period map is a neighbourhood of the origin in a nilpotent orbit associated to N β,ρ . Hence the monodromy of S is, after finite order base change, a power of exp(N β,ρ ) [29,Theorem 4.6].…”

Section: P=w For Maximal Log Calabi-yau Pairs In Dimensionmentioning

confidence: 99%

“…from work of Shen and Yin [26] along with results of Soldatenkov [29]. The main result of [26] is that if M is a compact hyperkähler variety which admits a Lagrangian torus fibration ℓ : M → B, then (2) dim Gr P i H i+j (M ) = dim Gr i F H i+j (M ) for all i and j.…”

Section: Introductionmentioning

confidence: 98%

“…Let H i+j lim denote the limit mixed Hodge structure associated to the degeneration (M, π). In [29], Soldatenkov proves that if a monodromy operator N associated to a semistable degeneration (M, π) of compact hyperkähler manifolds has the property that N 2 | H 2 (M ) = 0, then the limit mixed Hodge structure of this degeneration is Hodge-Tate. This means that, for all i and j, Organization.…”

Section: Introductionmentioning

confidence: 99%

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Torus fibers and the weight filtration

Harder¹

2019

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We show that if (X, Y ) is a simple normal crossings log Calabi-Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of Y which can be used to construct W 2k−1 H k (X \Y ; Q) for all k. We show that an analogous result holds for degenerations of Calabi-Yau varieties. We use this to show that P=W type results hold for pairs (X, Y ) consisting of a rational surface X and a nodal anticanonical divisor Y , and for K3 surfaces. Contents 1. Introduction 1 2. The weight filtration for a snc log Calabi-Yau pair 4 3. The weight filtration for a Calabi-Yau degeneration 8 4. P=W type results for surfaces 12 References 18

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Section: P=w For Maximal Log Calabi-yau Pairs In Dimensionmentioning

confidence: 68%

Section: P=w For Maximal Log Calabi-yau Pairs In Dimensionmentioning

confidence: 99%

Section: P=w For Maximal Log Calabi-yau Pairs In Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Torus fibers and the weight filtration

Harder¹

2019

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Deformation principle and André motives of projective hyperkähler manifolds

Soldatenkov¹

2019

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Let X 1 and X 2 be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of X 1 is abelian if and only if the André motive of X 2 is abelian. Applying this to manifolds of K3 [n] , generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford-Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds. Contents1. Introduction 1.1. André motives and hyperkähler manifolds 1.2. Absolute Hodge classes 1.3. The Mumford-Tate conjecture 1.4. Organization of the paper 2. Motivated cohomology classes and André motives 2.1. Motivated cohomology classes 2.2. The André motives 3. Hyperkähler manifolds and the Kuga-Satake embedding 3.1. Hyperkähler manifolds 3.2. The Kuga-Satake construction 3.3. The Kuga-Satake construction in families 4. The manifolds of K3 [n] , generalized Kummer and OG6 deformation types 4.1. Hilbert schemes of points on K3 surfaces and generalized Kummer varieties 4.2. Manifolds of OG6 deformation type 5. Deformation principle 5.1. The setting 5.2. The statement and preliminary constructions 5.3. Proof of Proposition 5.1 6. Motives of hyperkähler manifolds 6.1. Constructing smooth families of hyperkähler manifolds 6.2. Proof of theorem 1.1

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The LLV decomposition of hyper-Kaehler cohomology

Green¹,

Kim²,

Laza³

et al. 2019

Preprint

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Looijenga-Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold X admits a natural action by the Lie algebra so(4, b 2 (X) − 2), generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we determine the Looijenga-Lunts-Verbitsky (LLV) decomposition for all known examples of compact hyper-Kähler manifolds. As an application, we compute the Hodge numbers of the exceptional OG10 example (recovering a recent result of de Cataldo-Rapagnetta-Saccà) starting only from the knowledge of the Euler number e(X), and the vanishing of the odd cohomology of X. In a different direction, we establish the so-called Nagai's conjecture for all known examples of hyper-Kähler manifolds. More importantly, we prove that, in general, Nagai's conjecture is equivalent to a representation theoretic condition on the LLV decomposition of the cohomology H * (X). We then notice that all known examples of hyper-Kähler manifolds satisfy a stronger, more natural condition on the LLV decomposition of H * (X): the Verbitsky component is the dominant representation in the LLV decomposition of H * (X).

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Limit mixed Hodge structures of hyperkähler manifolds

Cited by 5 publications

References 0 publications

Torus fibers and the weight filtration

Torus fibers and the weight filtration

Deformation principle and André motives of projective hyperkähler manifolds

The LLV decomposition of hyper-Kaehler cohomology

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