Lagrangian embeddings of cubic fourfolds containing a plane

Ouchi, Genki

doi:10.1112/s0010437x16008307

Cited by 16 publications

(10 citation statements)

References 23 publications

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“…By Proposition 20.11, M 0 (v) gives a descent datum for the stability conditions with central charge in A 2 to some that are generic on all fibers; the associated relative coarse moduli space extends M 0 (v) → S 0 to a proper morphism M(v) → S of algebraic spaces, with all fibers being smooth and projective. Over C 8 , it agrees with the moduli spaces of stable objects constructed by Ouchi in [Ouc17].…”

Section: Proofs Of the Main Resultssupporting

confidence: 82%

Stability conditions in families

et al. 2021

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We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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Section: Proofs Of the Main Resultssupporting

confidence: 82%

Stability conditions in families

et al. 2021

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“…injects into cohomology, via the cycle class map. Here, h is the natural polarization and Y ⊂ Z is the lagrangian embedding constructed in [Ouc17].…”

Section: Further Resultsmentioning

confidence: 99%

The generalized Franchetta conjecture for some hyper-Kähler varieties

Laterveer

Vial

et al. 2019

Journal de Mathématiques Pures et Appliquées

114

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We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten. As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macrì-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface. 1 2 LIE FU, ROBERT LATERVEER, AND CHARLES VIAL Hyper-Kähler varieties. It was first conjectured by O'Grady [O'G13] that the universal family of K3 surfaces of given genus over the corresponding moduli space satisfies the Franchetta property. By using Mukai models, this was proved for certain families of K3 surfaces of low genus by Pavic-Shen-Yin [PSY17]. By investigating the case of the Beauville-Donagi family [BD85] of Fano varieties of lines on smooth cubic fourfolds, we were led in [FLVS19] to ask whether O'Grady's conjecture holds more generally for hyper-Kähler varieties : Conjecture 1 (Generalized Franchetta conjecture for hyper-Kähler varieties [FLVS19]). Let F be the moduli stack of a locally complete family of polarized hyper-Kähler varieties. Then the universal family X → F satisfies the Franchetta property.It might furthermore be the case that, for some positive integers n, the relative n-powers X n /F → F satisfy the Franchetta property. This was proved for instance in the case n = 2 in [FLVS19] for the universal family of K3 surfaces of genus ≤ 12 (but different from 11) and for the Beauville-Donagi family of Fano varieties of lines on smooth cubic fourfolds.The first main object of study of this paper is about the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten [LLSvS17], subsequently referred to as LLSS eightfolds. An LLSS eightfold is constructed from the space of twisted cubic curves on a smooth cubic fourfold not containing a plane. The following result, which is the first main result of this paper, completes our previous work [FLVS19, Theorem 1.11] where the Franchetta property was established for 0-cycles and codimension-2 cycles on LLSS eightfolds. Theorem 1. The universal family of LLSS hyper-Kähler eightfolds over the moduli space of smooth cubic fourfolds not containing a plane satisfies the Franchetta property.As already observed in [FLVS19, Proposition 2.5], the generalized Franchetta conjecture for a family of hyper-Kähler varieties implies the Beauville-Voisin conjecture [Voi08] for the very general member of the family : Corollary 1. Let Z be an LLSS hyper-Kähler eightfold. Then the Q-subalgebra R * (Z) := h, c j (Z) ⊂ CH * (Z) generated by the natural polarization h and the C...

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“…In Section 3 we argue that if Y is Pfaffian then for a general [C] ∈ M , the projection of I C (2) into A ∼ = D(X) is the ideal sheaf of four points in X, again up to a shift. Rather than proving this directly, we observe that if [C] lies over j(y) After the preprint of this paper appeared, Ouchi [11] showed that for a very general Y containing a plane -that is, in the opposite situation to [10] -there is a Bridgeland stability condition on A under which the projection of the skyscraper sheaf O y into A is stable; hence Y embeds as a Lagrangian in a holomorphic symplectic eightfold deformation-equivalent to Hilb 4 (K3). Presumably his eightfold is the limit of Z as Y approaches the locus of cubics containing a plane.…”

Section: Introductionmentioning

confidence: 94%

On the symplectic eightfold associated to a Pfaffian cubic fourfold

Addington

Lehn

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.Comment: 9 pages. Minor changes; to appear in Crelle as an appendix to 1305.017

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Lagrangian embeddings of cubic fourfolds containing a plane

Cited by 16 publications

References 23 publications

Stability conditions in families

Stability conditions in families

The generalized Franchetta conjecture for some hyper-Kähler varieties

On the symplectic eightfold associated to a Pfaffian cubic fourfold

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