2-Cycles on Higher Fano Hypersurfaces

TL;DR

This paper proves the triviality of the Griffiths group for certain 2-Fano hypersurfaces, providing insights into their algebraic cycles and answering a question by Voisin.

Contribution

It establishes the Griffiths group Griff_1(F(X_d)) is trivial for 2-Fano hypersurfaces and shows CH_2(X_d)=Z for specific 3-Fano hypersurfaces, advancing understanding of their algebraic cycles.

Findings

Griffiths group Griff_1(F(X_d)) is trivial for 2-Fano hypersurfaces

CH_2(X_d)=Z for certain 3-Fano hypersurfaces

Positive answer to Voisin's question in some cases

Abstract

Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d 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 CH_2(X_d)=\mathbb{Z} for a complex smooth $3$-Fano hypersurface X_d whose Fano variety of lines is smooth.