A motivic study of generalized Burniat surfaces

TL;DR

This paper investigates generalized Burniat surfaces, proving a variant of the Bloch conjecture and showing their Chow motives are finite-dimensional, extending results to Sicilian surfaces.

Contribution

It introduces a new approach to the Bloch conjecture for generalized Burniat and Sicilian surfaces, establishing finite-dimensionality of their Chow motives.

Findings

Proves a variant of the Bloch conjecture for these surfaces.

Shows Chow motives are finite-dimensional in the sense of Kimura.

Extends methods to Sicilian surfaces introduced by Bauer, Catanese, and Frapporti.

Abstract

Generalized Burniat surfaces are surfaces of general type with $p_g=q$ and Euler number $e=6$ obtained by a variant of Inoue's construction method for the classical Burniat surfaces. I prove a variant of the Bloch conjecture for these surfaces. The method applies also to the so-called Sicilian surfaces introduced by Bauer, Catanese and Frapporti. This implies that the Chow motives of all of these surfaces are finite-dimensional in the sense of Kimura.