Homological periods and higher cycles
L. Barbieri-Viale
TL;DR
The paper constructs a homological regulator from Suslin homology to period homology and a higher cycle map from Bloch's higher Chow groups to period Borel-Moore homology, forming a bridge between motivic homology theories and period realizations. It develops a comprehensive motivic framework of regulators via de Rham and Betti–de Rham realizations and shows that 1-motivic period conjectures hold rationally, with explicit computations in degree 1 using the motivic Albanese. For j=0 the regulators are shown to be surjective and the period conjecture holds, while for j=1 the cokernels are described in terms of torsion in cdh and Suslin groups, yielding generalized Roitman conclusions under Grothendieck conjectures. The results illuminate how Grothendieck–type period theory implies Roitman-type theorems and provide a geometric description of period homologies in low degree, tying together higher cycles, motivic Albanese, and mixed realizations with precise dualities. Overall, the work advances understanding of how motivic structures govern period maps and the arithmetic of algebraic cycles.
Abstract
For any scheme which is algebraic over a subfield of the complex numbers we here construct an homological regulator from Suslin homology to period homology and a higher cycle class map from Bloch's higher Chow group to the period Borel-Moore homology. Over algebraic numbers, making use of the motivic Albanese, we provide a purely geometric description of these period homologies in degree 1 and we characterise the $\mathbb{Q}/\mathbb{Z}$-cokernel of these regulators in terms of torsion zero-cycles, showing that Grothendieck period conjectures imply generalised Roĭtman theorems.