Bloch's conjecture and valences of correspondences for K3 surfaces

Abstract

Bloch's conjecture for a surface $X$ over an algebraically closed field $k$ states that every homologically trivial correspondence $\Gamma$ acts as 0 on the Albanese kernel $T(X_{\Omega })$, where $\Omega$ is a universal domain containing $k$. Here we prove that, for a complex K3 surface $X$, Bloch's conjecture is equivalent to the existence of a valence for every correspondence. We also give applications of this result to the case of a correspondence associated to an automorphisms of finite order and to the existence of constant cycle curves on $X$. Finally we show that Franchetta's conjecture, as stated by K.O'Grady, holds true for the family of polarized K3 surfacees of genus $g$, if $3\le g\le 6$

References in corpus (1)

• Curves and cycles on K3 surfaces

Cited by in corpus (1)

• On O'Grady's generalized Franchetta conjecture