Construction of logarithmic cohomology theories II: On Chow groups
Doosung Park
TL;DR
The paper develops a toric framework to express Chow groups of complex varieties in terms of toric Chow groups, enabling a logarithmic refinement of cohomology theories. It introduces r-standard and very standard subdivisions, organizes Chow groups into $\mathrm{CH}_{\mathrm{sta}}^*(n,r)$ via colimits, and uses fine fans $\Theta_{n,r,\mathbf{d}}$ together with a carefully designed maximal-cone ordering to apply toric–Chow resolutions. A central technical achievement is the resolution of toric Chow homology and a robust lifting property that ensures exactness of the relevant complexes, underpinning the quasi-isomorphism linking $\mathrm{CH}^q(\mathbb{C})$ to toric data. These results provide a concrete toric-algebraic route to the claimed identification, furnishing essential tools for constructing logarithmic cohomology theories in the broader logSHF framework and for the toric computations behind the logarithmic cyclotomic trace.
Abstract
The purpose of this second part of the series is to show a technical result on Chow groups of toric varieties. This is a crucial ingredient for the first part.