Examples of irreducible symplectic varieties

Perego, Arvid

doi:10.48550/arxiv.1905.11720

Cited by 3 publications

(3 citation statements)

References 42 publications

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“…2 By [36, Theorem 3 (3)], a general fiber of a Lagrangian fibration associated with any irreducible symplectic variety is an abelian variety. 3 The fact that Matsushita's example is irreducible symplectic is verified by [34,Proposition 2.5].…”

Section: Introductionmentioning

confidence: 94%

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

Felisetti¹,

Shen²,

Yin³

2021

Preprint

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We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variety admits a symplectic resolution. These results generalize some earlier work by the second and third authors in the nonsingular case.

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Section: Introductionmentioning

confidence: 94%

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

Felisetti¹,

Shen²,

Yin³

2021

Preprint

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“…Such manifolds are intensively studied [BL,Me1,MaT,FuMe] because they can be seen as a natural generalization of smooth irreducible hyper-Kähler manifolds. There are only a few known families of these orbifolds, see [Me2,Pe]. A first nonsmooth example of irreducible holomorphic symplectic orbifold of dimension 4 is given as deformation of a partial resolution of the quotient of a fourfold of K3 [2] -type by a symplectic involution: we call this an orbifold of Nikulin type, in analogy with Nikulin surfaces in dimension two; the first examples were studied by [MaT].…”

Section: Introductionmentioning

confidence: 99%

Projective models of Nikulin orbifolds

Camere¹,

Garbagnati²,

Kapustka³

et al. 2021

Preprint

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We study projective fourfolds of K3 [2] -type with a symplectic involution and the deformations of their quotients, called orbifolds of Nikulin types; they are IHS orbifolds. We compute the Riemann-Roch formula for Weil divisors on such orbifolds and describe the first complete family of orbifolds of Nikulin type with a polarization of degree 2 as double covers of special complete intersections (3, 4) in P 6 .

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“…Finally "singular" irreducible symplectic varieties appear as building blocks of mildly singular projective varieties with trivial canonical class (see [19,13,14]). See the recent survey [32] for all the different definitions and results. As the Boucksom-Zariski decomposition proved to be a very useful tool in the theory of smooth irreducible symplectic varieties it is natural to ask whether it holds for some classes of singular irreducible symplectic varieties.…”

Section: Introductionmentioning

confidence: 99%

On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity

Kapustka¹,

Mongardi²,

Pacienza³

et al. 2019

Preprint

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Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. Boucksom showed that it also holds for irreducible symplectic manifolds. Different variants of singular holomorphic symplectic varieties have been extensively studied in recent years. In this note we first show that the Boucksom-Zariski decomposition holds in the largest possible framework of varieties with symplectic singularities. On the other hand in the case of surfaces, it was recently shown that there is a strict relation between the boundedness of coefficients of Zariski decompositions of pseudoeffective integral divisors and the bounded negativity conjecture. In the present note, we show that an analogous phenomenon can be observed in the case of irreducible symplectic manifolds. We furthermore prove an effective analog of the bounded negativity conjecture in the smooth case. Combining these results we obtain information on the denominators of Boucksom-Zariski decompositions for holomorphic symplectic manifolds. From such a bound we easily deduce a result of effective birationality for big line bundles on projective holomorphic symplectic manifolds, answering a question asked by Charles.

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Examples of irreducible symplectic varieties

Cited by 3 publications

References 42 publications

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

Projective models of Nikulin orbifolds

On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity

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