The generalized Franchetta conjecture for some hyper-K\'ahler varieties
| Autor: | Fu, Lie, Laterveer, Robert, Vial, Charles, Shen, Mingmin |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | The generalized Franchetta conjecture for hyper-K\"ahler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-K\"ahler varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville--Donagi family of Fano varieties of lines on cubic fourfolds and its relative square, and for $0$-cycles and codimension-$2$ cycles for the Lehn--Lehn--Sorger--van Straten family of hyper-K\"ahler eightfolds. We also draw many consequences in the direction of the Beauville--Voisin conjecture as well as Voisin's refinement involving coisotropic subvarieties. In the appendix, we establish a new relation among tautological cycles on the square of the Fano variety of lines of a smooth cubic fourfold and provide some applications. Comment: 35 pages; to appear in J. Math. Pures Appl |
| Databáze: | arXiv |
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