Bloch's conjecture for surfaces with involutions and of geometric genus zero

Abstract

Let $S$ be a smooth projective surface with $p_g=0$, let $\iota$ be a regular involution acting on $S$, and let $W$ be the resolution of singularities of the quotient surface $S/\iota$. In the paper we prove that Bloch's conjecture holds for the surface $S$ if and only if it holds for the surface $W$. This yields Bloch's conjecture for all surfaces $S$ whenever the same conjecture is true for the desingularized quotient $W$. In particular, Bloch's conjecture holds true for all numerical Godeaux surfaces with involutions, a "half" of Campedelli surfaces with involutions, the surface of Craighero and Gattazzo, some Catanese surfaces and other examples. Applying the same method to $K3$-surfaces, we prove that if a $K3$-surface $S$ admits a regular involution whose quotient is of Enriques type, then the motive $M(S)$ is finite-dimensional.

The paper has been withdrawn due to a crucial misconception about the monodromy argument used in the proofs of the main theorems

References in corpus (1)

• Bloch's conjecture for the surface of Craighero and Gattazzo

Cited by in corpus (3)

• Bloch's conjecture on surfaces of general type with an involution
• Mumford-Roitman argument on families
• Chow groups of conic bundles in ${\mathbb{P}}^5$ and the Generalised Bloch's conjecture