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

2017/04/13 by Vladimir Guletskiĭ, Guletskii, Vladimir
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1704.04187

openalex publication_date 2017/04/13 · openalex created_date 2017/04/28 · openalex updated_date 2026/07/28

Abstract

Let S be a smooth projective surface with p_g=0, let ι be a regular involution acting on S, and let W be the resolution of singularities of the quotient surface S/ι. 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.

Related

• Bloch's conjecture on surfaces of general type with an involution
• Some surfaces of general type for which Bloch's conjecture holds
• Bloch's conjecture on surfaces of general type with p_g=q=0, K²=3 and with an involution
• Bloch's conjecture on certain surfaces of general type with p_g=0 and with an involution: the Enriques case
• Involutions on algebraic surfaces and the Generalised Bloch's conjecture