General explicit expressions for higher discriminants that vanish on low-rank bundles

Develop explicit expressions for logarithmic Chern classes Ak, for all integers k ≥ 1, with the property that Ak(E) = 0 for every locally free sheaf E of rank r < k, extending the modified constructions exhibited for k = 4 and k = 5.

Background

The authors show that the naive higher logarithmic classes fail to vanish on bundles of rank less than the degree starting at k=4, but they construct modified classes for k=4 and k=5 that restore this vanishing property.

They report being unable to discern a general pattern to extend this modification to all higher degrees, and call for explicit formulas achieving the vanishing property for Ak across arbitrary k.