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Abstract: |
The Chow group of algebraic cycles of a smooth projective variety is an important subject in algebraic geometry, which in general, is too massive to grasp. In this talk, we show that the Chow group of a smooth cubic hypersurface X can be recovered by the algebraic cycles of its Fano variety of lines F(X). It generalises M. Shen’s previous work of 1-cycles on cubics. The proof relies on some birational geometry concerning the Hilbert square of cubic hypersurfaces recently studied by E. Shinder, S. Galikin and C. Voisin. As applications, when X is a complex smooth 4-fourfold, the result we obtained could prove the integral Hodge conjecture of 1-cycles on F(X), as a polarised hyper-K？hler variety. Unlike the Hodge conjecture, the integral Hodge conjecture fails in general. In the arithmetic aspect, C. Schoen addressed the integral analog of the Tate conjecture, which is predicted to be true for 1-cycles of any smooth projective variety defined over finite fields. We use our result to prove this conjecture for 1-cycles on the Fano variety F(X) if X is a smooth cubic 4-fold over a field finitely generated over its prime subfield. |
