Abstract
We prove that for any polarized symplectic automorphism of the Fano variety of lines of a smooth cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch–Beilinson conjecture. We also prove the same result for the Chow group of homologically trivial 2-cycles up to torsion.
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Notes
In the scheme-theoretic language, \(F(X)\) is defined to be the zero locus of \(s_T\in H^0\left( \mathop {\mathrm{Gr}}\nolimits (\mathop {\mathbf{P}}\nolimits ^1, \mathop {\mathbf{P}}\nolimits ^5), \mathop {\mathrm{Sym}}\nolimits ^3S^{\vee }\right) \), where \(S\) is the universal tautological subbundle on the Grassmannian, and \(s_T\) is the section induced by \(T\) using the morphism of vector bundles \(\mathop {\mathrm{Sym}}\nolimits ^3V^{\vee }\otimes {\fancyscript{O}}\rightarrow \mathop {\mathrm{Sym}}\nolimits ^3 S^{\vee }\) on \(\mathop {\mathrm{Gr}}\nolimits (\mathop {\mathbf{P}}\nolimits ^1, \mathop {\mathbf{P}}\nolimits ^5)\).
In fact easier, because we do not need to invoke Roitman theorem.
For rational coefficients it can be easily deduced by the argument in [13].
Recall that we are allowed to shrink \(B\) whenever we want, see Remark 4.2.
Here \({\tau ^o}^*\) is well-defined cause \(W^o\) and \({\fancyscript{X}}\times _B{\fancyscript{X}}\) are both smooth.
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Acknowledgments
I would like to express my gratitude to my thesis advisor Claire Voisin for bringing to me this interesting subject as well as many helpful suggestions. I also want to thank Mingmin Shen for pointing out the connection of our result and his joint work with Charles Vial [19], which motivates the last section of the paper. Finally, I thank the referee for his or her very helpful suggestions which improved the paper a lot.
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Fu, L. On the action of symplectic automorphisms on the \(\mathrm{CH}_0\)-groups of some hyper-Kähler fourfolds. Math. Z. 280, 307–334 (2015). https://doi.org/10.1007/s00209-015-1424-9
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DOI: https://doi.org/10.1007/s00209-015-1424-9