The Generalized Franchetta Conjecture for Some Hyperkähler Fourfolds

Laterveer, Robert

doi:10.4171/prims/55-4-8

Cited by 3 publications

(3 citation statements)

References 32 publications

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“…Recently, Beauville has shown in [1] that there exists for every g a hypersurface in F • g such that GFC holds for the corresponding family. We also refer the readers to [14,15,27] for the generalization to hyper-Kähler varieties. In [3], the authors have verified GFC in cohomology groups.…”

Section: Introductionmentioning

confidence: 99%

Deligne-Beilinson cohomology of the universal K3 surface

Li¹,

Zhang²

2022

Preprint

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O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In [3], this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4-th DBcohomology group of universal oriented polarized K3 surfaces with at worst an A1-singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O'Grady's original conjecture in DB cohomology.

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Section: Introductionmentioning

confidence: 99%

Deligne-Beilinson cohomology of the universal K3 surface

Li¹,

Zhang²

2022

Preprint

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show abstract

“…Recently, Beauville has shown in [1] that there exists for every g a hypersurface in ℱ • 𝑔 such that GFC holds for the corresponding family. We also refer readers to [15,16,28] for the generalisation to hyper-Kähler varieties. In [4], the authors have verified GFC in cohomology groups.…”

Section: Introductionmentioning

confidence: 99%

Deligne-Beilinson cohomology of the universal K3 surface

Zhang

2022

Forum of Mathematics, Sigma

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O’Grady’s generalised Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarised K3 surfaces. In [4], this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4th DB-cohomology group of universal oriented polarised K3 surfaces with at worst an $A_1$ -singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O’Grady’s original conjecture in DB cohomology.

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“…Then b is rationally trivial if and only if b has degree 0.) (The g = 5 case of theorem 4.2 was already done in [29].) The second series of examples consists of Hilbert cubes X = S [3] , where S is a general K3 surface of genus g. For g = 9, the Hilbert cube X admits a Lagrangian fibration φ : X → P 3 [21] (cf.…”

Section: Introductionmentioning

confidence: 99%