Teichmüller spaces and Torelli theorems for hyperkähler manifolds

Looijenga, Eduard

doi:10.1007/s00209-020-02594-6

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…However, this does not affect the results we are using, as they rely on Markman's formulation of the global Torelli theorem using marked moduli spaces instead of the Teichmuller space used by Verbitsky. We refer the reader to [Loo21, Theorem 3.1 and Remark 3.3] for the correct statement and a comment about the difference between the Teichmuller space and marked moduli spaces with respect to the global Torelli theorem (see also [Ver20]).…”

Section: Varieties Of -Type and Their Polarizationsmentioning

confidence: 99%

Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

Charles,

Mongardi,

Pacienza

2023

Compositio Math.

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We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$ -type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$ , we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$ -type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of $0$ -cycles on such varieties.

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Section: Varieties Of -Type and Their Polarizationsmentioning

confidence: 99%

Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

Charles,

Mongardi,

Pacienza

2023

Compositio Math.

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“…Looijenga [9] has defined a different moduli space, called the separated moduli space, and gives a complete proof of the global Torelli Theorem for this space. He further proves that this moduli space is equal to the one considered by Verbitsky (giving a different proof that the relation defining the moduli space is transitive).…”

Section: Remark 110mentioning

confidence: 99%

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

Kreck

2021

Math. Ann.

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The Torelli group $$\mathcal T(X)$$ T ( X ) of a closed smooth manifold X is the subgroup of the mapping class group $$\pi _0(\mathrm {Diff}^+(X))$$ π 0 ( Diff + ( X ) ) consisting of elements which act trivially on the integral cohomology of X. In this note we give counterexamples to Theorem 3.4 by Verbitsky (Duke Math J 162(15):2929–2986, 2013) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $$\ge 3$$ ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms $$J: \mathcal T(X) \rightarrow H^3(X;\mathbb Q)$$ J : T ( X ) → H 3 ( X ; Q ) and showing that for certain Kähler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in (Verbitsky in Duke Math J 162(15):2929–2986, 2013) where Verbitsky claims that the Torelli group of hyperkähler manifolds are finite. These examples are detected by the action of diffeomorphsims on $$\pi _4(X)$$ π 4 ( X ) . Finally we confirm the finiteness result for the special case of the hyperkähler manifold $$K^{[2]}$$ K [ 2 ] .

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“…Verbitsky's global Torelli theorem [Ver13] (also see [Huy12] and [Loo21]) for compact hyperkähler manifolds is the following.…”

Section: 7mentioning

confidence: 99%

Hyperkähler manifolds

Izadi¹,

Samir²,

Dutta³

et al. 2021

Preprint

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We give an elementary introduction to hyperkähler manifolds, survey some of their interesting properties and some open problems. Contents 1. C ∞ manifolds 3 2. Riemannian manifolds 7 3. Kähler manifolds 11 4. Holomorphic symplectic manifolds 14 5. Moduli of hyperkählers, the Beauville-Bogomolov form, the period domain and the period map 20 6. Twistor spaces and twistor conics 27 7. Examples of hyperkählers in dimension 2 and beyond, by Samir Canning 29 8. Basic properties of Lagrangian fibrations of Hyperkählers, by Yajnaseni Dutta 32 9. Rational curves on K3 surfaces and Euler characteristics of Moduli spaces, by David Stapleton 35 References 37

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Teichmüller spaces and Torelli theorems for hyperkähler manifolds

Cited by 6 publications

References 26 publications

Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

Hyperkähler manifolds

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