The Relative Chern Character and Regulators

In this thesis we construct a modified version of Karoubi’s relative Chern character for smooth varieties over the complex numbers or the ring of integers in a p-adic number field. Comparison results with the Deligne-Beilinson Chern character and the p-adic Borel regulator constructed by Huber and Kings are proven. As a corollary we obtain a new proof of Burgos’ theorem that Borel’s regulator is twice Beilinson’s regulator.

💡 Research Summary

The thesis develops a unified framework for comparing several important regulator maps that appear in algebraic K‑theory, Hodge theory, and p‑adic cohomology. Starting from Karoubi’s relative Chern character, the author constructs a modified version that works uniformly for smooth complex algebraic varieties and for schemes over the ring of integers O_K of a p‑adic number field.

In the complex setting, the construction proceeds by introducing a relative de Rham complex A^{rel}_X that captures the difference between algebraic differential forms on X and smooth forms on the associated complex manifold X(ℂ). The relative K‑group K^{rel}_n(X) is defined as the kernel of the natural map from algebraic K‑theory to topological K‑theory of X(ℂ). The modified relative Chern character
\