2-Cycles on Higher Fano Hypersurfaces
Abstract
Let F(Xd) be a smooth Fano variety of lines of a hypersurface Xd of degree d. In this paper, we prove the Griffiths group Griff1(F(Xd)) is trivial if the hypersurface Xd is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in some cases. Base on this result, we prove that CH2(Xd)=Z for a complex smooth 3-Fano hypersurface Xd whose Fano variety of lines is smooth.
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