Cohomological Chow Groups of codimension one of varieties with isolated singularities

We compute some particular examples of cohomological Chow groups for varieties with isolated singularities. For higher-dimensional varieties, we compute the cohomological Chow groups of codimension one, provided that the dual complex associated to the normal crossing divisor is contractible. For 3-dimensional varieties, we consider a weaker condition on the dual complex, namely $H^2(\mathrm{\Gamma }(E))=0$.